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-   -   Minimal set of the strings for primes with at least two digits (https://www.mersenneforum.org/showthread.php?t=24972)

sweety439 2021-10-19 11:04

Like [URL="http://www.wiskundemeisjes.nl/wp-content/uploads/2007/02/primes2.pdf"]http://www.wiskundemeisjes.nl/wp-content/uploads/2007/02/primes2.pdf[/URL], if you write down a prime > 10, then I can always strike out 0 or more digits to get a prime on this set: {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027}

In fact, if a primes which do not contain any of the 1st to the 76th primes in the set, then this prime must be of the form 5{0}27 = 5*10^(n+2)+27, and if a primes which do not contain any of the 1st to the 75th primes in the set, then the prime must be of one of these forms: 5{0}27 = 5*10^(n+2)+27, {5}1 = (5*10^(n+1)-41)/9, 8{5}1 = (77*10^(n+1)-41)/9, by our theorem, such primes must contain either the 76th prime or the 77th prime as subsequence, however any such prime cannot contain both the 76th prime and the 77th prime as subsequences, since the 76th prime contain "1" and the 77th prime contain "7", any prime (in fact, any number, need not to be prime) containing both the 76th prime and the 77th prime as subsequences will contain both "1" and "7", and hence contain either "17" or "71" (or both) as subsequence, but both 17 and 71 are primes > 10

sweety439 2021-10-19 13:04

A minimal prime (start with b+1) base b is called [B]infinite-minimal prime[/B] iff there are infinitely many primes which have this minimal prime (start with b+1) as subsequence (when written as base b string) but not have any other minimal prime (start with b+1) base b as subsequence (when written as base b string).

e.g. the largest minimal prime (start with b+1) 5000000000000000000000000000027 is likely infinite-minimal prime in base 10, since there are likely infinitely many primes of the form 5{0}27 in base 10 (reference: [URL="https://stdkmd.net/nrr/prime/primecount.txt"]https://stdkmd.net/nrr/prime/primecount.txt[/URL], search: “50w27”), so is the second-largest minimal prime (start with b+1) 555555555551, since there are likely infinitely many primes of the form {5}1 (search “5w1” in the reference page) or 8{5}1 (search “85w1” in the reference page), but the third-largest minimal prime (start with b+1) and the fourth-largest minimal prime (start with b+1), 80555551 and 66600049 are not infinite-minimal prime in base 10, since the only primes which have no minimal prime (start with b+1) base b=10 but 80555551 as subsequence are 80555551 and 8055555551, and the only prime which have no minimal prime (start with b+1) base b=10 but 66600049 as subsequence is 66600049.

In base 2, of course, 11 is infinite-minimal prime, since all odd primes have 11 as subsequence (at least, the first digit and the last digit of all odd primes are both 1), and there are infinitely many such primes (reference: [URL="https://primes.utm.edu/notes/proofs/infinite/euclids.html"]Euclid's Proof[/URL]), and in base 3, all the three minimal primes (start with b+1) are very likely infinite-minimal primes, since for 12, all primes of the form 1{2} ([URL="https://oeis.org/A003307"]A003307[/URL]) or 1{0}2 ([URL="https://oeis.org/A051783"]A051783[/URL]) have no subsequence 21 or 111, and for 21, all primes of the form {2}1 ([URL="https://oeis.org/A014224"]A014224[/URL]) or 2{0}1 ([URL="https://oeis.org/A003306"]A003306[/URL]) have no subsequence 12 or 111, and for 111, all primes of the form {1} ([URL="https://oeis.org/A028491"]A028491[/URL]) or 1{0}11 ([URL="https://oeis.org/A058958"]A058958[/URL]) or 11{0}1 ([URL="https://oeis.org/A005537"]A005537[/URL]) have no subsequence 12 or 21, and all these seven sequences are conjectured to be infinite.

[B]Problem: In base 10, how many of the 77 minimal primes (start with b+1) are infinite-minimal primes? (assume the conjecture in [URL="https://www.mersenneforum.org/showpost.php?p=529838&postcount=675"]this post[/URL])[/B]

In base 10, 1235607889460606009419 the smallest prime containing all 26 original minimal primes (i.e. p>b not needed) as subsequences (see [URL="https://www.primepuzzles.net/puzzles/puzz_178.htm"]https://www.primepuzzles.net/puzzles/puzz_178.htm[/URL] and [URL="https://primes.utm.edu/curios/page.php/1235607889460606009419.html"]https://primes.utm.edu/curios/page.php/1235607889460606009419.html[/URL]), [B]problem: Find the smallest prime containing all 77 minimal primes (start with b+1) in base b=10[/B], also this problem can be generalized to other bases, in base 2, this prime is clearly 11, and in base 3, this prime is 1121 (since this number is the smallest number containing both 21 and 111 as subsequences, and indeed this number is prime and containing 12 as subsequence).

sweety439 2021-10-23 20:29

Base b=72 has largest known minimal prime (start with b+1) for all bases 2<=b<=1024: 3(71^1119849), its value is [URL="https://primes.utm.edu/primes/page.php?id=122173"]4*72^1119849-1[/URL], it has 2079933 digits when written in decimal (this number is [I]proven[/I] prime, for unproven probable prime, the largest known minimal prime (start with b+1) for all bases 2<=b<=1024 is base b=23, the PRP 9(14^800873), its value is [URL="http://www.primenumbers.net/prptop/searchform.php?form=%28106*23%5En-7%29%2F11&action=Search"](106*23^800873-7)/11[/URL], it has 1090573 digits when written in decimal)

sweety439 2021-10-23 20:39

[QUOTE=sweety439;579853]New minimal prime (start with b+1) in base b is found for b=650: 3:{649}^(498101), see [URL="https://mersenneforum.org/showpost.php?p=579849&postcount=931"]https://mersenneforum.org/showpost.php?p=579849&postcount=931[/URL]

Added it to excel file [URL="https://docs.google.com/spreadsheets/d/e/2PACX-1vTKkSNKGVQkUINlp1B3cXe90FWPwiegdA07EE7-U7sqXntKAEQrynoI1sbFvvKriieda3LfkqRwmKME/pubhtml"]https://docs.google.com/spreadsheets/d/e/2PACX-1vTKkSNKGVQkUINlp1B3cXe90FWPwiegdA07EE7-U7sqXntKAEQrynoI1sbFvvKriieda3LfkqRwmKME/pubhtml[/URL]

Base 108 is an interesting base since ....

* For the family {1}, length 2 is prime, but the next prime is large (length 449)
* For the family 1{0}1, length 2 is prime, the next prime is not known
* For the family y{z}, first prime is large (length 411)
* For the family 11{0}1, first prime is large (length 400)
* For the family {y}z, first prime is large (length 492) (note that length 1 is also prime, but length 1 is not allowed in this project)
* For the family 6{0}1, first prime is large (length 16318)
* For the family #{z} (# = (base/2)-1)), first prime is large (length 7638)

This situation is not common in bases with many divisors, but although 108 has many divisors, this situation occurs in this base, this is why this base is interesting :))

Also base 282 ....

* For the family A{0}1, first prime is large (length 1474)
* For the family C{0}1, first prime is large (length 2957)
* For the family z{0}1, first prime is large (length 277)

[URL="http://www.noprimeleftbehind.net/crus/Sierp-conjecture-base282-reserve.htm"]http://www.noprimeleftbehind.net/crus/Sierp-conjecture-base282-reserve.htm[/URL] only tells you that all these three families have a prime with length <= 100001 ....

* For the family 7{z}, first prime is large (length 21413)
* For the family 10{z}, first prime is large (length 780)

[URL="http://www.noprimeleftbehind.net/crus/Riesel-conjecture-base282-reserve.htm"]http://www.noprimeleftbehind.net/crus/Riesel-conjecture-base282-reserve.htm[/URL] only tells you that the farmer family has a prime with length <= 100001, and the letter family has a prime with length <= 100002[/QUOTE]

Another interesting base is b=23, this base is [URL="https://raw.githubusercontent.com/curtisbright/mepn-data/master/data/minimal.23.txt"]SOLVED[/URL] when single-digit primes (i.e. the primes p<23) and the prime "10" (i.e. the prime p=23) are included, and this base is the solved base with the largest minimal prime ((106*23^800873-7)/11, or 9(E^800873), which has 1090573 digits when written in decimal) and the largest number of minimal primes (6021 primes or PRPs, while the second-largest number (base 42) has 4551 proven primes), this base seems be easy and high weight, but in fact this base is low-weight base (since it is == 5 mod 6) and this base has large primes for special forms listed in [URL="https://docs.google.com/spreadsheets/d/e/2PACX-1vTKkSNKGVQkUINlp1B3cXe90FWPwiegdA07EE7-U7sqXntKAEQrynoI1sbFvvKriieda3LfkqRwmKME/pubhtml"]https://docs.google.com/spreadsheets/d/e/2PACX-1vTKkSNKGVQkUINlp1B3cXe90FWPwiegdA07EE7-U7sqXntKAEQrynoI1sbFvvKriieda3LfkqRwmKME/pubhtml[/URL] ....

* [URL="https://oeis.org/A138066"]1{0}2[/URL] (b^n+2): 12 digits [a record value for bases b] (minimal prime in my project, but not minimal prime in original project (i.e. the primes p<=base are also included), since this prime has "10" and "2" as subsequences)
* 4{0}1 (4*b^n+1): 343 digits [a record value for bases b]
* {4}1: 13 digits [not a record value for bases b, since base b=11 requires 45 digits]
* 8{0}1 (8*b^n+1): 119216 digits [a record value for bases b]
* {K}L ([URL="https://docs.google.com/document/d/e/2PACX-1vSQlPrWZgVM1g5spyMs_USkKy3XEGcBsadeLc82JmQVbXCOWbbcSkuHMtO_EmspQME3ITGNvoCcffZt/pub"]extended Sierpinski problem[/URL] with k>base (k=10)): 3762 digits [a record value for base b for extended Sierpinski problem with k=10 (the prime (10*23^3762+1)/11, exponent is 3762), the previous record is b=17, the prime 10*17^1356+1, exponent is 1356]
* [URL="https://oeis.org/A122396"]y{z}[/URL] ((b-1)*b^n-1, [URL="http://www.bitman.name/math/table/484"]Williams prime of the 1st kind[/URL]): 56 digits [a record value for bases b]
* [URL="https://oeis.org/A087139"]z{0}1[/URL] ((b-1)*b^n+1, [URL="http://www.bitman.name/math/table/477"]Williams prime of the 2nd kind[/URL]): 15 digits [not a record value for bases b, since base b=19 requires 30 digits, also the same as base b=20, both require 15 digits]
* [URL="https://oeis.org/A113516"]{z}1[/URL] (b^n-(b-1), [URL="http://www.bitman.name/math/table/435"]dual Williams prime of the 1st kind[/URL]): 17 digits [a record value for bases b, the same as base b=20, both require 17 digits]
* [URL="https://oeis.org/A250200"]{z}y[/URL] (b^n-2): 24 digits [a record value for bases b]

sweety439 2021-10-26 21:07

Many families are ruled out as only contain composite numbers by trivial 1-cover, e.g.

* All families ending with digit not coprime to base
* All families whose digits have gcd > 1
* Base 3 family 1{0}1, trivial factor of 2
* Base 4 family 2{0}1, trivial factor of 3
* Base 5 family 1{0}1, trivial factor of 2
* Base 5 family 1{0}3, trivial factor of 2
* Base 5 family 3{0}1, trivial factor of 2
* Base 6 family 4{0}1, trivial factor of 5
...
* Base 10 family 28{0}7, trivial factor of 7
* Base 10 family 4{6}9, trivial factor of 7

and for families which are ruled out as only contain composite numbers by reasons other than trivial 1-cover:

* The first base which have family which are ruled out as only contain composite numbers by 2-cover (i.e. covering set {p,q} with primes p and q) is 9, such families are {1}5, 2{7}, {3}5, {3}8, 5{1}, 5{7}, 6{1}, {7}2, {7}5 (note: base 5 families {1}3, {1}4, 3{1}, 4{1} are not counted, since they will not produce minimal primes (start with b+1), since 111 (base 5) is prime)

* The first base which have family which are ruled out as only contain composite numbers by covering set with >=3 primes is 13, which have families 3{0}95 and 95{0}3, which have covering set {3, 5, 17}, period=4 (note: base 8 families 6{4}7 is not counted, since it will not produce minimal primes (start with b+1), since (4^220)7 (base 8) is prime)

* The first base which have family which are ruled out as only contain composite numbers by full algebra factorization is 8, which have family 1{0}1, which have full sum-of-cubes factorization

* The first base which have family which are ruled out as only contain composite numbers by partial algebra factorization / partial covering set is 12, which have family {B}9B, which have partial difference-of-squares factorization / partial 1-cover {13}

* The first base which have family which are ruled out as only contain composite numbers by full difference-of-squares factorization is 9, which have families {1}, 3{1}, 3{8}, {8}5

* The first base which have family which are ruled out as only contain composite numbers by full difference-of-cubes factorization (not full sum-of-cubes factorization) is 27, which have families 7{Q} and {Q}J (note: base 8 family {1} and base 27 family {1} are not counted, since although they have full difference-of-cubes factorization, they still have primes, 111 in base 8 and base 27 are both primes)

* The first base which have family which are ruled out as only contain composite numbers by full sum-of-5th-powers factorization is 32, which have family 1{0}1

* The first base which have family which are ruled out as only contain composite numbers by full difference-of-5th-powers factorization is 32, which have family {1}

* The first base which have family which are ruled out as only contain composite numbers by full Aurifeuillian factorization of x^4+4*y^4 is 16, which have families {C}D and {C}DD

sweety439 2021-10-26 21:40

[QUOTE=sweety439;581427]Formula of these families:

1{0}1: b^n+1 (b>=2) (n>=1) (length=n+1)
1{0}2: b^n+2 (b>=3) (n>=1) (length=n+1)
1{0}3: b^n+3 (b>=4) (n>=1) (length=n+1)
1{0}4: b^n+4 (b>=5) (n>=1) (length=n+1)
1{0}z: b^n+(b-1) (b>=2) (n>=1) (length=n+1)
{1}: (b^n-1)/(b-1) (b>=2) (n>=2) (length=n)
1{2}: ((b+1)*b^n-2)/(b-1) (b>=3) (n>=1) (length=n+1)
1{3}: ((b+2)*b^n-3)/(b-1) (b>=4) (n>=1) (length=n+1)
1{4}: ((b+3)*b^n-4)/(b-1) (b>=5) (n>=1) (length=n+1)
1{z}: 2*b^n-1 (b>=2) (n>=1) (length=n+1)
2{0}1: 2*b^n+1 (b>=3) (n>=1) (length=n+1)
2{0}3: 2*b^n+3 (b>=4) (n>=1) (length=n+1)
{2}1: (2*b^n-(b+1))/(b-1) (b>=3) (n>=2) (length=n)
2{z}: 3*b^n-1 (b>=3) (n>=1) (length=n+1)
3{0}1: 3*b^n+1 (b>=4) (n>=1) (length=n+1)
3{0}2: 3*b^n+2 (b>=4) (n>=1) (length=n+1)
3{0}4: 3*b^n+4 (b>=5) (n>=1) (length=n+1)
{3}1: (3*b^n-(2*b+1))/(b-1) (b>=4) (n>=2) (length=n)
3{z}: 4*b^n-1 (b>=4) (n>=1) (length=n+1)
4{0}1: 4*b^n+1 (b>=5) (n>=1) (length=n+1)
4{0}3: 4*b^n+3 (b>=5) (n>=1) (length=n+1)
{4}1: (4*b^n-(3*b+1))/(b-1) (b>=5) (n>=2) (length=n)
4{z}: 5*b^n-1 (b>=5) (n>=1) (length=n+1)
5{0}1: 5*b^n+1 (b>=6) (n>=1) (length=n+1)
5{z}: 6*b^n-1 (b>=6) (n>=1) (length=n+1)
6{0}1: 6*b^n+1 (b>=7) (n>=1) (length=n+1)
6{z}: 7*b^n-1 (b>=7) (n>=1) (length=n+1)
7{0}1: 7*b^n+1 (b>=8) (n>=1) (length=n+1)
7{z}: 8*b^n-1 (b>=8) (n>=1) (length=n+1)
8{0}1: 8*b^n+1 (b>=9) (n>=1) (length=n+1)
8{z}: 9*b^n-1 (b>=9) (n>=1) (length=n+1)
9{0}1: 9*b^n+1 (b>=10) (n>=1) (length=n+1)
9{z}: 10*b^n-1 (b>=10) (n>=1) (length=n+1)
A{0}1: 10*b^n+1 (b>=11) (n>=1) (length=n+1)
A{z}: 11*b^n-1 (b>=11) (n>=1) (length=n+1)
B{0}1: 11*b^n+1 (b>=12) (n>=1) (length=n+1)
B{z}: 12*b^n-1 (b>=12) (n>=1) (length=n+1)
C{0}1: 12*b^n+1 (b>=13) (n>=1) (length=n+1)
{#}$ (# = (b−1)/2, $ = (b+1)/2): (b^n+1)/2 (b>=3 is odd) (n>=2) (length=n)
{y}z: ((b-2)*b^n+1)/(b-1) (b>=3) (n>=2) (length=n)
y{z}: (b-1)*b^n-1 (b>=3) (n>=1) (length=n+1)
z{0}1: (b-1)*b^n+1 (b>=2) (n>=1) (length=n+1)
{z}1: b^n-(b-1) (b>=2) (n>=2) (length=n)
{z}w: b^n-4 (b>=5) (n>=2) (length=n)
{z}x: b^n-3 (b>=4) (n>=2) (length=n)
{z}y: b^n-2 (b>=3) (n>=2) (length=n)[/QUOTE]

These families can have prime only for these length (in any base): (since the corresponding polynomial of the base should be [URL="https://en.wikipedia.org/wiki/Irreducible_polynomial"]irreducible polynomial[/URL], and assuming [URL="https://en.wikipedia.org/wiki/Schinzel%27s_hypothesis_H"]Schinzel's hypothesis H[/URL], there are infinitely bases such that these families with these given length are prime)

1{0}1: 2^r+1 with integer r, since all other length are divisible by smaller number of the form 1{0}1
1{0}2: all length
1{0}3: all length
1{0}4: not == 1 mod 4, since length 4*m+1 are divisible by b^(2*m) - 2*b^m + 2
1{0}z: not == 0 mod 6, since length 6*m are divisible by b^2 - b + 1
{1}: prime, since all other length are divisible by smaller number of the form {1}
1{z}: all length
2{0}1: all length
2{z}: all length
3{0}1: all length
3{z}: even, since odd length have difference-of-squares factorization
4{0}1: not == 1 mod 4, since length 4*m+1 are divisible by 2*b^(2*m) - 2*b^m + 1
4{z}: all length
5{0}1: all length
5{z}: all length
6{0}1: all length
{#}$ (# = (b−1)/2, $ = (b+1)/2): 2^r+1 with integer r, since all other length are divisible by smaller number of the form {#}$
{y}z: all length
y{z}: not == 5 mod 6, since length 6*m+5 are divisible by b^2 - b + 1
z{0}1: not == 2 mod 6 (except 2), since length 6*m+2 are divisible by b^2 - b + 1
{z}1: not == 2 mod 6 (except 2), since length 6*m+2 are divisible by b^2 - b + 1
{z}w: even, since odd length have difference-of-squares factorization
{z}x: all length
{z}y: all length

also for the bases such that given form of given length is prime:

1{0}1: [URL="https://oeis.org/A006093"]A006093[/URL] (2) [URL="https://oeis.org/A005574"]A005574[/URL] (3) [URL="https://oeis.org/A000068"]A000068[/URL] (5) [URL="https://oeis.org/A006314"]A006314[/URL] (9) [URL="https://oeis.org/A006313"]A006313[/URL] (17) [URL="https://oeis.org/A006315"]A006315[/URL] (33)
1{0}2: [URL="https://oeis.org/A040976"]A040976[/URL] (2) [URL="https://oeis.org/A067201"]A067201[/URL] (3) [URL="https://oeis.org/A067200"]A067200[/URL] (4)
1{0}z: [URL="https://oeis.org/A006254"]A006254[/URL] (2) [URL="https://oeis.org/A045546"]A045546[/URL] (3) [URL="https://oeis.org/A236475"]A236475[/URL] (4)
{1}: [URL="https://oeis.org/A006093"]A006093[/URL] (2) [URL="https://oeis.org/A002384"]A002384[/URL] (3) [URL="https://oeis.org/A049409"]A049409[/URL] (5) [URL="https://oeis.org/A100330"]A100330[/URL] (7) [URL="https://oeis.org/A162862"]A162862[/URL] (11) [URL="https://oeis.org/A217070"]A217070[/URL] (13)
1{z}: [URL="https://oeis.org/A006254"]A006254[/URL] (2) [URL="https://oeis.org/A066049"]A066049[/URL] (3) [URL="https://oeis.org/A214289"]A214289[/URL] (4)
2{0}1: [URL="https://oeis.org/A005097"]A005097[/URL] (2) [URL="https://oeis.org/A089001"]A089001[/URL] (3) [URL="https://oeis.org/A168550"]A168550[/URL] (4)
{#}$ (# = (b−1)/2, $ = (b+1)/2): [URL="https://oeis.org/A002731"]A002731[/URL] (2) [URL="https://oeis.org/A096169"]A096169[/URL] (4)
y{z}: [URL="https://oeis.org/A002328"]A002328[/URL] (2) [URL="https://oeis.org/A162293"]A162293[/URL] (3)
z{0}1: [URL="https://oeis.org/A055494"]A055494[/URL] (2) [URL="https://oeis.org/A111501"]A111501[/URL] (3)
{z}1: [URL="https://oeis.org/A055494"]A055494[/URL] (2) [URL="https://oeis.org/A236477"]A236477[/URL] (3)
{z}y: [URL="https://oeis.org/A028870"]A028870[/URL] (2) [URL="https://oeis.org/A038599"]A038599[/URL] (3) [URL="https://oeis.org/A154831"]A154831[/URL] (4)

sweety439 2021-10-31 06:13

[QUOTE=sweety439;587491]Some unsolved families are obvious through [URL="https://en.wikiversity.org/wiki/Quasi-minimal_prime"]data for minimal primes (start with b+1) base b up to certain limit[/URL]

Example 1: 80555551 is minimal prime (start with b+1) base b for b=10, thus these are unsolved families when searched to given limit if they [I]possible[/I] contain primes:

* 80{5} searched to 5 5's (this family cannot have primes since all such numbers are divisible by 5)
* 8{5}1 searched to 5 5's (the smallest prime in this family is 8(5^20)1, but this prime is not minimal prime (start with b+1) for base b=10 since (5^11)1 is prime in this base and 20 >= 11)
* 0{5}1 searched to 5 5's (not considered, since this family has [URL="https://en.wikipedia.org/wiki/Leading_zero"]leading zeros[/URL])
* 8{5} searched to 5 5's (this family cannot have primes since all such numbers are divisible by 5)
* 0{5} searched to 5 5's (not considered, since this family has leading zeros)
* {5}1 searched to 5 5's (the smallest prime in this family is (5^11)1, and indeed minimal prime (start with b+1) for base b=10)
* {5} searched to 5 5's (this family cannot have primes since all such numbers are divisible by 5)

Example 2: 55555025 is minimal prime (start with b+1) base b for b=8, thus these are unsolved families when searched to given limit if they [I]possible[/I] contain primes:

* {5}02 searched to 5 5's (this family cannot have primes since all such numbers are divisible by 2)
* {5}05 searched to 5 5's (this family cannot have primes since all such numbers are divisible by 5)
* {5}25 searched to 5 5's (the smallest prime in this family is (5^13)25, and indeed minimal prime (start with b+1) for base b=8)
* {5}0 searched to 5 5's (not considered, since this family has [URL="https://en.wikipedia.org/wiki/Trailing_zero"]trailing zeros[/URL])
* {5}2 searched to 5 5's (this family cannot have primes since all such numbers are divisible by 2)
* {5}5 searched to 5 5's = {5} searched to 6 5's (this family cannot have primes since all such numbers are divisible by 5)

Example 3: 33333301 is minimal prime (start with b+1) base b for b=7, thus these are unsolved families when searched to given limit if they [I]possible[/I] contain primes:

* {3}0 searched to 6 3's (not considered, since this family has trailing zeros)
* {3}1 searched to 6 3's (the smallest prime in this family is (3^16)1, and indeed minimal prime (start with b+1) for base b=7)
* {3} searched to 6 3's (this family cannot have primes since all such numbers are divisible by 3)

Example 4: 100000000000507 is minimal prime (start with b+1) base b for b=9, thus these are unsolved families when searched to given limit if they [I]possible[/I] contain primes:

* 1{0}50 searched to 11 0's (not considered, since this family has trailing zeros)
* 1{0}57 searched to 11 0's (the smallest prime in this family is 1(0^25)57, and indeed minimal prime (start with b+1) for base b=9)
* 1{0}07 searched to 11 0's = 1{0}7 searched to 12 0's (this family cannot have primes since all such numbers are divisible by 2)
* 1{0}5 searched to 11 0's (this family cannot have primes since all such numbers are divisible by 2)
* 1{0}0 searched to 11 0's = 1{0} searched to 12 0's (not considered, since this family has trailing zeros)

Example 5: BBBBBB99B is minimal prime (start with b+1) base b for b=12, thus these are unsolved families when searched to given limit if they [I]possible[/I] contain primes:

* {B}99 searched to 6 B's (this family cannot have primes since all such numbers are divisible by 3)
* {B}9B searched to 6 B's (this family cannot have primes since such numbers with even length are factored as difference of squares and such numbers with odd length are divisible by 13)
* {B}9 searched to 6 B's (this family cannot have primes since all such numbers are divisible by 3)
* {B}B searched to 6 B's = {B} searched to 7 B's (this family cannot have primes since all such numbers are divisible by 11)

Example 6: 500025 is minimal prime (start with b+1) base b for b=8, thus these are unsolved families when searched to given limit if they [I]possible[/I] contain primes:

* 5{0}2 searched to 3 0's (this family cannot have primes since all such numbers are divisible by 2)
* 5{0}5 searched to 3 0's (this family cannot have primes since all such numbers are divisible by 5)
* {0}25 searched to 3 0's (not considered, since this family has leading zeros)
* 5{0} searched to 3 0's (not considered, since this family has trailing zeros)
* {0}2 searched to 3 0's (not considered, since this family has leading zeros)
* {0}5 searched to 3 0's (not considered, since this family has leading zeros)
* {0} searched to 3 0's (not considered, since this family has leading zeros and trailing zeros)

Example 7: 77774444441 is minimal prime (start with b+1) base b for b=8, thus these are unsolved families when searched to given limit if they [I]possible[/I] contain primes:

* {7}1 searched to 4 7's
* {7}41 searched to 4 7's
* {7}441 searched to 4 7's
* {7}4441 searched to 4 7's
* {7}44441 searched to 4 7's
* {7}444441 searched to 4 7's
* {4}1 searched to 6 4's
* 7{4}1 searched to 6 4's
* 77{4}1 searched to 6 4's
* 777{4}1 searched to 6 4's

Example 8: 88888888833335 is minimal prime (start with b+1) base b for b=9, thus these are unsolved families when searched to given limit if they [I]possible[/I] contain primes:

* {8}5 searched to 9 8's
* {8}35 searched to 9 8's
* {8}335 searched to 9 8's
* {8}3335 searched to 9 8's
* {3}5 searched to 4 3's
* 8{3}5 searched to 4 3's
* 88{3}5 searched to 4 3's
* 888{3}5 searched to 4 3's
* 8888{3}5 searched to 4 3's
* 88888{3}5 searched to 4 3's
* 888888{3}5 searched to 4 3's
* 8888888{3}5 searched to 4 3's
* 88888888{3}5 searched to 4 3's

Example 9: A44444777 is minimal prime (start with b+1) base b for b=11, thus these are unsolved families when searched to given limit if they [I]possible[/I] contain primes:

* A{4} searched to 5 4's
* A{4}7 searched to 5 4's
* A{4}77 searched to 5 4's
* A{7} searched to 3 7's
* A4{7} searched to 3 7's
* A44{7} searched to 3 7's
* A444{7} searched to 3 7's
* A4444{7} searched to 3 7's

Example 10: 96664444 is minimal prime (start with b+1) base b for b=13, thus these are unsolved families when searched to given limit if they [I]possible[/I] contain primes:

* 9{6} searched to 3 6's
* 9{6}4 searched to 3 6's
* 9{6}44 searched to 3 6's
* 9{6}444 searched to 3 6's
* 9{4} searched to 4 4's
* 96{4} searched to 4 4's
* 966{4} searched to 4 4's

Example 11: 88828823 is minimal prime (start with b+1) base b for b=11, thus these are unsolved families when searched to given limit if they [I]possible[/I] contain primes:

* {8} searched to 5 8's
* {8}2 searched to 5 8's
* {8}3 searched to 5 8's
* 8882{8} searched to 5 8's
* {8}288 searched to 5 8's
* {8}23 searched to 5 8's
* 8882{8}2 searched to 5 8's
* 8882{8}3 searched to 5 8's
* {8}2882 searched to 5 8's
* {8}2883 searched to 5 8's

Example 12: B0BBB05BB is minimal prime (start with b+1) base b for b=13, thus these are unsolved families when searched to given limit if they [I]possible[/I] contain primes:

* {B} searched to 6 B's
* B0{B} searched to 6 B's
* {B}0BB searched to 6 B's
* {B}5BB searched to 6 B's
* BBBB0{B} searched to 6 B's
* BBBB5{B} searched to 6 B's
* B0{B}0BB searched to 6 B's
* B0{B}5BB searched to 6 B's
* BBBB05{B} searched to 6 B's
* B0BBB0{B} searched to 6 B's
* B0BBB5{B} searched to 6 B's[/QUOTE]

Examples with large minimal primes (start with b+1):

* 2(7^686)07 is minimal prime (start with b+1) for base b=9, thus these are unsolved families for base b=9 when searched to given limit if they [I]possible[/I] contain primes:

** 2(7^n)0 searched to n=686 (has trailing zeros)
** 2(7^n)7 searched to n=686 equivalent to 2(7^n) searched to n=687 (covering set {2,5})
** (7^n)07 searched to n=686 (trivial factor of 7)
** (7^n)0 searched to n=686 (has trailing zeros)
** (7^n)7 searched to n=686 equivalent to (7^n) searched to n=687 (trivial factor of 7)

* 3(0^1158)11 is minimal prime (start with b+1) for base b=9, thus these are unsolved families for base b=9 when searched to given limit if they [I]possible[/I] contain primes:

** 3(0^n)1 searched to n=1158 (trivial factor of 2)
** (0^n)11 searched to n=1158 (has leading zeros)
** 3(0^n) searched to n=1158 (has trailing zeros)
** (0^n)1 searched to n=1158 (has leading zeros)
** (0^n) searched to n=1158 (has both leading zeros and trailing zeros)

* (7^759)44 is minimal prime (start with b+1) for base b=11, thus these are unsolved families for base b=11 when searched to given limit if they [I]possible[/I] contain primes:

** (7^n)4 searched to n=759 (covering set {2,3})
** (7^n) searched to n=759 (trivial factor of 7)

* 55(7^1011) is minimal prime (start with b+1) for base b=11, thus these are unsolved families for base b=11 when searched to given limit if they [I]possible[/I] contain primes:

** 5(7^n) searched to n=1011 (indeed an unsolved family, and in fact searched to n=50000 without any prime or PRP found)
** (7^n) searched to n=1011 (trivial factor of 7)

sweety439 2021-10-31 08:16

1 Attachment(s)
these are the bases b such that a given prime p is minimal prime (start with b+1)

since all two-digit primes base b (except b itself, when b is prime) are minimal prime (start with b+1) base b, thus every prime p is minimal prime (start with b+1) base b for all sqrt(p)<b<p (not for b>=p, since for these bases b, p is not prime >= b+1)

Conjecture: For all primes p other than 2, 3, 5, 7, 11, 17, 19, 23, 37, 47, 53, 67, 167, 233, there is base b<sqrt(p) such that p is minimal prime (start with b+1) base b

sweety439 2021-10-31 08:36

unsolved families for large bases b which has already extensive searched:

b=38 family 1:{0}:31, already searched to length 185001 by Peter Košinár, see [URL="https://math.stackexchange.com/questions/597234/least-prime-of-the-form-38n31"]https://math.stackexchange.com/questions/597234/least-prime-of-the-form-38n31[/URL]

b=93 family {92}:1, already searched to length 60000 by T. D. Noe, see [URL="https://oeis.org/A113516"]https://oeis.org/A113516[/URL]

b=167 family 1:{0}:2, already searched to length 100001 by Ray Chandler, see [URL="https://oeis.org/A138066"]https://oeis.org/A138066[/URL]

b=243 family 40:{121}, already searched to length 443060 by Paul Bourdelais, see [URL="https://oeis.org/A028491"]https://oeis.org/A028491[/URL]

b=305 family {304}:303, already searched to length 30000 by me, see [URL="https://mersenneforum.org/showpost.php?p=472886&postcount=13"]https://mersenneforum.org/showpost.php?p=472886&postcount=13[/URL]

b=353 family {352}:351, already searched to length 38000 by Robert Price, see [URL="https://oeis.org/A292201"]https://oeis.org/A292201[/URL]

b=9409 family {9312}:9313, already searched to length 250000 by Fan Ming, see [URL="https://mersenneforum.org/showpost.php?p=545152&postcount=223"]https://mersenneforum.org/showpost.php?p=545152&postcount=223[/URL]

b=67607 family 2:{0}:1, already searched to length 412001 by Keller, see [URL="https://mersenneforum.org/showpost.php?p=383979&postcount=2"]https://mersenneforum.org/showpost.php?p=383979&postcount=2[/URL]

and family {1} for bases b = {185, 269, 281, 380, 384, 385, 394, 396, 452, 465, 511, 574, 598, 601, 631, 632, 636, 711, 713, 759, 771, 795, 861, 866, 881, 938, 948, 951, 956, 963, 1005, 1015}, all searched to length 100000 by Michael Stocker

and family 1:{0}:1 for bases b = {38, 50, 62, 68, 86, 92, 98, 104, 122, 144, 168, 182, 186, 200, 202, 212, 214, 218, 244, 246, 252, 258, 286, 294, 298, 302, 304, 308, 322, 324, 338, 344, 354, 356, 362, 368, 380, 390, 394, 398, 402, 404, 410, 416, 422, 424, 446, 450, 454, 458, 468, 480, 482, 484, 500, 514, 518, 524, 528, 530, 534, 538, 552, 558, 564, 572, 574, 578, 580, 590, 602, 604, 608, 620, 622, 626, 632, 638, 648, 650, 662, 666, 668, 670, 678, 684, 692, 694, 698, 706, 712, 720, 722, 724, 734, 744, 746, 752, 754, 762, 766, 770, 792, 794, 802, 806, 812, 814, 818, 836, 840, 842, 844, 848, 854, 868, 870, 872, 878, 888, 896, 902, 904, 908, 922, 924, 926, 932, 938, 942, 944, 948, 954, 958, 964, 968, 974, 978, 980, 988, 994, 998, 1002, 1006, 1014, 1016}, all searched to length 8388608 by PrimGrid, see [URL="http://www.primegrid.com/stats_genefer.php"]http://www.primegrid.com/stats_genefer.php[/URL]

and family 1:{b-1} for bases b = {581, 992, 1019, 1079, 1152, 1193, 1217, 1283, 1593, 1646, 1670, 1760, 1762, 1829, 1901, 1909, 1997, 2012}, all searched to length 200001 by LaurV, see [URL="https://mersenneforum.org/showthread.php?t=24576"]https://mersenneforum.org/showthread.php?t=24576[/URL]

and many families for [URL="http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm"]Sierpinski problems[/URL] and [URL="http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm"]Riesel problems[/URL], search limits are shown in these pages (convert: Sierpinski problem is k:{0}:1 in base b, Riesel problem is k-1:{b-1} in base b) (such primes are [I]always[/I] minimal primes (start with b+1) base b if k<b, some k>=b primes are also minimal primes (start with b+1) base b, but most such primes are not)

and families for [URL="https://www.rieselprime.de/ziki/Williams_prime_MM_table"]Williams primes MM[/URL] and [URL="https://www.rieselprime.de/ziki/Williams_prime_MP_table"]Williams primes MP[/URL], search limits are shown in these pages (convert: Williams primes MM is b-2:{b-1} in base b, Williams primes MP is b-1:{0}:1 in base b)

sweety439 2021-10-31 09:10

This problem is an extension of the [URL="https://cs.uwaterloo.ca/~cbright/reports/mepn.pdf"]original minimal prime problem[/URL] to include [URL="http://www.noprimeleftbehind.net/crus/"]CRUS[/URL] [URL="http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm"]Sierpinski[/URL]/[URL="http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm"]Riesel[/URL] conjectures base b with k-values < b, i.e. the smallest prime of the form k*b^n+1 and k*b^n−1 for all k < b. The original minimal prime base b puzzle does not cover CRUS Sierpinski/Riesel conjectures base b with [URL="http://www.noprimeleftbehind.net/crus/tab/CRUS_tab.htm"]CK[/URL] < b, since in Riesel side, the prime is not minimal prime in original definition if either k−1 or b−1 (or both) is prime, and in Sierpinski side, the prime is not minimal prime in original definition if k is prime (e.g. 25*30^34205−1 is not minimal prime in base 30 in original definition, since it is O(T^34205) in base 30, and T (= 29 in decimal) is prime, but it is minimal prime in base 30 if only primes > base are counted), but this extended version of minimal prime base b problem does, this requires a restriction of prime > b, and the primes ≤ b (including the k−1, b−1, k) are not allowed, in fact, to include these conjectures, we only need to exclude the single-digit primes (i.e. the primes < b), also, in fact, I create this problem because I think that the single-digit primes are [URL="https://en.wikipedia.org/wiki/Triviality_(mathematics)"]trivial[/URL], thus I do not count them, however, including the base (b) itself results in automatic elimination of all possible extension numbers with "0 after 1" from the set (when the base is prime, if the base is composite, then there is no difference to include the base (b) itself or not), which is quite restrictive, thus, we also exclude the prime = b (you may ask me why we do not exclude the prime = b+1? Since b+1 is "11" in base b, this is generalized repunit number base b, if we exclude it ("11" in base b), then should we exclude "111", "1111", "11111", etc. in base b? This is hard to answer, and if we exclude them all, the result will not be "primes > m" for some integer m, thus we do not exclude "11" in base b but exclude "10" in base b)

sweety439 2021-10-31 12:57

[QUOTE=sweety439;585193]* Case (2,1):

** Since 23, 25, 41, 61, [B]221[/B] are primes, we only need to consider the family 2{0,1}1 (since any digits 2, 3, 4, 5, 6 between them will produce smaller primes)

*** Since [B]2111[/B] is prime, we only need to consider the families 2{0}1 and 2{0}1{0}1

**** All numbers of the form 2{0}1 are divisible by 3, thus cannot be prime.

**** All numbers of the form 2{0}1{0}1 are divisible by 2, thus cannot be prime.

* Case (2,2):

** Since 23, 25, 32, 52, [B]212[/B] are primes, we only need to consider the family 2{0,2,4,6}2 (since any digits 1, 3, 5 between them will produce smaller primes)

*** All numbers of the form 2{0,2,4,6}2 are divisible by 2, thus cannot be prime.

* Case (2,3):

** [B]23[/B] is prime, and thus the only minimal prime in this family.

* Case (2,4):

** Since 23, 25, 14 are primes, we only need to consider the family 2{0,2,4,6}4 (since any digits 1, 3, 5 between them will produce smaller primes)

*** All numbers of the form 2{0,2,4,6}4 are divisible by 2, thus cannot be prime.

* Case (2,5):

** [B]25[/B] is prime, and thus the only minimal prime in this family.

* Case (2,6):

** Since 23, 25, 16, 56 are primes, we only need to consider the family 2{0,2,4,6}6 (since any digits 1, 3, 5 between them will produce smaller primes)

*** All numbers of the form 2{0,2,4,6}6 are divisible by 2, thus cannot be prime.[/QUOTE]

* Case (3,1):

** Since 32, 41, 61 are primes, we only need to consider the family 3{0,1,3,5}1 (since any digits 2, 4, 6 between them will produce smaller primes)

*** Since 551 is prime, we only need to consider the family 3{0,1,3}1 and 3{0,1,3}5{0,1,3}1 (since any digits combo 55 between (3,1) will produce smaller primes)

**** For the 3{0,1,3}1 family, since [B]3031[/B] and 131 are primes, we only need to consider the families 3{0,1}1 and 3{3}3{0,1}1 (since any digits combo 03, 13 between (3,1) will produce smaller primes, thus for the digits between (3,1), all 3's must be before all 0's and 1's, and thus we can let the red [COLOR="Red"]3[/COLOR] in 3{3}[COLOR="red"]3[/COLOR]{0,1}1 be the rightmost 3 between (3,1), all digits before this 3 must be 3's, and all digits after this 3 must be either 0's or 1's)

***** For the 3{0,1}1 family:

****** If there are >=2 0's and >=1 1's between (3,1), then at least one of [B]30011[/B], [B]30101[/B], [B]31001[/B] will be a subsequence.

****** If there are no 1's between (3,1), then the form will be 3{0}1

******* All numbers of the form 3{0}1 are divisible by 2, thus cannot be prime.

****** If there are no 0's between (3,1), then the form will be 3{1}1

******* The smallest prime of the form 3{1}1 is [B]31111[/B]

****** If there are exactly 1 0's between (3,1), then there must be <3 1's between (3,1), or [B]31111[/B] will be a subsequence.

******* If there are 2 1's between (3,1), then the digit sum is 6, thus the number is divisible by 6 and cannot be prime.

******* If there are 1 1's between (3,1), then the number can only be either 3101 or 3011

******** Neither 3101 nor 3011 is prime.

******* If there are no 1's between (3,1), then the number must be 301

******** 301 is not prime.

***** For the 3{3}3{0,1}1 family:

****** If there are at least one 3 between (3,3{0,1}1) and at least one 1 between (3{3}3,1), then [B]33311[/B] will be a subsequence.

****** If there are no 3 between (3,3{0,1}1), then the form will be 33{0,1}1

******* If there are at least 3 1's between (33,1), then 31111 will be a subsequence.

******* If there are exactly 2 1's between (33,1), then the digit sum is 12, thus the number is divisible by 3 and cannot be prime.

******* If there are exactly 1 1's between (33,1), then the digit sum is 11, thus the number is divisible by 2 and cannot be prime.

******* If there are no 1's between (33,1), then the form will be 33{0}1

******** The smallest prime of the form 33{0}1 is [B]33001[/B]

****** If there are no 1 between (3{3}3,1), then the form will be 3{3}3{0}1

******* If there are at least 2 0's between (3{3}3,1), then 33001 will be a subsequence.

******* If there are exactly 1 0's between (3{3}3,1), then the form is 3{3}301

******** The smallest prime of the form 3{3}301 is [B]33333301[/B]

******* If there are no 0's between (3{3}3,1), then the form is 3{3}31

******** The smallest prime of the form 3{3}31 is [B]33333333333333331[/B]

**** For the 3{0,1,3}5{0,1,3}1 family, since 335 is prime, we only need to consider the family 3{0,1}5{0,1,3}1

***** Numbers containing 3 between (3{0,1}5,1):

****** The form is 3{0,1}5{0,1,3}3{0,1,3}1

******* Since 3031 and 131 are primes, we only need to consider the family 35{3}3{0,1,3}1 (since any digits combo 03, 13 between (3,1) will produce smaller primes)

******** Since 533 is prime, we only need to consider the family 353{0,1}1 (since any digits combo 33 between (35,1) will produce smaller primes)

********* Since 5011 is prime, we only need to consider the family 353{1}{0}1 (since any digits combo 01 between (353,1) will produce smaller primes)

********** If there are at least 3 1's between (353,{0}1), then 31111 will be a subsequence.

********** If there are exactly 2 1's between (353,{0}1), then the digit sum is 20, thus the number is divisible by 2 and cannot be prime.

********** If there are exactly 1 1's between (353,{0}1), then the form is 3531{0}1

*********** The smallest prime of the form 3531{0}1 is 3531001, but it is not minimal prime since 31001 is prime.

********** If there are no 1's between (353,{0}1), then the digit sum is 15, thus the number is divisible by 6 and cannot be prime.

***** Numbers not containing 3 between (3{0,1}5,1):

****** The form is 3{0,1}5{0,1}1

******* If there are >=2 0's and >=1 1's between (3,1), then at least one of 30011, 30101, 31001 will be a subsequence.

******* If there are no 1's between (3,1), then the form will be 3{0}5{0}1

******** All numbers of the form 3{0}5{0}1 are divisible by 3, thus cannot be prime.

******* If there are no 0's between (3,1), then the form will be 3{1}5{1}1

******** If there are >=3 1's between (3,1), then 31111 will be a subsequence.

******** If there are exactly 2 1's between (3,1), then the number can only be 31151, 31511, 35111

********* None of 31151, 31511, 35111 are primes.

******** If there are exactly 1 1's between (3,1), then the digit sum is 13, thus the number is divisible by 2 and cannot be prime.

******** If there are no 1's between (3,1), then the number is 351

********* 351 is not prime.

******* If there are exactly 1 0's between (3,1), then the form will be 3{1}0{1}5{1}1 or 3{1}5{1}0{1}1

******** No matter 3{1}0{1}5{1}1 or 3{1}5{1}0{1}1, if there are >=3 1's between (3,1), then 31111 will be a subsequence.

******** If there are exactly 2 1's between (3,1), then the number can only be 311051, 310151, 310511, 301151, 301511, 305111, 311501, 315101, 315011, 351101, 351011, 350111

********* Of these numbers, 311051, 301151, 311501, 351101, 350111 are primes.

********** However, 311051, 301151, 311501 have 115 as subsequence, and 350111 has 5011 as subsequence, thus only [B]351101[/B] is minimal prime.

******** No matter 3{1}0{1}5{1}1 or 3{1}5{1}0{1}1, if there are exactly 1 1's between (3,1), then the digit sum is 13, thus the number is divisible by 2 and cannot be prime.

******** If there are no 1's between (3,1), then the number is 3051 for 3{1}0{1}5{1}1 or 3501 for 3{1}5{1}0{1}1

********* Neither 3051 nor 3501 is prime.

sweety439 2021-11-11 16:34

[URL="https://sites.google.com/view/data-of-minimal-primes"]https://sites.google.com/view/data-of-minimal-primes[/URL]

sweety439 2021-11-14 02:05

[QUOTE=sweety439;586885]We have properties for bases 2<=b<=1024:[/QUOTE]

e.g.

b=491:

Primes with period length 1: 2, 5, 7
Primes with period length 2: 3, 41
Primes with period length 3: 37, 6529
Primes with period length 4: 149, 809
Primes with period length 5: 101, 191, 603791
Primes with period length 6: 13, 31, 199
Unique period lengths: 16, 22, 30, 31, 56, 67, 96, 204, ...
The smallest prime p such that znorder(Mod((this base,p)) = (p-1)/n for n = 1, 2, ...: 2, 13, 79, 5, 31, 7, 2339, 97, 1063, 151, 3257, 37, ...
Known generalized Wieferich primes: 7, 79, 661763933, 121261604419, ...
Known lengths of generalized repunit primes: 31, 67, 1201, ...
Known generalized (half) Fermat primes: F3, ...
Known Williams primes: n = 1, 5, 9, 75, 109, 2171, 3575, ...
CK for Sierpinski problem: 40 (covering set: {3,41})
CK for Riesel problem: 40 (covering set: {3,41})
Status of Sierpinski problem: proven (largest prime: 26*491^767+1)
Status of Riesel problem: proven (largest prime: 4*491^1683-1)

b=492:

Primes with period length 1: 491
Primes with period length 2: 17, 29
Primes with period length 3: 7, 34651
Primes with period length 4: 5, 48413
Primes with period length 5: 58714318141
Primes with period length 6: 37, 6529
Unique period lengths: 1, 5, 8, 22, 28, 50, 70, 78, 95, 100, 113, 170, 290, 452, 485, ...
The smallest prime p such that znorder(Mod((this base,p)) = (p-1)/n for n = 1, 2, ...: 5, 7, 31, 53, 881, 37, 71, 17, 127, 131, 6029, 373, ...
Known generalized Wieferich primes: 1034239, 211096013, 85750417002241, ...
Known lengths of generalized repunit primes: 5, 113, ...
Known generalized (half) Fermat primes: F2, ...
Known Williams primes: n = 29, 283, 1829, 3283, 4223, ...
CK for Sierpinski problem: 86 (covering set: {17,29})
CK for Riesel problem: 86 (covering set: {17,29})
Status of Sierpinski problem: k = 69 remain at n = 600K (largest known prime: 10*492^42842+1)
Status of Riesel problem: proven (largest prime: 81*492^399095-1)

sweety439 2021-11-15 02:41

we can consider the set of "strings of base b digits" which have no "prime strings > base" as subsequences.

In base 2, such strings are {0} (i.e. 000...000 with any number of 0's) and {0}1{0} (i.e. 000...0001000...000 with any number of 0's before the 1 and any number of 0's after the 1), i.e. all strings with at most one 1

In base 3, such strings are {0}1{0} and {0}1{0}1{0} and {0,2} (i.e. any combinations of any number of 0's and any number of 2's)

In base 10, the set of such strings are not simply to write, however, if "primes > base" is not needed, then such strings are any strings n such that [URL="https://oeis.org/A039995"]A039995[/URL](n) = 0 (not [URL="https://oeis.org/A062115"]A062115[/URL], since [URL="https://oeis.org/A062115"]A062115[/URL] is for [URL="https://en.wikipedia.org/wiki/Substring"]substring[/URL] instead of [URL="https://en.wikipedia.org/wiki/Subsequence"]subsequence[/URL], i.e. [URL="https://oeis.org/A062115"]A062115[/URL] is the numbers n such that [URL="https://oeis.org/A039997"]A039997[/URL](n) = 0 instead of the numbers n such that [URL="https://oeis.org/A039995"]A039995[/URL](n) = 0) with any number (including 0) of leading zeros.

Such strings are called [B]primefree strings[/B] in this post.

There are some clear theorems:

* All subsequences of any primefree strings are primefree strings.
* All [I]proper[/I] subsequences of any minimal prime (start with b+1) are primefree strings.
* All strings with "GCD of digits" <> 1 are primefree strings.
* All strings with at most one nonzero digit are primefree strings.
* All strings with no digits coprime to base are primefree strings.
* All strings of the form (combinations of only digits not coprime to base).(primefree string).(combinations of only digits not coprime to base) (where "." means concatenation) are primefree strings.
* The sequence of primefree strings (without leading zeros, which do not change the number values of the strings) numbers in base b is b-[URL="https://en.wikipedia.org/wiki/Automatic_sequence"]automatic sequence[/URL].
* If x and y are digits and family x{0}y can be ruled out as only contain composites (only count numbers > base) (including the cases: GCD(x,y) <> 1, GCD(y,base) <> 1, GCD(x+y,base-1) <> 1, etc.), then all strings of the form {0}x{0}y{0} are primefree strings.
* Equivalently, primefree strings are strings containing no minimal primes (start with b+1) as subsequences.
* In base 10, 4(infinitely many 6's)9 and 28(infinitely many 0's)7 are primefree strings, since all their subsequences are divisible by at least one prime in set {2,3,7}
* In base 10, 5(n 0's)27 is primefree string for n<=27, but not for n>=28, since the smallest prime in family 5{0}27 is 5(28 0's)27
* In base 9, (infinitely many 3's)(infinitely many 0's)5 is primefree string, since all their subsequences are divisible by at least one prime in set {2,5}
* In base 9, (infinitely many 1's)6(infinitely many 1's) is primefree string, since all their subsequences are either "divisible by at least one prime in set {2,5}" or "factored as difference of squares divided by 8"
* In base 8, 6(n 4's)7 is primefree string for n<=219, but not for n>=220, since all numbers in family 6{4}7 are divisible by at least one prime in set {3,5,13}, all numbers in family 6{4} or {4} are divisible by 2, and the smallest prime in family {4}7 is (220 4's)7
* Unsolved families are families whose repeating digit replaced to infinitely many this digits, to get a string, whether this string is primefree string are open problems, e.g. base 11 family 5{7} is unsolved family, thus whether base 11 string 5(infinitely many 7's) is primefree string is open problem (such families are usually conjectured to contain infinitely many "prime strings > base" as subsequences) (all unsolved families are families whether they contain at least one prime > base are open problems, but the converse is not true, if a family contain a sub-family which has known primes > base (e.g. base 22 family A{1} and base 33 family 8{1} and base 40 family 4{1}, whether they contain at least one prime > base are open problems, but since they have a sub-family {1}, which have known primes (there are known repunit primes in all non-perfect-power bases b<185), thus these families is not unsolved family, some classic examples are base 2 family 1(16 0's){0}1 (equivalent to whether 65537 is the largest Fermat prime), base 2 family 101001010111101{0}1 (equivalent to whether 21181 is Sierpinski number), and base 2 family 101110001110100{1} (equivalent to whether 23669 is Riesel number), whether they contain at least one prime > base are open problems, but since they have sub-family {1} or sub-family 1{0}1, which have known prime "11" (decimal 3) > base (2))
* All sub-families of all unsolved families are either can be ruled out as only contain composites (only consider numbers > base) or also unsolved families

[B]** Definition of "sub-family" in this post (like [URL="https://en.wikipedia.org/wiki/Subset"]subset[/URL] and [URL="https://en.wikipedia.org/wiki/Subsequence"]subsequence[/URL] and [URL="https://en.wikipedia.org/wiki/Substring"]substring[/URL])[/B]: family x is sub-family of family y if and only if: (the repeating digit of x and y are same) and (let x' be string that change family the repeating digit of x to a new character "#" (must not be the same of any digit in this base) and let y' be string that change family the repeating digit of y to a new character "#" (must not be the same of any digit in this base), x' is subsequence of y'), e.g. for the [URL="https://github.com/curtisbright/mepn-data/blob/master/data/minimal.25.txt"]unsolved families for the original minimal prime problem (i.e. prime > base is not need) for base b=25[/URL], "O{L}8" is sub-family of "LO{L}8" (since the repeating digit of them are both "L", and "O#8" is subsequence of "LO#8"), thus the smallest prime in family LO{L}8 may or may not be minimal prime, it is minimal prime if and only if there is no prime with (fewer or equal) repeating digit "L" of O{L}8, but all other families has no sub-family [URL="https://en.wikipedia.org/wiki/Relation_(mathematics)"]relation[/URL], i.e., all other families are pairwise [URL="https://en.wikipedia.org/wiki/Comparability"]incomparable[/URL], e.g. F0{K}O and F{0}KO are incomparable families, since neither is sub-family of another, since the repeating digit of them are not the same ("K" and "0") and "F0#O" and "F#KO" are incomparable sequences (related to "subsequence", not to "substring"), and for base 23, when searched to length 10000, all of these families 8{0}1, 8{0}81, 96{E}, 9{E}, K9A{E} are unsolved families, and for these families, "8{0}1" is sub-family of "8{0}81" (since the repeating digit of them are both "0", and "8#1" is subsequence of "8#81"), thus the smallest prime in family 8{0}81 may or may not be minimal prime, it is minimal prime if and only if there is no prime with (fewer or equal) repeating digit "0" of 8{0}1, also, both "96{E}" and "K9A{E}" are sub-families of "9{E}" (but "96{E}" and "K9A{E}" are incomparable families, since neither is sub-family of another, since although the repeating digit of them are not the same (both are "E"), but "96#" and "K9A#" are incomparable sequences (related to "subsequence", not to "substring")) (since the repeating digit of them are both "E", and both "96#" and "K9A#" are subsequence of "9#"), thus the smallest prime in families 96{E} and K9A{E} may or may not be minimal prime, it is minimal prime if and only if there is no prime with (fewer or equal) repeating digit "E" of 9{E})

(in this research, only consider [B]simple families[/B] for sub-families, for the definition of simple families, see [URL="https://cs.uwaterloo.ca/~cbright/reports/mepn.pdf"]this article[/URL], i.e. families with only one "repeating digits")

sweety439 2021-11-16 01:34

Using "sub-family", some unsolved families are obvious through [URL="https://en.wikiversity.org/wiki/Quasi-minimal_prime"]data for minimal primes (start with b+1) base b up to certain limit[/URL]: ("proper sub-family" is defined like "proper subset", i.e. sub-families other than itself)

Example 1: 80555551 = 80(5^5)1 is minimal prime (start with b+1) base b for b=10, thus all proper sub-families of 80{5}1 are unsolved families when searched to 5 5's if they [I]possible[/I] contain primes

Example 2: 55555025 = (5^5)25 is minimal prime (start with b+1) base b for b=8, thus all proper sub-families of {5}025 are unsolved families when searched to 5 5's if they [I]possible[/I] contain primes

Example 3: 33333301 = (3^6)01 is minimal prime (start with b+1) base b for b=7, thus all proper sub-families of {3}01 are unsolved families when searched to 6 3's if they [I]possible[/I] contain primes

Example 4: 100000000000507 = 1(0^11)507 is minimal prime (start with b+1) base b for b=9, thus all proper sub-families of 1{0}507 are unsolved families when searched to 11 0's if they [I]possible[/I] contain primes

Example 5: BBBBBB99B = (B^6)99B is minimal prime (start with b+1) base b for b=12, thus all proper sub-families of {B}99B are unsolved families when searched to 6 B's if they [I]possible[/I] contain primes

Example 6: 500025 = 5(0^3)25 is minimal prime (start with b+1) base b for b=8, thus all proper sub-families of 5{0}25 are unsolved families when searched to 3 0's if they [I]possible[/I] contain primes

Example 7: 77774444441 = (7^4)(4^6)1 is minimal prime (start with b+1) base b for b=8, thus:

* all proper sub-families of 7777{4}1 are unsolved families when searched to 6 4's if they [I]possible[/I] contain primes
* all proper sub-families of {7}4444441 are unsolved families when searched to 4 7's if they [I]possible[/I] contain primes

Example 8: 88888888833335 = (8^9)(3^4)5 is minimal prime (start with b+1) base b for b=9, thus:

* all proper sub-families of 888888888{3}5 are unsolved families when searched to 4 3's if they [I]possible[/I] contain primes
* all proper sub-families of {8}33335 are unsolved families when searched to 9 8's if they [I]possible[/I] contain primes

(Note: families with either [URL="https://en.wikipedia.org/wiki/Leading_zero"]leading zeros[/URL] or [URL="https://en.wikipedia.org/wiki/Trailing_zero"]trailing zeros[/URL] (or both) can be excluded, as they never produce primes > base, see [URL="https://mersenneforum.org/showpost.php?p=531660&postcount=10"]this related post[/URL])

sweety439 2021-11-20 10:39

[QUOTE=sweety439;571731]{z0}z1: (almost cannot be minimal prime (start with b+1), since this is not simple family, but always minimal prime (start with b'+1) in base b'=b^2)

[URL="http://www.fermatquotient.com/PrimSerien/GenRepuP.txt"]http://www.fermatquotient.com/PrimSerien/GenRepuP.txt[/URL]
[URL="https://mersenneforum.org/attachment.php?attachmentid=19026&d=1535956171"]https://mersenneforum.org/attachment.php?attachmentid=19026&d=1535956171[/URL]
[URL="https://cs.uwaterloo.ca/journals/JIS/VOL3/DUBNER/dubner.pdf"]https://cs.uwaterloo.ca/journals/JIS/VOL3/DUBNER/dubner.pdf[/URL]
[URL="http://www.primenumbers.net/Henri/us/MersFermus.htm"]http://www.primenumbers.net/Henri/us/MersFermus.htm[/URL]
[URL="http://www.bitman.name/math/table/488"]http://www.bitman.name/math/table/488[/URL]
[URL="https://oeis.org/A084742"]https://oeis.org/A084742[/URL]
[URL="https://oeis.org/A084741"]https://oeis.org/A084741[/URL] (corresponding primes)
[URL="https://oeis.org/A065507"]https://oeis.org/A065507[/URL] (prime bases)[/QUOTE]

Although {z0}z1 almost cannot be minimal prime (start with b+1), but always minimal prime (start with b'+1) in base b'=b^2, its formula is (b^n+1)/(b+1), like the generalized repunit {1} = (b^n-1)/(b-1), if this number is prime, then n must be prime, besides, n must be odd since for even n (b^n+1)/(b+1) is not integer, also, n=3 will make {z0}z1 = z1 which is a single-digit number in base b'=b^2, which is not allowed in my project (my project is finding and proving the minimal set of the primes > b written in base b, for bases 2<=b<=1024), thus we start with n=5

An interesting minimal (probable) prime (start with b'+1) in base b'=b^2 is b=34, it is (34^294277+1)/35, see references [URL="http://www.primenumbers.net/prptop/searchform.php?form=%2834%5En%2B1%29%2F35&action=Search"]http://www.primenumbers.net/prptop/searchform.php?form=%2834%5En%2B1%29%2F35&action=Search[/URL] and [URL="http://www.fermatquotient.com/PrimSerien/GenRepuP.txt"]http://www.fermatquotient.com/PrimSerien/GenRepuP.txt[/URL], this (probable) prime is (1122^147137):1123 in base b=1156, and this (probable) prime is minimal (probable) prime (start with b+1) in this base, also, for some bases, like 4, 8, 27, 32, such primes do not exist because of algebra factors, thus the families {12}:13 in base 16, {56}:57 in base 64, {702}:703 in base 729, {992}:993 in base 1024, etc. are ruled out as only contain composite numbers (only consider the numbers > base), for the smallest such prime (always minimal prime (start with b'+1) in base b' = b^2) (its formula is (#^n)$, where # = b^2-b and $ = b^2-b+1), see this list (for the first 100 bases b' = b^2, list the value n of the minimal prime (#^n)$):

[CODE]
b'=b^2 the n of (#^n)$
4,1
9,1
16,algebra
25,1
36,4
49,7
64,algebra
81,28
100,1
121,1
144,1
169,4
196,2
225,2
256,1
289,2
324,2
361,7
400,1
441,1
484,1
529,4
576,2
625,2
676,4
729,algebra
784,8
841,2
900,68
961,53
1024,algebra
1089,1
1156,147137
1225,4
1296,14
1369,1
1444,1
1521,5
1600,25
1681,7
1764,353
1849,1
1936,2
2025,50
2116,2
2209,1
2304,1
2401,2
2500,575
2601,73
2704,2
2809,10970
2916,2
3025,1
3136,17
3249,25
3364,7
3481,7
3600,467
3721,2
3844,4
3969,17
4096,algebra
4225,8
4356,2
4489,1172
4624,377
4761,4
4900,29
5041,1
5184,2
5329,2
5476,5
5625,1
5776,1
5929,17
6084,2
6241,52
6400,1
6561,1
6724,145
6889,8
7056,2
7225,82
7396,2
7569,2
7744,353
7921,5
8100,22
8281,4
8464,17
8649,43
8836,34
9025,20
9216,17
9409,>249998
9604,8
9801,2
10000,145
10201,2
[/CODE]

(note: base 9409 (i.e. b=97) has been searched to length 249998 by Fan Ming without any prime or PRP found, see [URL="https://mersenneforum.org/showpost.php?p=545152&postcount=223"]this post[/URL])

sweety439 2021-11-20 10:57

This problem is better than the original minimal prime problem since this problem is regardless [URL="https://primes.utm.edu/notes/faq/one.html"]whether 1 is considered as prime or not[/URL], i.e. [URL="https://primefan.tripod.com/Prime1ProCon.html"]no matter 1 is considered as prime or not prime[/URL] ([URL="https://cs.uwaterloo.ca/journals/JIS/VOL15/Caldwell2/cald6.pdf"]in the beginning of the 20th century, 1 is regarded as prime[/URL]), the sets M(Lb) in this problem are the same, while the sets M(Lb) in the original minimal prime problem are different, e.g. in base 10, if 1 is considered as prime, then the set M(Lb) in the original minimal prime problem is {1, 2, 3, 5, 7, 89, 409, 449, 499, 6469, 6949, 9049, 9649, 9949, 60649, 666649, 946669, 60000049, 66000049, 66600049}, while if 1 is not considered as prime, then the set M(Lb) in the original minimal prime problem is {2, 3, 5, 7, 11, 19, 41, 61, 89, 409, 449, 499, 881, 991, 6469, 6949, 9001, 9049, 9649, 9949, 60649, 666649, 946669, 60000049, 66000049, 66600049}, however, in base 10, the set M(Lb) in this problem is always {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027}, no matter 1 is considered as prime or not prime.

Besides, this problem is better than the original minimal prime problem since this problem (if we extend the problem to any larger and larger base) will cover "is there a prime of the form k*b^n+1 and k*b^n-1 for fixed integers k and b" (see [URL="http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm"]http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm[/URL] and [URL="http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm"]http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm[/URL]), we can extend this problem to a large base which is [URL="https://en.wikipedia.org/wiki/Exponentiation"]power[/URL] of b, while the original minimal prime problem will not, e.g. consider the form 67607*2^n+1, since if 67607*2^n+1 is prime then n == 3, 11 (mod 24) (for all other n, 67607*2^n+1 is divisible by at least one of {3, 5, 13, 17}) we can extend this problem to base b = 2^24 = 16777216, since 2^24 > 67607, thus the prime "67607" will be single-digit prime in base b = 2^24, which is excluded in this problem, and now we can consider the unsolved family (540856):{0}:1 (where 540856 = 67607 * 2^3) in base b = 16777216, whether there is a prime of this form is equivalent to whether there is a prime of the form 67607*2^n+1 with n == 3 (mod 24), for other forms like 21181*2^n+1 and 23669*2^n-1, we can choose a large power of 2 as the base (b) to make 21181 or 23669 be a single-digit number, i.e. choose a base b=2^r such that 21181 or 23669 are <b, while for the original minimal prime problem (i.e. prime > base is not required), there is not always a base such that the original minimal prime problem covers "is there a prime of the form k*b^n+1 and k*b^n-1 for fixed integers k and b", since if k is prime (for the k*b^n+1 case) or k-1 is prime (for the k*b^n-1 case), then this prime must be subsequence of k*b^n+1 or k*b^n-1, thus the original minimal prime problem cannot cover it, but for the problem in this forum (i.e. the minimal prime (start with b+1) problem), we can choose enough large r such that k or k-1 is single-digit prime in base b^r (i.e. choose r such that b^r > k), then this problem can cover "finding the smallest prime of the form k*b^n+1 or k*b^n-1 for fixed integer k".

sweety439 2021-11-21 15:27

For the forms in [URL="https://docs.google.com/spreadsheets/d/e/2PACX-1vTKkSNKGVQkUINlp1B3cXe90FWPwiegdA07EE7-U7sqXntKAEQrynoI1sbFvvKriieda3LfkqRwmKME/pubhtml"]https://docs.google.com/spreadsheets/d/e/2PACX-1vTKkSNKGVQkUINlp1B3cXe90FWPwiegdA07EE7-U7sqXntKAEQrynoI1sbFvvKriieda3LfkqRwmKME/pubhtml[/URL], it is conjectured that every base has infinitely many primes of these forms:

1{0}z
1{z}
y{z}
z{0}1
{z}1

However ....

* For the form 2{0}1, all bases b == 1 mod 3 can be ruled out as only contain composite numbers (by trivial 1-cover {3}), the smallest base b != 1 mod 3 such that family 2{0}1 has no primes is conjectured to be 201446503145165177 (with covering set {3, 5, 17, 257, 641, 65537, 6700417}), but this conjecture is very hard to prove (however, since the smallest prime of the form 2{0}1 (if exists) is always minimal prime (start with b+1), the "minimal primes (start with b+1) problem" in all bases b<201446503145165177 covers the conjecture that 201446503145165177 is the smallest base b != 1 mod 3 such that there is no prime of the form 2{0}1
* Since in odd bases, 1{0}2 is the dual form of 2{0}1 (in even bases, form 1{0}2 can be ruled out as only contain composite numbers (by trivial 1-cover {2})), 201446503145165177 is also conjectured to be the smallest base b == 3, 5 mod 6 (i.e. b is odd AND b != 1 mod 3) (with covering set {3, 5, 17, 257, 641, 65537, 6700417}), but again, this conjecture is also very hard to prove (however, since the smallest prime of the form 1{0}2 (if exists) is always minimal prime (start with b+1), the "minimal primes (start with b+1) problem" in all bases b<201446503145165177 covers the conjecture that 201446503145165177 is the smallest base b == 3, 5 mod 6 such that there is no prime of the form 1{0}2
* For the form {1}, all bases in [URL="https://oeis.org/A096059"]A096059[/URL] (all are perfect powers) can be ruled out as only contain composite numbers (by difference-of-r-th-power factorization), but there is
* For the form 4{0}1, all bases b == 1 mod 5 can be ruled out as only contain composite numbers (by trivial 1-cover {5}), the smallest base b != 1 mod 5 such that family 4{0}1 has no primes is [I]proven[/I] to be 14, for bases 5, 8, 12, the 3-digit number 401 is prime, and for all other base 4<b<14 (b>4 is required since bases b<=4 have no digit "4" and hence no family "4{0}1") which not == 1 mod 5, the 2-digit number 41 is prime
* For the form 1{0}4, all bases b == 1 mod 5 and all even bases can be ruled out as only contain composite numbers (by trivial 1-cover, {5} for bases b == 1 mod 5 and {2} for even bases), the smallest base which is neither == 1 mod 5 nor even base is [I]proven[/I] to be 29, while the largest prime is 1000004 (with 7 digits) in base 23, which is equal to 23^6+4

sweety439 2021-11-23 15:45

Every base b family is the [B]dual[/B] (for the definition, see [URL="http://www.kurims.kyoto-u.ac.jp/EMIS/journals/INTEGERS/papers/i61/i61.pdf"]http://www.kurims.kyoto-u.ac.jp/EMIS/journals/INTEGERS/papers/i61/i61.pdf[/URL] and [URL="https://oeis.org/A076336/a076336c.html"]https://oeis.org/A076336/a076336c.html[/URL] and [URL="https://mersenneforum.org/showthread.php?t=10761"]https://mersenneforum.org/showthread.php?t=10761[/URL] and [URL="https://mersenneforum.org/showthread.php?t=6545"]https://mersenneforum.org/showthread.php?t=6545[/URL] and [URL="https://mersenneforum.org/showthread.php?t=26328"]https://mersenneforum.org/showthread.php?t=26328[/URL] and [URL="https://mersenneforum.org/showthread.php?t=21954"]https://mersenneforum.org/showthread.php?t=21954[/URL], also see [URL="https://mersenneforum.org/forumdisplay.php?f=86"]Five or Bust forum[/URL], the dual form has same [URL="https://www.rieselprime.de/ziki/Nash_weight"]Nash weight[/URL] (or [URL="https://stdkmd.net/nrr/prime/primedifficulty.txt"]difficulty[/URL]), the [URL="https://stdkmd.net/nrr/prime/primedifficulty.txt"]difficulty page[/URL] has many forms shown as having same difficulty and next to each other (e.g. (107*10^n-71)/9 and (71*10^n-107)/9, 4*10^n+3 and 75*10^n+1, (53*10^n+37)/9 and (37*10^n+53)/9, (5*10^n-437)/9 and (874*10^n-1)/9), they are exactly the dual forms each other), for the forms in base b, these forms are dual forms each other:

1{0}1 (b^n+1) is the dual of itself (if b is even)
1{0}2 (b^n+2) and 2{0}1 (2*b^n+1) (if b is odd)
#{0}1 (# = b/2) ((b/2)*b^n+1) and 2{0}1 (2*b^n+1) (if b is even)
1{0}3 (b^n+3) and 3{0}1 (3*b^n+1) (if b == 2, 4 mod 6)
#{0}1 (# = b/3) ((b/3)*b^n+1) and 3{0}1 (3*b^n+1) (if b == 0 mod 6)
1{0}z (b^n+(b-1)) and z{0}1 ((b-1)*b^n+1)
1{0}11 (b^n+(b+1)) and 11{0}1 ((b+1)*b^n+1)
{1} ((b^n-1)/(b-1)) is the dual of itself
1{z} (2*b^n-1) and {z}y (b^n-2) (if b is odd)
1{z} (2*b^n-1) and #{z} (# = b/2-1) ((b/2)*b^n-1) (if b is even)
2{z} (3*b^n-1) and {z}x (b^n-3) (if b == 2, 4 mod 6)
2{z} (3*b^n-1) and #{z} (# = b/3-1) ((b/3)*b^n-1) (if b == 0 mod 6)
2{1} (((2*b-1)*b^n-1)/(b-1)) and {1}0z ((b^n-(2*b-1))/(b-1))
{y}z (((b-2)*b^n+1)/(b-1)) and {1}2 ((b^n+(b-2))/(b-1)) (if b is odd)
y{z} ((b-1)*b^n-1) and {z}1 (b^n-(b-1))
10{z} ((b+1)*b^n-1) and {z}yz (b^n-(b+1))

also .... for [I]any[/I] digits X and Y:

X{0}Y and Y{0}X (if gcd(X,b) = 1, gcd(Y,b) = 1)
X{0}1 and Y{0}1 (if X*Y = b)
X{z} and Y{z} (if (X+1)*(Y+1) = b)

sweety439 2021-11-23 18:13

See [URL="https://stdkmd.net/nrr/prime/primesize.txt"]https://stdkmd.net/nrr/prime/primesize.txt[/URL], sequences of the form k*10^n+1 includes Generalized Cullen prime numbers, and sequences of the form k*10^n-1 includes Generalized Woodall prime numbers, thus, sequences of the form k*b^n+1 (of the form *{0}1 in base b) includes Generalized Cullen prime numbers base b (numbers of the form n*b^n+1, i.e. the special case that k=n), and sequences of the form k*b^n-1 (of the form *{z} in base b) includes Generalized Woodall prime numbers base b (numbers of the form n*b^n-1, i.e. the special case that k=n), for such numbers, see:

Generalized Cullen primes:

[URL="http://www.prothsearch.net/cullen.html"]http://www.prothsearch.net/cullen.html[/URL] (broken link: [URL="http://web.archive.org/web/20161028015144/http://www.prothsearch.net/cullen.html"]from wayback machine[/URL] [URL="https://docs.google.com/document/d/e/2PACX-1vTo4s_9GlGDlEFr3LVododSgO3ursWockhzOkDxLxLzUxPoaZMy70T7XSdrRYLL9wPt8D8Ts8rHlWeE/pub"]cached copy[/URL]) (b=2)
[URL="http://guenter.loeh.name/gc/status.html"]http://guenter.loeh.name/gc/status.html[/URL] (3<=b<=100)
[URL="https://harvey563.tripod.com/GClist.txt"]https://harvey563.tripod.com/GClist.txt[/URL] (101<=b<=10000)
[URL="http://www.primzahlenarchiv.de/index.html"]http://www.primzahlenarchiv.de/index.html[/URL] (broken link: [URL="http://web.archive.org/web/20070626212719/http://www.primzahlenarchiv.de/index.html"]from wayback machine[/URL] [URL="https://docs.google.com/document/d/e/2PACX-1vRzUm818ShXVRh62wZauDMEwpJO9PdPY_ILhGMBC5rn6_KdjcLkrcvc3ogaU7mDc-6piP_wvv5zQ1BM/pub"]cached copy[/URL]) (101<=b<=200)
[URL="http://www.primzahlenarchiv.de/news.html"]http://www.primzahlenarchiv.de/news.html[/URL] (broken link: [URL="http://web.archive.org/web/20070213035517/http://www.primzahlenarchiv.de/news.html"]from wayback machine[/URL] [URL="https://docs.google.com/document/d/e/2PACX-1vSeeUop6-R9J2M1nl6gUM_Kz6M_qx4FuFwajpvodxCaJJRPWioayPJm5nu_EBixbp-dGRlaOFONjqPa/pub"]cached copy[/URL]) (101<=b<=200)
[URL="http://www.primzahlenarchiv.de/status.html"]http://www.primzahlenarchiv.de/status.html[/URL] (broken link: [URL="http://web.archive.org/web/20070211162945/http://www.primzahlenarchiv.de/status.html"]from wayback machine[/URL] [URL="https://docs.google.com/document/d/e/2PACX-1vSvOaq-Bl6yIp3F4Tn-4xd_OTSxjm84L_OtbxD2D7Qw5viA1sQsRSSn6H8jty57E4BS5Z9ihgbC77zp/pub"]cached copy[/URL]) (101<=b<=200)
[URL="https://www.rieselprime.de/ziki/Gen_Cullen_prime_table"]https://www.rieselprime.de/ziki/Gen_Cullen_prime_table[/URL] (all available bases b)
[URL="http://www.primegrid.com/forum_thread.php?id=7073"]http://www.primegrid.com/forum_thread.php?id=7073[/URL] (primegrid search page)
[URL="http://www.primegrid.com/stats_cullen_llr.php"]http://www.primegrid.com/stats_cullen_llr.php[/URL] (primegrid status page, b=2)
[URL="http://www.primegrid.com/stats_gcw_llr.php"]http://www.primegrid.com/stats_gcw_llr.php[/URL] (primegrid status page, b>=3)
[URL="https://primes.utm.edu/top20/page.php?id=6"]https://primes.utm.edu/top20/page.php?id=6[/URL] (top 20 page, b=2)
[URL="https://primes.utm.edu/top20/page.php?id=42"]https://primes.utm.edu/top20/page.php?id=42[/URL] (top 20 page, b>=3)
[URL="https://oeis.org/A240234"]https://oeis.org/A240234[/URL] (smallest n for given base b)
[URL="https://oeis.org/A327660"]https://oeis.org/A327660[/URL] (smallest n>b-2 for given base b)

Generalized Woodall primes:

[URL="http://www.prothsearch.net/woodall.html"]http://www.prothsearch.net/woodall.html[/URL] (broken link: [URL="http://web.archive.org/web/20161028080439/http://www.prothsearch.net/woodall.html"]from wayback machine[/URL] [URL="https://docs.google.com/document/d/e/2PACX-1vRJh4b0EwlNS9fY0Axf5vZXfyHVSSYmFP48NuVqjA7wpyu_-u0C4cRcNsrZlqXy9kV_vKWM6zm5EjIA/pub"]cached copy[/URL]) (b=2)
[URL="https://harvey563.tripod.com/GWlist.txt"]https://harvey563.tripod.com/GWlist.txt[/URL] (3<=b<=10000)
[URL="https://harvey563.tripod.com/GeneralizedWoodallPrimes.txt"]https://harvey563.tripod.com/GeneralizedWoodallPrimes.txt[/URL] (3<=b<=10000)
[URL="http://science.kennesaw.edu/~jdemaio/generali.htm"]http://science.kennesaw.edu/~jdemaio/generali.htm[/URL] (broken link: [URL="https://web.archive.org/web/20170416094910/http://science.kennesaw.edu/~jdemaio/generali.htm"]from wayback machine[/URL] [URL="https://docs.google.com/document/d/e/2PACX-1vQYAlrkt98ED9knW8YdN9CccR6kyj85sC14UKV9OgbeARtJq35I9lLtbuMd4R4LElq3oosviyZsxsFU/pub"]cached copy[/URL]) (3<=b<=100)
[URL="https://www.rieselprime.de/ziki/Gen_Woodall_prime_table"]https://www.rieselprime.de/ziki/Gen_Woodall_prime_table[/URL] (all available bases b)
[URL="http://www.primegrid.com/forum_thread.php?id=7073"]http://www.primegrid.com/forum_thread.php?id=7073[/URL] (primegrid search page)
[URL="http://www.primegrid.com/stats_woodall_llr.php"]http://www.primegrid.com/stats_woodall_llr.php[/URL] (primegrid status page, b=2)
[URL="http://www.primegrid.com/stats_gcw_llr.php"]http://www.primegrid.com/stats_gcw_llr.php[/URL] (primegrid status page, b>=3)
[URL="https://primes.utm.edu/top20/page.php?id=7"]https://primes.utm.edu/top20/page.php?id=7[/URL] (top 20 page, b=2)
[URL="https://primes.utm.edu/top20/page.php?id=45"]https://primes.utm.edu/top20/page.php?id=45[/URL] (top 20 page, b>=3)
[URL="https://oeis.org/A240235"]https://oeis.org/A240235[/URL] (smallest n for given base b)
[URL="https://oeis.org/A327661"]https://oeis.org/A327661[/URL] (smallest n>b-2 for given base b)

Also, since the dual for n*b^n+1 and n*b^n-1 are b^n+n and b^n-n, respectively (we assume that gcd(n,b) = 1) (since the dual of k*b^n+1 and k*b^n-1 are b^n+k and b^n-k, respectively), and n*b^n+1 is including in *{0}1, n*b^n-1 is including in *{z}, b^n+n is including in 1{0}*, b^n-n is including in z{*}, thus the OEIS sequence:

[URL="https://oeis.org/A093324"]https://oeis.org/A093324[/URL] (b^n+n)
[URL="https://oeis.org/A084743"]https://oeis.org/A084743[/URL] (b^n+n, corresponding primes)
[URL="https://oeis.org/A084746"]https://oeis.org/A084746[/URL] (b^n-n)
[URL="https://oeis.org/A084745"]https://oeis.org/A084745[/URL] (b^n-n, corresponding primes)

sweety439 2021-11-24 18:45

3 Attachment(s)
[QUOTE=sweety439;568817]All GFN base b and GRU base b are strong-probable-primes to base b, so don't test with this base (see [URL="https://mersenneforum.org/showthread.php?t=10476&page=2"]https://mersenneforum.org/showthread.php?t=10476&page=2[/URL], [URL="https://oeis.org/A171381"]https://oeis.org/A171381[/URL], [URL="https://oeis.org/A028491"]https://oeis.org/A028491[/URL]), in fact, all Zsigmondy numbers Zs(n,b,1) ant all their factors are strong-probable-primes to base b, so don't test with this base.[/QUOTE]

Our result about minimal primes (start with b+1) assume that a number which has passed [URL="https://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test"]Miller–Rabin tests[/URL] to the first 9 prime bases is in fact prime, since in some cases (e.g. b = 13 and b = 16) some candidate elements of M(Lb) are too long to be [URL="https://primes.utm.edu/prove/prove4.html"]proven prime[/URL] rigorously (and neither [URL="https://primes.utm.edu/prove/prove3_1.html"]N−1[/URL] nor [URL="https://primes.utm.edu/prove/prove3_2.html"]N+1[/URL] can be ≥33.3333% [URL="https://en.wikipedia.org/wiki/Integer_factorization"]factored[/URL]). The smallest [I]composite[/I] number which passed Miller–Rabin tests to the first n prime bases are listed in [URL="https://oeis.org/A014233"]https://oeis.org/A014233[/URL], most composite numbers which passed Miller–Rabin tests to many prime bases are of the form (m+1)*(2*m+1) with m+1 and 2*m+1 both primes (such numbers are [URL="https://en.wikipedia.org/wiki/Triangular_number"]triangular numbers[/URL], and such numbers N have 8*N+1 square numbers), or of the form (m+1)*(3*m+1) with m+1 and 3*m+1 both primes (such numbers are [URL="https://en.wikipedia.org/wiki/Octagonal_number"]octagonal numbers[/URL], and such numbers N have 3*N+1 square numbers (the fully N-values with 3*N+1 square numbers are the [URL="https://oeis.org/A001082"]generalized octagonal numbers[/URL])), or of the form (2*m+1)*(3*m+1) with 2*m+1 and 3*m+1 both primes (such numbers are [URL="https://oeis.org/A033570"]pentagonal numbers with odd index[/URL], and such numbers N have 24*N+1 square numbers (the fully N-values with 24*N+1 square numbers are the [URL="https://oeis.org/A001318"]generalized pentagonal numbers[/URL])) (note that 24 is the largest number k such that all squares of all numbers coprime to k are of the form k*N+1, and number k having this property if and only if k is [URL="https://oeis.org/A018253"]divisor or 24[/URL], and generalized pentagonal numbers are important for the [URL="https://en.wikipedia.org/wiki/Pentagonal_number_theorem"]pentagonal number theorem[/URL])

e.g. [URL="https://oeis.org/A014233"]A014233[/URL](12) = [URL="https://oeis.org/A014233"]A014233[/URL](13) = 318665857834031151167461 = 399165290221 * 798330580441 = (399165290220+1) * (2*399165290220+1), and 8*318665857834031151167461+1 = 1596661160883^2 is a square, and [URL="http://factordb.com/prooffailed.php"]for the currently only "probable primes that failed primality proof or had a factor" with order 21 in factordb[/URL], 1396981702787004809899378463251 = 835757651112751 * 1671515302225501 = (835757651112750+1) * (2*835757651112750+1), and 8*1396981702787004809899378463251+1 = 3343030604451003^2 is a square. (both of them are [URL="https://en.wikipedia.org/wiki/Triangular_number"]triangular numbers[/URL])

Factorization of [URL="https://oeis.org/A014233"]A014233[/URL](n):

[CODE]
1: 2047 = 23 * 89 (form (m+1) * (4*m+1) with m+1 and 4*m+1 both primes)
2: 1373653 = 829 * 1657 (form (m+1) * (2*m+1) with m+1 and 2*m+1 both primes)
3: 25326001 = 2251 * 11251 (form (m+1) * (5*m+1) with m+1 and 5*m+1 both primes)
4: 3215031751 = 151 * 751 * 28351 (can also be written as 28351 * 113401, is still form (m+1) * (4*m+1), although 4*m+1 is not prime)
5: 2152302898747 = 6763 * 10627 * 29947 (form (7*m+1) * (11*m+1) * (31*m+1) with 7*m+1, 11*m+1 and 31*m+1 all primes)
6: 3474749660383 = 1303 * 16927 * 157543 (form (m+1) * (13*m+1) * (121*m+1) with m+1, 13*m+1 and 121*m+1 all primes)
7,8: 341550071728321 = 10670053 * 32010157 (form (m+1) * (3*m+1) with m+1 and 3*m+1 both primes)
9,10,11: 3825123056546413051 = 149491 * 747451 * 34233211 (form (m+1) * (5*m+1) * (229*m+1) with m+1, 5*m+1 and 229*m+1 all primes)
12,13: 318665857834031151167461 = 399165290221 * 798330580441 (form (m+1) * (2*m+1) with m+1 and 2*m+1 both primes)
[/CODE]

Most of these "[URL="http://dx.doi.org/10.1090/S0025-5718-1993-1192971-8"]strong pseudoprimes to several bases[/URL]" are of the form (m+1)*(k*m+1) with m+1 and k*m+1 both primes, since (m+1)*(k*m+1) is pseudoprime to all bases coprime to it which are k-th power residue mod k*m+1, and note that numbers of the form (m+1)*(k*m+1) are (2*k+2)-gonal numbers, e.g. numbers of the form (m+1)*(2*m+1) are [URL="https://en.wikipedia.org/wiki/Hexagonal_number"]hexagonal numbers[/URL], numbers of the form (m+1)*(3*m+1) are [URL="https://en.wikipedia.org/wiki/Octagonal_number"]octagonal numbers[/URL], numbers of the form (m+1)*(4*m+1) are [URL="https://en.wikipedia.org/wiki/Decagonal_number"]decagonal numbers[/URL], and numbers of the form (m+1)*(5*m+1) are [URL="https://en.wikipedia.org/wiki/Dodecagonal_number"]dodecagonal number[/URL].

In fact, the hexagonal number (m+1)*(2*m+1) (if both m+1 and 2*m+1 are primes) are [URL="https://en.wikipedia.org/wiki/Fermat_pseudoprime"]Fermat pseudoprime[/URL] to all bases which are coprime to this triangular number and [URL="https://en.wikipedia.org/wiki/Quadratic_residue"]quadratic residue[/URL] mod 2*m+1 ([URL="https://oeis.org/A129521"]reference for a special example that base = 4, which is quadratic residue mod all numbers since it is square number[/URL])

Since the [URL="https://en.wikipedia.org/wiki/Fermat_primality_test"]Fermat primality test[/URL] is b^(n-1) == 1 mod n, composite number n is Fermat pseudoprime base b if and only if [URL="https://en.wikipedia.org/wiki/Multiplicative_order"]znorder[/URL](Mod(b,n)) divides n-1, thus, composite number n is Fermat pseudoprime base b if and only if these three conditions are all satisfied:

* gcd(b,n) = 1
* For every prime p dividing n, b is ((p-1)/gcd(p-1,n-1))-th power residue ([URL="https://en.wikipedia.org/wiki/Quadratic_residue"]quadratic residue[/URL] for 2, [URL="https://en.wikipedia.org/wiki/Cubic_residue"]cubic residue[/URL] for 3, [URL="https://en.wikipedia.org/wiki/Quartic_reciprocity"]quartic residue[/URL] for 4, [URL="https://en.wikipedia.org/wiki/Octic_reciprocity"]octic residue[/URL] for 8, ...) mod p
* For every prime power p^r (r>1) dividing n, p is [URL="http://www.fermatquotient.com/FermatQuotienten/FermQ_Sorg.txt"]Wieferich prime base b[/URL] with [URL="http://www.fermatquotient.com/FermatQuotienten/FermatQ3.txt"]order[/URL] >= r-1 (i.e. b^(p-1) == 1 mod p^r)

(for [URL="https://en.wikipedia.org/wiki/Carmichael_number"]Carmichael numbers[/URL] n, [I]all[/I] bases b such that gcd(b,n) = 1 satisfy these three conditions, thus Carmichael numbers n are Fermat pseudoprimes to [I]all[/I] bases b coprime to n)

(if (p-1)/gcd(p-1,n-1) = 1, i.e. p-1 divides n-1, then every b is ((p-1)/gcd(p-1,n-1))-th power residue mod p, thus this part of conditions (e.g. p=7 for 91, which 7-1 divides 91-1) are not listed below, since every b satisfies this condition)

e.g.

* 15 is Fermat pseudoprime base b if and only if gcd(b,15) = 1 and b is 2nd power residue (quadratic residue) mod 5
* 21 is Fermat pseudoprime base b if and only if gcd(b,21) = 1 and b is 3rd power residue (cubic residue) mod 7
* 22 is Fermat pseudoprime base b if and only if gcd(b,22) = 1 and b is 10th power residue mod 11
* 24 is Fermat pseudoprime base b if and only if gcd(b,24) = 1 and b is 2nd power residue (quadratic residue) mod 3 and 2 is Wieferich prime with order >=2 base b
* 25 is Fermat pseudoprime base b if and only if gcd(b,25) = 1 and 5 is Wieferich prime base b
* 27 is Fermat pseudoprime base b if and only if gcd(b,27) = 1 and 3 is Wieferich prime with order >=2 base b
* 28 is Fermat pseudoprime base b if and only if gcd(b,28) = 1 and b is 2nd power residue (quadratic residue) mod 7 and 2 is Wieferich prime base b
* 35 is Fermat pseudoprime base b if and only if gcd(b,35) = 1 and b is 2nd power residue (quadratic residue) mod 5 and b is 3rd power residue (quadratic residue) mod 7
* 65 is Fermat pseudoprime base b if and only if gcd(b,65) = 1 and b is 3rd power residue (cubic residue) mod 13
* 87 is Fermat pseudoprime base b if and only if gcd(b,87) = 1 and b is 14th power residue mod 29
* 91 is Fermat pseudoprime base b if and only if gcd(b,91) = 1 and b is 2nd power residue (quadratic residue) mod 13
* 117 is Fermat pseudoprime base b if and only if gcd(b,117) = 1 and b is 3rd power residue (cubic residue) mod 13 and 3 is Wieferich prime base b
* 148 is Fermat pseudoprime base b if and only if gcd(b,148) = 1 and b is 12th power residue mod 37 and 2 is Wieferich prime base b
* 169 is Fermat pseudoprime base b if and only if gcd(b,169) = 1 and 13 is Wieferich prime base b
* 205 is Fermat pseudoprime base b if and only if gcd(b,205) = 1 and b is 10th power residue mod 41
* 231 is Fermat pseudoprime base b if and only if gcd(b,231) = 1 and b is 3rd power residue (cubic residue) mod 7
* 285 is Fermat pseudoprime base b if and only if gcd(b,285) = 1 and b is 9th power residue mod 19
* 325 is Fermat pseudoprime base b if and only if gcd(b,325) = 1 and 5 is Wieferich prime base b
* 341 is Fermat pseudoprime base b if and only if gcd(b,341) = 1 and b is 3rd power residue (cubic residue) mod 31
* 493 is Fermat pseudoprime base b if and only if gcd(b,493) = 1 and b is 4th power residue (quartic residue) mod 17 and b is 7th power residue (septic residue) mod 29

(for more informations, see attached text files and [URL="https://de.wikibooks.org/wiki/Pseudoprimzahlen:_Tabelle_Fermatsche_Pseudoprimzahlen"]https://de.wikibooks.org/wiki/Pseudoprimzahlen:_Tabelle_Fermatsche_Pseudoprimzahlen[/URL] and [URL="https://de.wikibooks.org/wiki/Pseudoprimzahlen:_Tabelle_Pseudoprimzahlen_%2815_-_4999%29"]https://de.wikibooks.org/wiki/Pseudoprimzahlen:_Tabelle_Pseudoprimzahlen_%2815_-_4999%29[/URL])

Thus, if n is hexagonal number (m+1)*(2*m+1) with m+1 and 2*m+1 both primes, then n is Fermat pseudoprime base b for all bases b which is coprime to n and 2nd power residue (quadratic residue) mod 2*m+1, which has m*m = m^2 bases in all n (= (m+1)*(2*m+1)) modular classes of n, which has about a half modular classes of n, thus these number are easily to be Fermat pseudoprime (if the base b is random chosen), and if n is octagonal number (m+1)*(3*m+1) with m+1 and 3*m+1 both primes, then n is Fermat pseudoprime base b for all bases b which is coprime to n and 3rd power residue (cubic residue) mod 3*m+1, which has m*m = m^2 bases in all n (= (m+1)*(3*m+1)) modular classes of n, which has about one third modular classes of n, and if n is pentagonal number (2*m+1)*(3*m+1) with 2*m+1 and 3*m+1 both primes, then n is Fermat pseudoprime base b for all bases b which is coprime to n and 2nd power residue (quadratic residue) mod 2*m+1 and 3rd power residue (cubic residue) mod 3*m+1, which has m*m = m^2 bases in all n (= (2*m+1)*(3*m+1)) modular classes of n, which has about one sixth modular classes of n

Note that no [URL="https://en.wikipedia.org/wiki/Polygonal_number"]polygonal numbers[/URL] can be primes (except the trivial case, i.e. every n is the second n-gonal number), see [URL="https://en.wikipedia.org/wiki/Centered_polygonal_number#Formula"]this reference[/URL], for generalized polygonal numbers, only index 2 (n-3), 3 (n), and 4 (3*n-8) can be primes, no generalized polygonal numbers with index > 4 can be primes.

Thus, for a number to test primality, suggest that if this number passes the strong test to first few prime bases (strong pseudoprimes are always Fermat pseudoprimes to the same bases), test if 3*n+1, 8*n+1, 24*n+1 are squares (this is very easily to test), if at least one of them are squares and n>7, then n must be composite because of the algebraic factorization of difference of two square numbers, if none of these three numbers are squares, then n is very likely to be prime.

sweety439 2021-11-24 18:59

Since triangular number * 9 + 1 are always triangular numbers, and no triangular numbers > 3 are primes, thus these families in base 9 contain no primes (only count numbers > base), since they can be factored as difference of squares divided by 8:

1{1} (={1}), 3{1}, 6{1}, 11{1} (={1}), 16{1}, 23{1}, 31{1} (=3{1}), 40{1}, ...

(all numbers in these families are [URL="https://oeis.org/A000217"]triangular numbers[/URL], and thus all such numbers except 3 are composite, but the prime 3 is not allowed since prime must be > base)

they are the families (triangular number){1}, in fact, families 6{1} and 16{1} are already ruled out as only contain composite by covering set {2,5}

Since generalized octagonal numbers * 4 + 1 are always generalized octagonal numbers, and no generalized octagonal numbers > 5 are primes, thus these families in base 4 contain no primes (only count numbers > base), since they can be factored as difference of squares divided by 3:

1{1} (={1}) (except the number 11 = 5 in decimal), 11{1} (={1}), 20{1}, 100{1}, 111{1} (={1}), 201{1} (=20{1}), 220{1}, ...

(all numbers in these families are [URL="https://oeis.org/A001082"]generalized octagonal numbers[/URL], and thus all such numbers except 11 in base 4 (= 5 in decimal) are composite, the prime 5 [B]is[/B] allowed in base 4, but not allowed in bases > 4 which are powers of 4 since prime must be > base)

they are the families (generalized octagonal number){1}, and since base b families can be converted to base b^n families for n>1 (the repeating digit (i.e. the digit in {}) will be multiple of Rn(b), where Rn(b) is the base b [URL="https://en.wikipedia.org/wiki/Repunit"]repunit[/URL] with length n), these base 4 families can be converted to base 16 (=4^2) families, {1} in base 4 = 1{5} in base 16, 20{1} in base 4 = 8{5} in base 16, 100{1} in base 4 = 10{5} in base 16, 220{1} in base 4 = A1{5} in base 16, etc. (one base b^n digit is equivalent to n base b digits, and thus minimal prime (start with b+1) base b is always minimal prime (start with b'+1) base b' = b^n if this prime is > b' (references: [URL="https://www.rose-hulman.edu/~rickert/Compositeseq/#b9d4"]the case base 3 families *{1} converted to base 9=3^2 families *{4}[/URL] [URL="https://en.wikipedia.org/wiki/Truncatable_prime#Other_bases"]the case n base 10 digits is equivalent to one base 10^n digit, in research of truncatable primes (instead of minimal primes) to other bases[/URL] [URL="https://en.wikipedia.org/wiki/Talk:P-adic_number/Archive_1#%22The_reason_for_this_property_turns_out_to_be_that_10_is_a_composite_number_which_is_not_a_power_of_a_prime.%22"]the case of p-adic numbers, one base p^n digit is equivalent to n base p digits[/URL], also see the [URL="https://en.wikipedia.org/wiki/Automorphic_number"]automorphic numbers[/URL] to base b and base b^n, they are equivalent)

Since generalized pentagonal numbers * 25 + 1 are always generalized pentagonal numbers, and no generalized pentagonal numbers > 7 are primes, thus these families in base 25 contain no primes (only count numbers > base), since they can be factored as difference of squares divided by 24:

1{1} (={1}), 2{1}, 5{1}, 7{1}, C{1}, F{1}, M{1}, 11{1} (={1}), 1A{1}, 1F{1}, 21{1} (=2{1}), 27{1}, 2K{1}, 32{1}, 3H{1}, 40{1}, ...

(all numbers in these families are [URL="https://oeis.org/A001318"]generalized pentagonal numbers[/URL], and thus all such numbers except 2, 5, 7 are composite, but the primes 2, 5, 7 are not allowed since prime must be > base)

they are the families (generalized pentagonal number){1}, in fact, families 1F{1} and 3H{1} are already ruled out as only contain composite by covering set {2,13}

Thus, according to [URL="https://mersenneforum.org/showpost.php?p=593783&postcount=215"]post #215[/URL], if the numbers in these families (i.e. 1{1} (={1}), 3{1}, 6{1}, 11{1} (={1}), 16{1}, 23{1}, 31{1} (=3{1}), 40{1}, ... in base 9, 1{1} (={1}), 11{1} (={1}), 20{1}, 100{1}, 111{1} (={1}), 201{1} (=20{1}), 220{1}, ... in base 4 (converted to 1{5}, 1{5}, 8{5}, 10{5}, 15{5}, 8{5}, A1{5}, ... in base 16), 1{1} (={1}), 2{1}, 5{1}, 7{1}, C{1}, F{1}, M{1}, 11{1} (={1}), 1A{1}, 1F{1}, 21{1} (=2{1}), 27{1}, 2K{1}, 32{1}, 3H{1}, 40{1}, ... in base 25) are [URL="https://en.wikipedia.org/wiki/Semiprime"]semiprimes[/URL], then they will pass the [URL="https://en.wikipedia.org/wiki/Fermat_primality_test"]Fermat primality test[/URL] to many bases, i.e. there is an unexpectedly large probability that the numbers will pass a Fermât test although it is composite (for the base 4 families, the probability is 1/3, for the base 9 families, the probability is 1/2, for the base 25 families, the probability is 1/6, if the base is randomly chosen in the set of the integers coprime to the numbers). It is clear that the practice of using small test bases should be questioned. It would be desirable if a criterion could be established for choosing optimum test bases. See page 4 of [URL="https://www.ams.org/journals/mcom/1993-61-204/S0025-5718-1993-1185243-9/S0025-5718-1993-1185243-9.pdf"]this article[/URL] ([URL="https://oeis.org/A028491/a028491.pdf"]scanned copy of this article[/URL]). Besides, we should use [URL="https://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test"]Miller–Rabin primality test[/URL], [URL="https://en.wikipedia.org/wiki/Lucas_primality_test"]Lucas primality test[/URL], and [URL="https://en.wikipedia.org/wiki/Baillie%E2%80%93PSW_primality_test"]Baillie–PSW primality test[/URL] for such numbers, see [URL="https://en.wikipedia.org/wiki/Baillie%E2%80%93PSW_primality_test#The_danger_of_relying_only_on_Fermat_tests"]The danger of relying only on Fermat tests[/URL].

[CODE]
Family lengths of [URL="https://en.wikipedia.org/wiki/Semiprime"]semiprimes[/URL] in the family
[URL="http://factordb.com/index.php?query=%289%5En-1%29%2F8&use=n&n=2&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show"]{1} in base 9[/URL] 2, 3, 7, 13, ...
[URL="http://factordb.com/index.php?query=%2825*9%5En-1%29%2F8&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show"]3{1} in base 9[/URL] 3, 7, 49, ...
[URL="http://factordb.com/index.php?query=%2849*9%5En-1%29%2F8&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show"]6{1} in base 9[/URL] 2 (this is all, because the covering set {2,5})
[URL="http://factordb.com/index.php?query=%28121*9%5En-1%29%2F8&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show"]16{1} in base 9[/URL] none (because the covering set {2,5})
[URL="http://factordb.com/index.php?query=%284*16%5En-1%29%2F3&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show"]1{5} in base 16[/URL] 1, 2, 3, 6, 8, 9, 15, 30, 63, ...
[URL="http://factordb.com/index.php?query=%2825*16%5En-1%29%2F3&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show"]8{5} in base 16[/URL] 2, 7, 37, ...
[URL="http://factordb.com/index.php?query=%2849*16%5En-1%29%2F3&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show"]10{5} in base 16[/URL] 3, 4, ...
[URL="http://factordb.com/index.php?query=%28484*16%5En-1%29%2F3&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show"]A1{5} in base 16[/URL] 3, 12, 42, ...
[URL="http://factordb.com/index.php?query=%2825%5En-1%29%2F24&use=n&n=2&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show"]{1} in base 25[/URL] 2, ...
[URL="http://factordb.com/index.php?query=%2849*25%5En-1%29%2F24&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show"]2{1} in base 25[/URL] 2, 6, 12, ...
[URL="http://factordb.com/index.php?query=%28121*25%5En-1%29%2F24&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show"]5{1} in base 25[/URL] 3, 9, ...
[URL="http://factordb.com/index.php?query=%28169*25%5En-1%29%2F24&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show"]7{1} in base 25[/URL] (no semiprimes are known, nor can be proven that no semiprimes exist)
[/CODE]

For these families, it is conjectured that only finite many semiprimes exist (it is unlikely any other semiprime in these families exist besides the semiprimes listed above), like that for every k divisible by 3, it is conjectured that there are only finitely many [URL="https://en.wikipedia.org/wiki/Twin_prime"]twin primes[/URL] of the form k*2^n+-1 (equivalent to that for every square k divisible by 3, it is conjectured that there are only finitely many semiprimes of the form k*4^n-1, see [URL="https://mersenneforum.org/showthread.php?t=19209"]https://mersenneforum.org/showthread.php?t=19209[/URL]), references: [URL="https://www.primepuzzles.net/problems/prob_049.htm"]https://www.primepuzzles.net/problems/prob_049.htm[/URL] [URL="https://www.rieselprime.de/Related/RieselTwinSG.htm"]https://www.rieselprime.de/Related/RieselTwinSG.htm[/URL] [URL="http://www.noprimeleftbehind.net/gary/twins100K.htm"]http://www.noprimeleftbehind.net/gary/twins100K.htm[/URL] [URL="http://www.noprimeleftbehind.net/gary/twins1M.htm"]http://www.noprimeleftbehind.net/gary/twins1M.htm[/URL], but except for the k with covering set (e.g. k = 237 and k = 807), currently it cannot be proven that there are only finitely many such twin primes. (unlikely the [URL="https://en.wikipedia.org/wiki/Schinzel%27s_hypothesis_H"]Schinzel's hypothesis H[/URL], for polynomial functions a_0+a_1*n+a_2*n^2+a_3*n^3+... rather than exponential functions (a*b^n+c)/gcd(a+c,b-1), it is conjectured that there are infinitely many n such that all functions give prime numbers simultaneously)

These numbers are [URL="https://en.wikipedia.org/wiki/Fermat_pseudoprime"]Fermat pseudoprimes[/URL] to these bases 2<=b<=64:

[CODE]
number bases ([URL="https://en.wikipedia.org/wiki/Fermat_primality_test#Concept"]Fermat liars[/URL])
[URL="http://factordb.com/index.php?id=317733228541"](1^13) in base 9[/URL] 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 18, 19, 20, 21, 23, 24, 25, 26, 27, 28, 29, 30, 32, 35, 36, 37, 38, 39, 40, 41, 42, 45, 46, 48, 49, 50, 52, 54, 56, 57, 58, 59, 60, 61, 63, 64
[URL="http://factordb.com/index.php?id=1100000000922492667"]3(1^48) in base 9[/URL] none (but this n is [URL="https://en.wikipedia.org/wiki/Lucas_pseudoprime"]Lucas pseudoprimes[/URL] to many (P,Q) pair, since its form is (k-1)*(2*k-1) with k-1 and 2*k-1 both primes)
[URL="http://factordb.com/index.php?id=1000000000043569151"]1(5^63) in base 16[/URL] 2, 3, 4, 6, 8, 9, 11, 12, 13, 16, 17, 18, 19, 22, 24, 26, 27, 29, 31, 32, 33, 34, 36, 37, 38, 39, 41, 44, 47, 48, 51, 52, 54, 57, 58, 59, 61, 62, 64
[URL="http://factordb.com/index.php?id=1100000000348829387"]8(5^36) in base 16[/URL] 3, 5, 8, 9, 13, 15, 17, 22, 24, 25, 27, 28, 29, 39, 40, 41, 45, 46, 47, 51, 53, 62, 64
[URL="http://factordb.com/index.php?id=1100000000265576402"]A1(5^40) in base 16[/URL] none (but this n is [URL="https://en.wikipedia.org/wiki/Lucas_pseudoprime"]Lucas pseudoprimes[/URL] to many (P,Q) pair, since its form is (k-1)*(3*k-1) with k-1 and 3*k-1 both primes)
[URL="http://factordb.com/index.php?id=4867712656656901"]2(1^11) in base 25[/URL] none
[/CODE]

Thus, if we assume a number which has passed the Fermat primality tests to many bases is in fact prime, our data will be wrong for these bases b, sometimes other primality tests (such as the Miller-Rabin tests and the Lucas tests) are needed.

[URL="https://en.wikipedia.org/wiki/Polygonal_number"]Polygonal numbers[/URL] are often Fermat pseudoprimes if they are semiprimes, and generalized polygonal numbers (the formula of polygonal numbers, but with negative n) are often Lucas pseudoprimes if they are semiprimes. Fermat pseudoprimes tend to fall into the residue class +1 (mod m) for many small m, whereas Lucas pseudoprimes tend to fall into the residue class -1 (mod m) for many small m. Fermat pseudoprimes to many bases are numbers of the form [URL="https://oeis.org/A129521"](k+1)*(2*k+1)[/URL] or [URL="https://oeis.org/A259677"](k+1)*(3*k+1)[/URL], with both factors primes, whereas Lucas pseudoprimes to many bases are numbers of the form [URL="https://oeis.org/A156592"](k-1)*(2*k-1)[/URL] or (k-1)*(3*k-1), with both factors primes. Also see [URL="https://oeis.org/A002997"]Carmichael numbers[/URL] and [URL="https://oeis.org/A006972"]Lucas-Carmichael numbers[/URL]

For the examples of that if we assume a number which has passed the Fermat primality tests to many bases is in fact prime, our data will be wrong for some bases, for solving the original minimal prime problem (i.e. prime > base is not required), see [URL="https://github.com/curtisbright/mepn-data/commit/7565d197d7b438b437871bf71614a6f8914397f7"]https://github.com/curtisbright/mepn-data/commit/7565d197d7b438b437871bf71614a6f8914397f7[/URL], 444669 in base 17 (= 6035009 = 449*13441) and I0901 in base 26 (= 8231653 = 2029*4057) are Fermat pseudoprimes to many bases.

These two numbers are [URL="https://en.wikipedia.org/wiki/Fermat_pseudoprime"]Fermat pseudoprimes[/URL] to these bases 2<=b<=64:

[CODE]
number bases ([URL="https://en.wikipedia.org/wiki/Fermat_primality_test#Concept"]Fermat liars[/URL])
6035009 27, 50
8231653 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 16, 18, 21, 22, 23, 24, 25, 26, 27, 28, 32, 33, 36, 39, 41, 42, 43, 44, 46, 48, 49, 50, 52, 53, 54, 56, 59, 63, 64
[/CODE]

For other examples see [URL="https://mersenneforum.org/showthread.php?t=10476"]https://mersenneforum.org/showthread.php?t=10476[/URL] and [URL="https://www.ams.org/journals/mcom/1993-61-204/S0025-5718-1993-1185243-9/S0025-5718-1993-1185243-9.pdf"]https://www.ams.org/journals/mcom/1993-61-204/S0025-5718-1993-1185243-9/S0025-5718-1993-1185243-9.pdf[/URL], if we assume a number which has passed the Fermat primality tests to many bases is in fact prime, our list for base 16 minimal primes (start with b+1) would include the composite 1(5^63) (its value is (4*16^63-1)/3), and our list for base 9 minimal primes (start with b+1) would include the composite (1^13) (its value is (9^13-1)/8) (and hence would exclude the minimal prime (start with b+1) 56(1^36), since this prime has (1^13) as subsequence), although their corresponding families (1{5} in base 16, {1} in base 9, respectively) can be ruled out as only contain composite numbers (only count the numbers > base), and our data will be wrong for these bases (thus, for this minimal prime (start with b+1) problem in base b, especially for square base b, we should not assume a number which has passed the Fermat primality tests to many bases is in fact prime, we need to combine with Lucas tests, to do Baillie–PSW tests)

If k+1 and 2*k+1 are both primes, then (k+1)*(2*k+1) is Fermat pseudoprime to base b (assume b is coprime to (k+1)*(2*k+1)) if and only if b is [URL="https://en.wikipedia.org/wiki/Quadratic_residue"]quadratic residue[/URL] mod 2*k+1, i.e. x^2 == b mod 2*k+1 has solutions, and if k+1 and 3*k+1 are both primes, then (k+1)*(3*k+1) is Fermat pseudoprime to base b (assume b is coprime to (k+1)*(3*k+1)) if and only if b is [URL="https://en.wikipedia.org/wiki/Cubic_residue"]cubic residue[/URL] mod 3*k+1, i.e. x^3 == b mod 3*k+1 has solutions

[URL="https://oeis.org/A247074"]A247074[/URL]((k+1)*(2*k+1)) = 2 if k+1 and 2*k+1 are both primes, and [URL="https://oeis.org/A247074"]A247074[/URL]((k+1)*(3*k+1)) = 3 if k+1 and 3*k+1 are both primes, note that [URL="https://oeis.org/A247074"]A247074[/URL](n) = 1 if and only if n is (1, prime, or [URL="https://oeis.org/A002997"]Carmichael number[/URL]), and if 6*k+1, 12*k+1, 18*k+1 are all primes, then (6*k+1)*(12*k+1)*(18*k+1) is Carmichael number, see [URL="https://oeis.org/A087788"]https://oeis.org/A087788[/URL] and [URL="https://oeis.org/A033502"]https://oeis.org/A033502[/URL]

(also, do not use base b (Fermat or Miller-Rabin) test for the GFN's (numbers of the form (b^n+1)/gcd(b-1,2), i.e. numbers in family 1{0}1 for even b or family {#}$ for odd b) and GRU's (numbers of the form (b^n-1)/(b-1), i.e. numbers in family {1}) in base b, since all these numbers (with n is power of 2 and n is prime, respectively, for other n such numbers cannot be prime, because of algebraic factorization) are strong-probable-primes to base b, see post [URL="https://mersenneforum.org/showpost.php?p=568817&postcount=116"]#116[/URL])

References: [URL="https://oeis.org/A063994"]https://oeis.org/A063994[/URL] [URL="https://oeis.org/A064234"]https://oeis.org/A064234[/URL] [URL="https://oeis.org/A247074"]https://oeis.org/A247074[/URL] [URL="https://oeis.org/A181780"]https://oeis.org/A181780[/URL] [URL="https://oeis.org/A211455"]https://oeis.org/A211455[/URL] [URL="http://www.numericana.com/answer/pseudo.htm"]http://www.numericana.com/answer/pseudo.htm[/URL] [URL="https://de.wikibooks.org/wiki/Pseudoprimzahlen:_Tabelle_Pseudoprimzahlen_%2815_-_4999%29"]https://de.wikibooks.org/wiki/Pseudoprimzahlen:_Tabelle_Pseudoprimzahlen_%2815_-_4999%29[/URL] [URL="https://de.wikibooks.org/wiki/Pseudoprimzahlen:_Tabelle_Fermatsche_Pseudoprimzahlen"]https://de.wikibooks.org/wiki/Pseudoprimzahlen:_Tabelle_Fermatsche_Pseudoprimzahlen[/URL]

[URL="https://oeis.org/A063994"]A063994[/URL](n) is the number of bases 0<=b<=n-1 such that n is (Fermat) pseudoprime (or prime) base b, and [URL="https://oeis.org/A063994"]A063994[/URL](n) must be a divisor of [URL="https://oeis.org/A000010"]eulerphi[/URL](n) (the number of numbers 0<=b<=n-1 which are coprime to n), and [URL="https://oeis.org/A247074"]A247074[/URL](n) is their ratio (i.e. [URL="https://oeis.org/A000010"]eulerphi[/URL](n)/[URL="https://oeis.org/A063994"]A063994[/URL](n)), if the base b is [URL="https://en.wikipedia.org/wiki/Randomness"]random[/URL] chosen in the numbers coprime to n, then the [URL="https://en.wikipedia.org/wiki/Probability"]probability[/URL] that n is (Fermat) pseudoprime base b (i.e. return [URL="https://en.wikipedia.org/wiki/False_positive"]false positive[/URL], or return "probable prime" for the composite n) is 1/[URL="https://oeis.org/A247074"]A247074[/URL](n)

sweety439 2021-11-25 11:48

Now I consider to add these families to [URL="https://docs.google.com/spreadsheets/d/e/2PACX-1vTKkSNKGVQkUINlp1B3cXe90FWPwiegdA07EE7-U7sqXntKAEQrynoI1sbFvvKriieda3LfkqRwmKME/pubhtml"]the list[/URL], although they do not always provide minimal primes (start with b+1):

* 10{z}
* {z}yz
* 11{0}1
* 1{0}11
* {1}2
* {1}3
* {1}4
* {1}0z
* 2{1}
* 3{1}
* 4{1}
* {z0}z1

sweety439 2021-11-26 14:56

[QUOTE=sweety439;593116]
* All sub-families of all unsolved families are either can be ruled out as only contain composites (only consider numbers > base) or also unsolved families[/QUOTE]

e.g. (list all sub-families of given unsolved family, in the same base)

base 11 unsolved family 5{7}:

{7}: divisible by 7

base 13 unsolved family A{3}A:

A{3}: covering set {2,7}
{3}A: covering set {2,7}
{3}: divisible by 3

base 16 unsolved family {3}AF:

{3}A: divisible by 2
{3}F: divisible by 3
{3}: divisible by 3

base 16 unsolved family {4}DD:

{4}D: covering set {3,7,13}
{4}: divisible by 2

base 17 unsolved family F1{9}:

F{9}: divisible by 3
1{9}: odd length: difference of squares divided by 16; even length: divisible by 2
{9}: divisible by 9

base 19 unsolved family EE1{6}:

EE{6}: divisible by 2
E1{6}: divisible by 3
E{6}: divisible by 2
1{6}: odd length: difference of squares divided by 3; even length: divisible by 5
{6}: divisible by 6

base 25 unsolved family CM{1}:

C{1}: difference of squares divided by 24
M{1}: difference of squares divided by 24
{1}: difference of squares divided by 24

base 25 unsolved family E{1}E:

E{1}: covering set {2,13}
{1}E: covering set {2,13}
{1}: difference of squares divided by 24

base 25 unsolved family F{1}F1:

F{1}F: divisible by 5
F{1}1 = F{1}: difference of squares divided by 24
{1}F1: covering set {2,13}
{1}F: divisible by 5
{1}1 = {1}: difference of squares divided by 24

base 25 family LO{L}8:

LO{L}: divisible by 3
L{L}8 = {L}8: covering set {2,13}
O{L}8: (another unsolved family)
L{L} = {L}: divisible by 21
O{L}: divisible by 3

base 27 family 999{G}:

99{G}: divisible by 2
9{G}: difference of cubes divided by 13
{G}: divisible by 16

base 33 family FFF{W}:

FF{W}: divisible by 2
F{W}: odd length: difference of squares; even length: divisible by 17
{W}: divisible by 32

base 47 family 8{0}FF1:

8{0}FF: divisible by 8
8{0}F1: divisible by 2
8{0}F: divisible by 23
8{0}1: covering set {3, 5, 13}
(other sub-families have either [URL="https://en.wikipedia.org/wiki/Leading_zero"]leading zeros[/URL] or [URL="https://en.wikipedia.org/wiki/Trailing_zero"]trailing zeros[/URL] (or both))

sweety439 2021-11-26 17:44

2 Attachment(s)
[QUOTE=sweety439;593853]Now I consider to add these families to [URL="https://docs.google.com/spreadsheets/d/e/2PACX-1vTKkSNKGVQkUINlp1B3cXe90FWPwiegdA07EE7-U7sqXntKAEQrynoI1sbFvvKriieda3LfkqRwmKME/pubhtml"]the list[/URL], although they do not always provide minimal primes (start with b+1):

* 10{z}
* {z}yz
* 11{0}1
* 1{0}11
* {1}2
* {1}3
* {1}4
* {1}0z
* 2{1}
* 3{1}
* 4{1}
* {z0}z1[/QUOTE]

Upload text files, searched up to length 5000

For 11{0}1 (the dual of 1{0}11), see [URL="https://www.rieselprime.de/ziki/Williams_prime_PP_least"]https://www.rieselprime.de/ziki/Williams_prime_PP_least[/URL]
For 10{z} (the dual of {z}yz), see [URL="https://www.rieselprime.de/ziki/Williams_prime_PM_least"]https://www.rieselprime.de/ziki/Williams_prime_PM_least[/URL]

sweety439 2021-11-26 22:14

For families X{Y}Z[SUB]b[/SUB] = (a*b^n+c)/gcd(a+c,b-1) (a>=1, b>=2, c != 0, gcd(a,c) = 1, gcd(b,c) = 1), which can be ruled out as only containing composite, which are by covering congruence, and which are by algebra factors, and which are by combine of them?

Hint:

* If by covering congruence, then all X{Y}Z[SUB]b[/SUB] numbers must have small prime factor (say less than 10^6, if b is small), especially, if there is (positive or negative) integer n such that all prime factors of [URL="https://en.wikipedia.org/wiki/Numerator"]numerator[/URL]([URL="https://en.wikipedia.org/wiki/Absolute_value"]abs[/URL]((a*b^n+c)/gcd(a+c,b-1))) are prime factors of b (including the case that [URL="https://en.wikipedia.org/wiki/Numerator"]numerator[/URL]([URL="https://en.wikipedia.org/wiki/Absolute_value"]abs[/URL]((a*b^n+c)/gcd(a+c,b-1))) = 1), then (a*b^n+c)/gcd(a+c,b-1) cannot have covering congruence.

* If by (full or partial) algebra factors, then there must be integers n>=1 and r>1 such that a*b^n and -c are both r-th powers, or there must be integer n>=1 such that a*b^n*c is of the form 4*m^4 with integer m

* If by full algebra factors, then there must be integer r>1 such that a, b, -c are all r-th powers, or a*c is of the form 4*m^4 with integer m and b is 4-th power

Thus ....

* Base 25 family [URL="http://factordb.com/index.php?query=%28289*25%5En-1%29%2F24&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show"]C{1}[/URL] must be only algebra factors, not by covering congruence, since

[CODE]
length factors
6 19 · 101 · 233 · 263
12 69173177 · 415039063
18 71 · 91337821853 · 1080830891927
36 277 · 383 · 4253 · 9157 · 305497 · 2649281 · 5391097 · 141581477 · 39961446977
72 337 · 499 · 1051 · 4598699002567 · 5169269868350933 · 132787369068406184845865339939 · 38714450336182811476979541935153363
84 103 · 139 · 197 · 9277 · 500921 · 13288909 · (94-digit composite with no small factor)
[/CODE]

and there is not appear any covering set, thus C{1} can only by algebra factors, indeed, C(1^n) = (289*25^n-1)/24 = (17*5^n-1) * (17*5^n+1) / 24

* Base 25 family [URL="http://factordb.com/index.php?query=%28337*25%5En-1%29%2F24&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show"]E{1}[/URL] must be only covering congruence, not by algebra factors, since E(1^n) = (337*25^n-1)/24, and there is no n such that 337*25^n is perfect power (since the exponent of the prime 337 in prime factorization of 337*25^n is always 1), thus there is no n>=1 and r>1 such that 337*25^n and -1 are both r-th powers

thus, E{1} can only by covering congruence, indeed, E(1^n) is divisible by 2 if n is even, and divisible by 13 if n is odd

For the base 9 families ....

* [URL="http://factordb.com/index.php?query=%289%5En-1%29%2F8&use=n&n=2&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show"]{1}[/URL] must be only algebra factors, not by covering congruence, since (9^n-1)/8 is [URL="https://en.wikipedia.org/wiki/Divisibility_sequence"]divisibility sequence[/URL], by [URL="https://en.wikipedia.org/wiki/Zsigmondy%27s_theorem"]Zsigmondy's theorem[/URL], (9^n-1)/8 has primitive prime factor for all n>=1, and cannot have covering congruence

thus {1} can only by algebra factors, indeed, (1^n) = (9^n-1)/8 = (3^n-1) * (3^n+1) / 8

* [URL="http://factordb.com/index.php?query=%2825*9%5En-1%29%2F8&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show"]3{1}[/URL] must be only algebra factors, not by covering congruence, since all prime factors of (25*9^n-1)/8 are prime factors of 9 when n=0, thus 3{1} cannot have covering congruence

thus 3{1} can only by algebra factors, indeed, 3(1^n) = (25*9^n-1)/8 = (5*3^n-1) * (5*3^n+1) / 8

* [URL="http://factordb.com/index.php?query=%2841*9%5En-1%29%2F8&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show"]5{1}[/URL] must be only covering congruence, not by algebra factors, since 5(1^n) = (41*9^n-1)/8, and there is no n such that 41*9^n is perfect power (since the exponent of the prime 41 in prime factorization of 41*9^n is always 1), thus there is no n>=1 and r>1 such that 41*9^n and -1 are both r-th powers

thus, 5{1} can only by covering congruence, indeed, 5(1^n) is divisible by 5 if n is even, and divisible by 2 if n is odd

* However, [URL="http://factordb.com/index.php?query=%2849*9%5En-1%29%2F8&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show"]6{1}[/URL] can be considered as by both covering congruence and algebra factors (in this situation, we consider as by covering congruence rather than by algebra factors, our order is: covering congruence --> algebra factors --> combine of covering congruenct and algebra factors), both of them can prove that 6{1} contain no primes, since:

** 6(1^n) is divisible by 2 if n is even, and divisible by 5 if n is odd
** 6(1^n) = (49*9^n-1)/8 = (7*3^n-1) * (7*3^n+1) / 8

See [URL="https://github.com/curtisbright/mepn-data/commit/c9c68f40b1f8c1bbbbaa0ce8e6e2f14f6b8aab19"]https://github.com/curtisbright/mepn-data/commit/c9c68f40b1f8c1bbbbaa0ce8e6e2f14f6b8aab19[/URL] for the researching of the original minimal prime problem (i.e. prime > base is not required), this difference includes families 1{6} in base 19, 89{6} in base 19, 1{8} in base 25, L{8} in base 25, 9{G} in base 27, they are ruled out as only contain composite numbers not by covering congruence, and instead by all or partial algebraic factorization (1{6} in base 19 and 89{6} in base 19 are by partial algebraic factorization, while other families are by full algebraic factorization), thus, families which are ruled out as only contain composite numbers by all or partial algebraic factorization are more difficult to remove then families which are ruled out as only contain composite numbers by covering congruence, by computer programs.

sweety439 2021-11-28 17:20

1 Attachment(s)
These are all minimal primes (start with b+1) in base b=25 up to 2^32

Base 25 is a very hard base (however, of course, bases > 25 which are coprime to 6 are harder than it), we can imagine an alien force, vastly more powerful than us, landing on Earth and demanding the set of minimal primes (start with b+1) in base b=17 (or 19, 21, 22, 23, 28, 30, 36) (including primality proving of all primes in this set) or they will destroy our planet. In that case, I claim, we should marshal all our computers and all our mathematicians and attempt to find the set and to prove the primality of all numbers in this set. But suppose, instead, that they ask for the set of minimal primes (start with b+1) in base b=25 (or 26, 27, 29, 31, 32, 33, 34, 35). In that case, I believe, we should attempt to destroy the aliens.

sweety439 2021-11-28 17:23

1 Attachment(s)
[QUOTE=sweety439;593958]Upload text files, searched up to length 5000

For 11{0}1 (the dual of 1{0}11), see [URL="https://www.rieselprime.de/ziki/Williams_prime_PP_least"]https://www.rieselprime.de/ziki/Williams_prime_PP_least[/URL]
For 10{z} (the dual of {z}yz), see [URL="https://www.rieselprime.de/ziki/Williams_prime_PM_least"]https://www.rieselprime.de/ziki/Williams_prime_PM_least[/URL][/QUOTE]

This is the text file for {z0}z1 (i.e. generalized Wagstaff primes, see [URL="https://oeis.org/A084742"]A084742[/URL], but exclude p=3, and use the length of the primes (i.e. use p-1 instead of p)), also searched to length 5000

very important note: they are minimal primes (start with b'+1) in base b'=b^2 instead of base b'=b

sweety439 2021-11-30 19:59

3 Attachment(s)
more datas

sweety439 2021-11-30 20:41

[QUOTE=sweety439;593116]In base 10, the set such strings are not simply to write, however, if "primes > base" is not needed, then such strings are any strings n such that [URL="https://oeis.org/A039995"]A039995[/URL](n) = 0 (not [URL="https://oeis.org/A062115"]A062115[/URL], since [URL="https://oeis.org/A062115"]A062115[/URL] is for [URL="https://en.wikipedia.org/wiki/Substring"]substring[/URL] instead of [URL="https://en.wikipedia.org/wiki/Subsequence"]subsequence[/URL], i.e. [URL="https://oeis.org/A062115"]A062115[/URL] is the numbers n such that [URL="https://oeis.org/A039997"]A039997[/URL](n) = 0 instead of the numbers n such that [URL="https://oeis.org/A039995"]A039995[/URL](n) = 0) with any number (including 0) of leading zeros.

Such strings are called [B]primefree strings[/B] in this post.[/QUOTE]

In base 10, a sting is primefree string if and only if:

The string is of the form {0}S{0,2,4,5,6,8} and S is of one of these forms:

* empty string
* single-digit number
* "gcd of its digits" <> 1 (note: gcd(0,n) = n for all n, including n=0)
* X{0}Y with X+Y divisible by 3 (this includes: 2{0}1, 2{0}7, 5{0}1, 5{0}7, 8{0}1, 8{0}7)
* 28{0}7
* 4{6}9
* 221
* 2021
* 2201
* 22001
* 220001
* 2200001
* (5^n)1 with n<11
* 581
* 5(0^n)27 with n<28
* 5207
* 52007
* 520007
* 649
* 6649
* 66649
* 6049
* 60049
* 600049
* 6000049
* 66049
* 660049
* 6600049
* 666049
* 6660049
* 8(5^n)1 with n<11
* 8051
* 80551
* 805551
* 8055551
* 91
* 901
* 921
* 951
* 981
* 9021
* 9051
* 9081
* 9201
* 9501
* 9581
* 9801
* 90581
* 949
* 9469
* 94669

reference: [URL="https://math.stackexchange.com/questions/58292/how-to-check-if-an-integer-has-a-prime-number-in-it"]https://math.stackexchange.com/questions/58292/how-to-check-if-an-integer-has-a-prime-number-in-it[/URL]

sweety439 2021-11-30 20:52

1 Attachment(s)
Update data for minimal primes, see [URL="https://sites.google.com/view/data-of-minimal-primes"]https://sites.google.com/view/data-of-minimal-primes[/URL]

sweety439 2021-12-06 10:14

[QUOTE=sweety439;584481]Related search for minimal primes (generalized form: (a*b^n+c)/d) in [URL="http://www.primenumbers.net/prptop/prptop.php"]PRP top[/URL]:

[URL="http://www.primenumbers.net/prptop/searchform.php?form=b%5En%2Bc&action=Search"]b^n+c[/URL]

[URL="http://www.primenumbers.net/prptop/searchform.php?form=b%5En-c&action=Search"]b^n-c[/URL]

[URL="http://www.primenumbers.net/prptop/searchform.php?form=a*b%5En%2Bc&action=Search"]a*b^n+c[/URL]

[URL="http://www.primenumbers.net/prptop/searchform.php?form=a*b%5En-c&action=Search"]a*b^n-c[/URL]

[URL="http://www.primenumbers.net/prptop/searchform.php?form=%28b%5En%2Bc%29%2Fd&action=Search"](b^n+c)/d[/URL]

[URL="http://www.primenumbers.net/prptop/searchform.php?form=%28b%5En-c%29%2Fd&action=Search"](b^n-c)/d[/URL]

[URL="http://www.primenumbers.net/prptop/searchform.php?form=%28a*b%5En%2Bc%29%2Fd&action=Search"](a*b^n+c)/d[/URL]

[URL="http://www.primenumbers.net/prptop/searchform.php?form=%28a*b%5En-c%29%2Fd&action=Search"](a*b^n-c)/d[/URL]

Also for the special case c = +-1 and d = 1, they are [I]proven[/I] primes, the search page in [URL="https://primes.utm.edu/primes/"]top 5000 primes[/URL]:

[URL="https://primes.utm.edu/primes/search_proth.php"]https://primes.utm.edu/primes/search_proth.php[/URL][/QUOTE]

Forms in [URL="https://docs.google.com/spreadsheets/d/e/2PACX-1vTKkSNKGVQkUINlp1B3cXe90FWPwiegdA07EE7-U7sqXntKAEQrynoI1sbFvvKriieda3LfkqRwmKME/pubhtml"]https://docs.google.com/spreadsheets/d/e/2PACX-1vTKkSNKGVQkUINlp1B3cXe90FWPwiegdA07EE7-U7sqXntKAEQrynoI1sbFvvKriieda3LfkqRwmKME/pubhtml[/URL]:

1{0}2 b^n+2: [URL="http://www.primenumbers.net/prptop/searchform.php?form=b%5En%2B2&action=Search"]http://www.primenumbers.net/prptop/searchform.php?form=b%5En%2B2&action=Search[/URL]

1{0}3 b^n+3: [URL="http://www.primenumbers.net/prptop/searchform.php?form=b%5En%2B3&action=Search"]http://www.primenumbers.net/prptop/searchform.php?form=b%5En%2B3&action=Search[/URL]

1{0}4 b^n+4: [URL="http://www.primenumbers.net/prptop/searchform.php?form=b%5En%2B4&action=Search"]http://www.primenumbers.net/prptop/searchform.php?form=b%5En%2B4&action=Search[/URL]

{z}w b^n-4: [URL="http://www.primenumbers.net/prptop/searchform.php?form=b%5En-4&action=Search"]http://www.primenumbers.net/prptop/searchform.php?form=b%5En-4&action=Search[/URL]

{z}x b^n-3: [URL="http://www.primenumbers.net/prptop/searchform.php?form=b%5En-3&action=Search"]http://www.primenumbers.net/prptop/searchform.php?form=b%5En-3&action=Search[/URL]

{z}y b^n-2: [URL="http://www.primenumbers.net/prptop/searchform.php?form=b%5En-2&action=Search"]http://www.primenumbers.net/prptop/searchform.php?form=b%5En-2&action=Search[/URL]

MattcAnderson 2021-12-06 13:17

another message _ message transfer to small corner of net
 
Apprciation is felt for your effort @sweety439 and all

sweety439 2021-12-07 14:15

[QUOTE=sweety439;569293]The families which are excepted as contain no primes, but undecidable at this point in time, for these 369 bases are: (totally 377 families)

* 4:{0}:1, 16:{0}:1 for b = 32
* 12:{62}:63 for b = 125 (Note: {62}:63 for b = 125 can be ruled out as contain no primes > base, by sum-of-cubes factorization, thus the smallest prime of the form 12:{62}:63 for b = 125 (if exists) must be minimal prime (start with b+1) in base b = 125)
* 16:{0}:1 for b = 128
* 36:{0}:1 for b = 216
* 24:{171}:172 for b = 343 (Note: {171}:172 for b = 343 can be ruled out as contain no primes > base, by sum-of-cubes factorization, thus the smallest prime of the form 24:{171}:172 for b = 343 (if exists) must be minimal prime (start with b+1) in base b = 343)
* 2:{0}:1, 4:{0}:1, 16:{0}:1, 32:{0}:1, 256:{0}:1 for b = 512
* 10:{0}:1, 100:{0}:1 for b = 1000
* 4:{0}:1, 16:{0}:1, 256:{0}:1 for b = 1024
* 1:{0}:1 for other even bases b
* {((b-1)/2)}:((b+1)/2) for other odd bases b[/QUOTE]

For GFN families in bases b <= 1024 which are odd powers (i.e. in [URL="https://oeis.org/A070265"]https://oeis.org/A070265[/URL]) (other bases already have information in [URL="http://jeppesn.dk/generalized-fermat.html"]http://jeppesn.dk/generalized-fermat.html[/URL] and [URL="http://www.noprimeleftbehind.net/crus/GFN-primes.htm"]http://www.noprimeleftbehind.net/crus/GFN-primes.htm[/URL] and [URL="http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt"]http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt[/URL], which are 1:{0}:1 for even bases and {((b-1)/2)}:((b+1)/2) for odd bases):

b=8:

1:{0}:1, ruled out as only contain composite numbers
2:{0}:1, prime at length 2
4:{0}:1, prime at length 3

b=27:

{13}:14, ruled out as only contain composite numbers
1:{13}:14, prime at length 2, also 1:{13} has prime at length 3
4:{13}:14, prime at length 11, also 4:{13} has prime at length 24

b=32:

1:{0}:1, ruled out as only contain composite numbers
2:{0}:1, prime at length 4
[B]4:{0}:1, unsolved family[/B]
8:{0}:1, prime at length 2
[B]16:{0}:1, unsolved family[/B]

b=64:

1:{0}:1, ruled out as only contain composite numbers
4:{0}:1, prime at length 2
16:{0}:1, prime at length 3

b=125:

{62}:63, ruled out as only contain composite numbers
2:{62}:63, prime at length 2
[B]12:{62}:63, unsolved family[/B]

b=128:

1:{0}:1, ruled out as only contain composite numbers
2:{0}:1, prime at length 2
4:{0}:1, prime at length 3
8:{0}:1, ruled out as only contain composite numbers
[B]16:{0}:1, unsolved family[/B]
32:{0}:1, ruled out as only contain composite numbers
64:{0}:1, ruled out as only contain composite numbers

b=216:

1:{0}:1, ruled out as only contain composite numbers
6:{0}:1, prime at length 2
[B]36:{0}:1, unsolved family[/B]

b=243:

{121}:122, ruled out as only contain composite numbers
1:{121}:122, prime at length 4, also 1:{121} has prime at length 15
4:{121}:122, prime at length 7, also 4:{121} has prime at length 2
13:{121}:122, although no known prime or PRP in this family, but 13:{121} has prime at length 3, thus 13:{121}:122 is still not unsolved family since the smallest prime (if exists) in this family will not be minimal prime (start with b+1)
40:{121}:122, prime at length 13, however 40:{121} has no known prime or PRP, and [B]40:{121} is unsolved family[/B] (however, 40:{121} is GRU family instead of GFN family, for reference of this family, see [URL="https://oeis.org/A028491"]https://oeis.org/A028491[/URL], there are no single known number in [URL="https://oeis.org/A028491"]https://oeis.org/A028491[/URL] which is == 4 mod 5)

b=343:

{171}:172, ruled out as only contain composite numbers
3:{171}:172, prime at length 2
[B]24:{171}:172, unsolved family[/B]

b=512:

1:{0}:1, ruled out as only contain composite numbers
[B]2:{0}:1, unsolved family[/B]
[B]4:{0}:1, unsolved family[/B]
8:{0}:1, ruled out as only contain composite numbers
[B]16:{0}:1, unsolved family[/B]
[B]32:{0}:1, unsolved family[/B]
64:{0}:1, ruled out as only contain composite numbers
128:{0}:1, prime at length 2
[B]256:{0}:1, unsolved family[/B]

b=729:

{364}:365, ruled out as only contain composite numbers
4:{364}:365, prime at length 6
40:{364}:365, prime at length 3

b=1000:

1:{0}:1, ruled out as only contain composite numbers
[B]10:{0}:1, unsolved family[/B]
[B]100:{0}:1, unsolved family[/B]

b=1024:

1:{0}:1, ruled out as only contain composite numbers
[B]4:{0}:1, unsolved family[/B]
[B]16:{0}:1, unsolved family[/B]
64:{0}:1, prime at length 2
[B]256:{0}:1, unsolved family[/B]

sweety439 2021-12-08 08:21

[QUOTE=sweety439;594226]In base 10, a sting is primefree string if and only if:

The string is of the form {0}S{0,2,4,5,6,8} and S is of one of these forms:

* empty string
* single-digit number
* "gcd of its digits" <> 1 (note: gcd(0,n) = n for all n, including n=0)
* X{0}Y with X+Y divisible by 3 (this includes: 2{0}1, 2{0}7, 5{0}1, 5{0}7, 8{0}1, 8{0}7)
* 28{0}7
* 4{6}9
* 221
* 2021
* 2201
* 22001
* 220001
* 2200001
* (5^n)1 with n<11
* 5(0^n)27 with n<28
* 5207
* 52007
* 520007
* 649
* 6649
* 66649
* 6049
* 60049
* 600049
* 6000049
* 66049
* 660049
* 6600049
* 666049
* 6660049
* 8(5^n)1 with n<11
* 8051
* 80551
* 805551
* 8055551
* 91
* 901
* 921
* 951
* 981
* 9021
* 9051
* 9081
* 9201
* 9501
* 9581
* 9801
* 90581
* 949
* 9469
* 94669[/QUOTE]

Like [URL="https://en.wikipedia.org/wiki/Maximal_and_minimal_elements"]minimal element[/URL] of prime numbers > b in base b, we can find the [URL="https://en.wikipedia.org/wiki/Maximal_and_minimal_elements"]maximal element[/URL] of primefree strings:

{0}S{0,2,4,5,6,8} with these sets S:

* {0,3,6,9}
* {0,7}
* 2{0}1
* 5{0}1
* 5{0}7
* 8{0}1
* 28{0}7
* 4{6}9
* 2021
* 2200001
* (5^10)1
* 581
* 5(0^27)27
* 520007
* 6000049
* 6600049
* 6660049
* 8(5^10)1
* 8055551
* 9021
* 9201
* 9801
* 90581
* 94669

(note: {0,2,4,6,8} and {0,5} are already included in the digits after S, i.e. {0,2,4,5,6,8})
(note: 2{0}7 and 8{0}7 are already sub-families of 28{0}7)
(note: 66649 is already subsequence of 6660049)
(note: 581, 9051, 9081, 9501, 9581 are already subsequences of 90581 or/and 95081)

sweety439 2021-12-08 08:29

and we have these "maximal primefree strings"

b=2:

* {0}1{0}

b=3:

* {0}1{0}1{0}
* {0,2}

b=4:

* {0}2{0}1{0,2}
* {0}{0,3}{0,2}

b=10:

* {0}{0,3,6,9}{0,2,4,5,6,8}
* {0}{0,7}{0,2,4,5,6,8}
* {0}2{0}1{0,2,4,5,6,8}
* {0}5{0}1{0,2,4,5,6,8}
* {0}5{0}7{0,2,4,5,6,8}
* {0}8{0}1{0,2,4,5,6,8}
* {0}28{0}7{0,2,4,5,6,8}
* {0}4{6}9{0,2,4,5,6,8}
* {0}2021{0,2,4,5,6,8}
* {0}2200001{0,2,4,5,6,8}
* {0}55555555551{0,2,4,5,6,8}
* {0}500000000000000000000000000027{0,2,4,5,6,8}
* {0}520007{0,2,4,5,6,8}
* {0}6000049{0,2,4,5,6,8}
* {0}6600049{0,2,4,5,6,8}
* {0}6660049{0,2,4,5,6,8}
* {0}855555555551{0,2,4,5,6,8}
* {0}8055551{0,2,4,5,6,8}
* {0}9021{0,2,4,5,6,8}
* {0}9201{0,2,4,5,6,8}
* {0}9801{0,2,4,5,6,8}
* {0}90581{0,2,4,5,6,8}
* {0}95081{0,2,4,5,6,8}
* {0}94669{0,2,4,5,6,8}

sweety439 2021-12-10 16:59

These factor pattern can show that such families contain no primes > base:

Reference: the [URL="https://en.wikipedia.org/wiki/Divisibility_rule"]divisibility rule[/URL] for base b: (also see [URL="http://www.urticator.net/essay/5/567.html"]http://www.urticator.net/essay/5/567.html[/URL] and [URL="http://www.numericana.com/answer/numeration.htm#divisibility"]http://www.numericana.com/answer/numeration.htm#divisibility[/URL])

* For prime p dividing b, the number is divisible by p if and only if the last digit of this number is divisible by p.
* For prime p dividing b-1, the number is divisible by p if and only if the sum of the digits of this number is divisible by p.
* For prime p dividing b+1, the number is divisible by p if and only if the [URL="https://en.wikipedia.org/wiki/Alternating_sum"]alternating sum[/URL] of the digits of this number is divisible by p. (this can also show that all [URL="https://en.wikipedia.org/wiki/Palindromic_prime"]palindromic primes[/URL] in any base b have an odd number of digits, the only possible exception is "11" in base b)

(in these examples, only list the numbers > base)

Example 1: base 10, family 4{6}9 (formula: (14*10^(n+1)+7)/3) ([URL="http://factordb.com/index.php?query=%2814*10%5E%28n%2B1%29%2B7%29%2F3&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show"]factordb[/URL])

[CODE]
49 = 7 * 7
469 = 7 * 67
4669 = 7 * 667
46669 = 7 * 6667
466669 = 7 * 66667
4666669 = 7 * 666667
[/CODE]

Example 2: base 10, family 28{0}7 (formula: 28*10^(n+1)+7) ([URL="http://factordb.com/index.php?query=28*10%5E%28n%2B1%29%2B7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show"]factordb[/URL])

[CODE]
287 = 7 * 41
2807 = 7 * 401
28007 = 7 * 4001
280007 = 7 * 40001
2800007 = 7 * 400001
28000007 = 7 * 4000001
[/CODE]

Example 3: base 9, family {1} (formula: (9^n-1)/8) ([URL="http://factordb.com/index.php?query=%289%5En-1%29%2F8&use=n&n=2&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show"]factordb[/URL])

[CODE]
11 = 2 * 5
111 = 7 * 14
1111 = 22 * 45
11111 = 67 * 144
111111 = 222 * 445
1111111 = 667 * 1444
11111111 = 2222 * 4445
111111111 = 6667 * 14444
1111111111 = 22222 * 44445
11111111111 = 66667 * 144444
111111111111 = 222222 * 444445
1111111111111 = 666667 * 1444444
[/CODE]

Example 4: base 9, family 3{8} (formula: 4*9^n-1) ([URL="http://factordb.com/index.php?query=4*9%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show"]factordb[/URL])

[CODE]
38 = 5 * 7
388 = 18 * 21
3888 = 58 * 61
38888 = 188 * 201
388888 = 588 * 601
3888888 = 1888 * 2001
38888888 = 5888 * 6001
388888888 = 18888 * 20001
3888888888 = 58888 * 60001
38888888888 = 188888 * 200001
388888888888 = 588888 * 600001
3888888888888 = 1888888 * 2000001
[/CODE]

Example 5: base 8, family 1{0}1 (formula: 8^(n+1)+1) ([URL="http://factordb.com/index.php?query=8%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show"]factordb[/URL])

[CODE]
11 = 3 * 3
101 = 5 * 15
1001 = 11 * 71
10001 = 21 * 361
100001 = 41 * 1741
1000001 = 101 * 7701
10000001 = 201 * 37601
100000001 = 401 * 177401
1000000001 = 1001 * 777001
10000000001 = 2001 * 3776001
100000000001 = 4001 * 17774001
1000000000001 = 10001 * 77770001
[/CODE]

Example 6: base 9, family {8}5 (formula: 9^(n+1)-4) ([URL="http://factordb.com/index.php?query=9%5E%28n%2B1%29-4&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show"]factordb[/URL])

[CODE]
85 = 7 * 12
885 = 87 * 102
8885 = 887 * 1002
88885 = 8887 * 10002
888885 = 88887 * 100002
8888885 = 888887 * 1000002
88888885 = 8888887 * 10000002
888888885 = 88888887 * 100000002
8888888885 = 888888887 * 1000000002
88888888885 = 8888888887 * 10000000002
888888888885 = 88888888887 * 100000000002
8888888888885 = 888888888887 * 1000000000002
[/CODE]

Example 7: base 11, family 2{5} (formula: (5*11^n-1)/2) ([URL="http://factordb.com/index.php?query=%285*11%5En-1%29%2F2&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show"]factordb[/URL])

[CODE]
25 = 3 * 9
255 = 2 * 128
2555 = 3 * 919
25555 = 2 * 12828
255555 = 3 * 91919
2555555 = 2 * 1282828
25555555 = 3 * 9191919
255555555 = 2 * 128282828
2555555555 = 3 * 919191919
25555555555 = 2 * 12828282828
255555555555 = 3 * 91919191919
2555555555555 = 2 * 1282828282828
[/CODE]

Example 8: base 12, family {B}9B (formula: 12^(n+2)-25) ([URL="http://factordb.com/index.php?query=12%5E%28n%2B2%29-25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show"]factordb[/URL])

[CODE]
9B = 7 * 15
B9B = 11 * AB
BB9B = B7 * 105
BBB9B = 11 * B0AB
BBBB9B = BB7 * 1005
BBBBB9B = 11 * B0B0AB
BBBBBB9B = BBB7 * 10005
BBBBBBB9B = 11 * B0B0B0AB
BBBBBBBB9B = BBBB7 * 100005
BBBBBBBBB9B = 11 * B0B0B0B0AB
BBBBBBBBBB9B = BBBBB7 * 1000005
BBBBBBBBBBB9B = 11 * B0B0B0B0B0AB
[/CODE]

Example 9: base 14, family B{0}1 (formula: 11*14^(n+1)+1) ([URL="http://factordb.com/index.php?query=11*14%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show"]factordb[/URL])

[CODE]
B1 = 5 * 23
B01 = 3 * 395
B001 = 5 * 22B3
B0001 = 3 * 39495
B00001 = 5 * 22B2B3
B000001 = 3 * 3949495
B0000001 = 5 * 22B2B2B3
B00000001 = 3 * 394949495
B000000001 = 5 * 22B2B2B2B3
B0000000001 = 3 * 39494949495
B00000000001 = 5 * 22B2B2B2B2B3
B000000000001 = 3 * 3949494949495
[/CODE]

Example 10: base 13, family 3{0}95 (formula: 3*13^(n+2)+122) ([URL="http://factordb.com/index.php?query=3*13%5E%28n%2B2%29%2B122&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show"]factordb[/URL])

[CODE]
395 = 14 * 2B
3095 = 7 * 58A
30095 = 5 * 7A71
300095 = 7 * 5758A
3000095 = 14 * 23A92B
30000095 = 7 * 575758A
300000095 = 5 * 7A527A71
3000000095 = 7 * 57575758A
30000000095 = 14 * 23A923A92B
300000000095 = 7 * 5757575758A
3000000000095 = 5 * 7A527A527A71
30000000000095 = 7 * 575757575758A
[/CODE]

Example 11: base 16, family {4}D (formula: (4*16^(n+1)+131)/15) ([URL="http://factordb.com/index.php?query=%284*16%5E%28n%2B1%29%2B131%29%2F15&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show"]factordb[/URL])

[CODE]
4D = 7 * B
44D = 3 * 16F
444D = D * 541
4444D = 7 * 9C0B
44444D = 3 * 16C16F
444444D = D * 540541
4444444D = 7 * 9C09C0B
44444444D = 3 * 16C16C16F
444444444D = D * 540540541
4444444444D = 7 * 9C09C09C0B
44444444444D = 3 * 16C16C16C16F
444444444444D = D * 540540540541
[/CODE]

Example 12: base 16, family {C}D (formula: (4*16^(n+1)+1)/5) ([URL="http://factordb.com/index.php?query=%284*16%5E%28n%2B1%29%2B1%29%2F5&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show"]factordb[/URL])

[CODE]
CD = 5 * 29
CCD = 71 * 1D
CCCD = 1E1 * 6D
CCCCD = 18D * 841
CCCCCD = 64D * 2081
CCCCCCD = 7F01 * 19CD
CCCCCCCD = 1FE01 * 66CD
CCCCCCCCD = 198CD * 80401
CCCCCCCCCD = 664CD * 200801
CCCCCCCCCCD = 7FF001 * 199CCD
CCCCCCCCCCCD = 1FFE001 * 666CCD
CCCCCCCCCCCCD = 1998CCD * 8004001
[/CODE]

Example 13: base 17, family 1{9} (formula: (25*17^n-9)/16) ([URL="http://factordb.com/index.php?query=%2825*17%5En-9%29%2F16&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show"]factordb[/URL])

[CODE]
19 = 2 * D
199 = B * 27
1999 = 2 * D4D
19999 = AB * 287
199999 = 2 * D4D4D
1999999 = AAB * 2887
19999999 = 2 * D4D4D4D
199999999 = AAAB * 28887
1999999999 = 2 * D4D4D4D4D
19999999999 = AAAAB * 288887
199999999999 = 2 * D4D4D4D4D4D
1999999999999 = AAAAAB * 2888887
[/CODE]

Example 14: base 19, family 1{6} (formula: (4*19^n-1)/3) ([URL="http://factordb.com/index.php?query=%284*19%5En-1%29%2F3&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show"]factordb[/URL])

[CODE]
16 = 5 * 5
166 = D * 1I
1666 = 5 * 515
16666 = CD * 1II
166666 = 5 * 51515
1666666 = CCD * 1III
16666666 = 5 * 5151515
166666666 = CCCD * 1IIII
1666666666 = 5 * 515151515
16666666666 = CCCCD * 1IIIII
166666666666 = 5 * 51515151515
1666666666666 = CCCCCD * 1IIIIII
[/CODE]

Example 15: base 25, family 2{1} (formula: (49*25^n-1)/24) ([URL="http://factordb.com/index.php?query=%2849*25%5En-1%29%2F24&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show"]factordb[/URL])

[CODE]
21 = 3 * H
211 = 14 * 1J
2111 = 2N * HC
21111 = 144 * 1IJ
211111 = 2MN * HCC
2111111 = 1444 * 1IIJ
21111111 = 2MMN * HCCC
211111111 = 14444 * 1IIIJ
2111111111 = 2MMMN * HCCCC
21111111111 = 144444 * 1IIIIJ
211111111111 = 2MMMMN * HCCCCC
2111111111111 = 1444444 * 1IIIIIJ
[/CODE]

Example 16: base 36, family O{Z} (formula: 25*36^n-1) ([URL="http://factordb.com/index.php?query=25*36%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show"]factordb[/URL])

[CODE]
OZ = T * V
OZZ = 4Z * 51
OZZZ = TZ * U1
OZZZZ = 4ZZ * 501
OZZZZZ = TZZ * U01
OZZZZZZ = 4ZZZ * 5001
OZZZZZZZ = TZZZ * U001
OZZZZZZZZ = 4ZZZZ * 50001
OZZZZZZZZZ = TZZZZ * U0001
OZZZZZZZZZZ = 4ZZZZZ * 500001
OZZZZZZZZZZZ = TZZZZZ * U00001
OZZZZZZZZZZZZ = 4ZZZZZZ * 5000001
[/CODE]

Some references of this, see:

[URL="http://www.worldofnumbers.com/wing.htm"]http://www.worldofnumbers.com/wing.htm[/URL] for:

{1}0{1} (base 10): (formula: (10^(2*n+1)-9*10^n-1)/9) ([URL="http://factordb.com/index.php?query=%2810%5E%282*n%2B1%29-9*10%5En-1%29%2F9&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show"]factordb[/URL])

[CODE]
101 = 1 * 101 (the only possible prime case)
11011 = 11 * 1001
1110111 = 111 * 10001
111101111 = 1111 * 100001
11111011111 = 11111 * 1000001
1111110111111 = 111111 * 10000001
[/CODE]

{1}2{1} (base 10): (formula: (10^(2*n+1)+9*10^n-1)/9) ([URL="http://factordb.com/index.php?query=%2810%5E%282*n%2B1%29%2B9*10%5En-1%29%2F9&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show"]factordb[/URL])

[CODE]
121 = 11 * 11
11211 = 101 * 111
1112111 = 1001 * 1111
111121111 = 10001 * 11111
11111211111 = 100001 * 111111
1111112111111 = 1000001 * 1111111
[/CODE]

{3}2{3} (base 10): (formula: (10^(2*n+1)-3*10^n-1)/3) ([URL="http://factordb.com/index.php?query=%2810%5E%282*n%2B1%29-3*10%5En-1%29%2F3&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show"]factordb[/URL])

[CODE]
323 = 17 * 19
33233 = 167 * 199
3332333 = 1667 * 1999
333323333 = 16667 * 19999
33333233333 = 166667 * 199999
3333332333333 = 1666667 * 1999999
[/CODE]

{3}4{3} (base 10): (formula: (10^(2*n+1)+3*10^n-1)/3) ([URL="http://factordb.com/index.php?query=%2810%5E%282*n%2B1%29%2B3*10%5En-1%29%2F3&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show"]factordb[/URL])

[CODE]
343 = 7 * 49
33433 = 67 * 499
3334333 = 667 * 4999
333343333 = 6667 * 49999
33333433333 = 66667 * 499999
3333334333333 = 666667 * 4999999
[/CODE]

[URL="http://www.worldofnumbers.com/deplat.htm"]http://www.worldofnumbers.com/deplat.htm[/URL] for:

1{2}1 (base 10): (formula: (11*10^(n+1)-11)/9) ([URL="http://factordb.com/index.php?query=%2811*10%5E%28n%2B1%29-11%29%2F9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show"]factordb[/URL])

[CODE]
11 = 11 * 1 (the only possible prime case)
121 = 11 * 11
1221 = 11 * 111
12221 = 11 * 1111
122221 = 11 * 11111
1222221 = 11 * 111111
12222221 = 11 * 1111111
[/CODE]

7{3}7 (base 10): (formula: (22*10^(n+1)+11)/3) ([URL="http://factordb.com/index.php?query=%2822*10%5E%28n%2B1%29%2B11%29%2F3&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show"]factordb[/URL])

[CODE]
77 = 11 * 7
737 = 11 * 67
7337 = 11 * 667
73337 = 11 * 6667
733337 = 11 * 66667
7333337 = 11 * 666667
73333337 = 11 * 6666667
[/CODE]

9{7}9 (base 10): (formula: (88*10^(n+1)+11)/9) ([URL="http://factordb.com/index.php?query=%2888*10%5E%28n%2B1%29%2B11%29%2F9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show"]factordb[/URL])

[CODE]
99 = 11 * 9
979 = 11 * 89
9779 = 11 * 889
97779 = 11 * 8889
977779 = 11 * 88889
9777779 = 11 * 888889
97777779 = 11 * 8888889
[/CODE]

9{4}9 (base 10): (formula: (85*10^(n+1)+41)/9) ([URL="http://factordb.com/index.php?query=%2885*10%5E%28n%2B1%29%2B41%29%2F9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show"]factordb[/URL])

[CODE]
99 = 11 * 9
949 = 13 * 73
9449 = 11 * 859
94449 = 3 * 31483
944449 = 11 * 85859
9444449 = 7 * 1349207
94444449 = 11 * 8585859
944444449 = 13 * 72649573
9444444449 = 11 * 858585859
94444444449 = 3 * 31481481483
944444444449 = 11 * 85858585859
9444444444449 = 7 * 1349206349207
94444444444449 = 11 * 8585858585859
[/CODE]

9{1}3 (base 10): (formula: (82*10^(n+1)+17)/9) ([URL="http://factordb.com/index.php?query=%2882*10%5E%28n%2B1%29%2B17%29%2F9&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show"]factordb[/URL])

[CODE]
93 = 3 * 31
913 = 11 * 83
9113 = 13 * 701
91113 = 11 * 8283
911113 = 7 * 130159
9111113 = 11 * 828283
91111113 = 3 * 30370371
911111113 = 11 * 82828283
9111111113 = 13 * 700854701
91111111113 = 11 * 8282828283
911111111113 = 7 * 130158730159
9111111111113 = 11 * 828282828283
[/CODE]

9{5}9 (base 10): (formula: (86*10^(n+1)+31)/9) ([URL="http://factordb.com/index.php?query=%2886*10%5E%28n%2B1%29%2B31%29%2F9&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show"]factordb[/URL])

[CODE]
99 = 11 * 9
959 = 7 * 137
9559 = 11 * 869
95559 = 3 * 31853
955559 = 11 * 86869
9555559 = 13 * 735043
95555559 = 11 * 8686869
955555559 = 7 * 136507937
9555555559 = 11 * 868686869
95555555559 = 3 * 31851851853
955555555559 = 11 * 86868686869
9555555555559 = 13 * 735042735043
[/CODE]

[URL="http://www.worldofnumbers.com/Appending%201s%20to%20n.txt"]http://www.worldofnumbers.com/Appending%201s%20to%20n.txt[/URL] for:

37{1} (base 10): (formula: (334*10^n-1)/9) ([URL="http://factordb.com/index.php?query=%28334*10%5En-1%29%2F9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show"]factordb[/URL])

[CODE]
37 = 37 * 1 (the only possible prime case)
371 = 7 * 53
3711 = 3 * 1237
37111 = 37 * 1003
371111 = 13 * 28547
3711111 = 3 * 1237037
37111111 = 37 * 1003003
371111111 = 7 * 53015873
3711111111 = 3 * 1237037037
37111111111 = 37 * 1003003003
371111111111 = 13 * 28547008547
3711111111111 = 3 * 1237037037037
37111111111111 = 37 * 1003003003003
[/CODE]

38{1} (base 10): (formula: (343*10^n-1)/9) ([URL="http://factordb.com/index.php?query=%28343*10%5En-1%29%2F9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show"]factordb[/URL])

[CODE]
38 = 2 * 19
381 = 3 * 127
3811 = 37 * 103
38111 = 23 * 1657
381111 = 3 * 127037
3811111 = 37 * 103003
38111111 = 233 * 163567
381111111 = 3 * 127037037
3811111111 = 37 * 103003003
38111111111 = 2333 * 16335667
381111111111 = 3 * 127037037037
3811111111111 = 37 * 103003003003
38111111111111 = 23333 * 1633356667
[/CODE]

176{1} (base 10): (formula: (1585*10^n-1)/9) ([URL="http://factordb.com/index.php?query=%281585*10%5En-1%29%2F9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show"]factordb[/URL])

[CODE]
176 = 11 * 16
1761 = 3 * 587
17611 = 11 * 1601
176111 = 13 * 13547
1761111 = 11 * 160101
17611111 = 7 * 2515873
176111111 = 11 * 16010101
1761111111 = 3 * 587037037
17611111111 = 11 * 1601010101
176111111111 = 13 * 13547008547
1761111111111 = 11 * 160101010101
17611111111111 = 7 * 2515873015873
176111111111111 = 11 * 16010101010101
[/CODE]

Sierpinski number 78557: 78557*2^(n+1)+1, 10011001011011101{0}1 in base 2 (period: 36) ([URL="http://factordb.com/index.php?query=78557*2%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show"]factordb[/URL])

Riesel number 509203: 509203*2^n-1, 1111100010100010010{1} in base 2 (period: 24) ([URL="http://factordb.com/index.php?query=509203*2%5En-1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show"]factordb[/URL])

sweety439 2021-12-12 08:57

3 Attachment(s)
[QUOTE=sweety439;593853]Now I consider to add these families to [URL="https://docs.google.com/spreadsheets/d/e/2PACX-1vTKkSNKGVQkUINlp1B3cXe90FWPwiegdA07EE7-U7sqXntKAEQrynoI1sbFvvKriieda3LfkqRwmKME/pubhtml"]the list[/URL], although they do not always provide minimal primes (start with b+1):

* 10{z}
* {z}yz
* 11{0}1
* 1{0}11
* {1}2
* {1}3
* {1}4
* {1}0z
* 2{1}
* 3{1}
* 4{1}
* {z0}z1[/QUOTE]

Searched families 2{1}, {1}0z, {1}2 to length 5000

Families {1}2 in even bases are divisible by 2, thus ruled out as only contain composites

The smallest primes in these three families [B]may or may not[/B] be minimal primes (start with b+1), 2{1} and {1}2 are minimal primes (start with b+1) if and only if there is not a prime with shorter length of the form {1}, and {1}0z is minimal prime (start with b+1) if and only if there is not a prime with shorter length of the form either {1} or {1}z

{1}0z have been extensively search for base b=6 by Paul Bourdelais: [URL="https://oeis.org/A199165"]https://oeis.org/A199165[/URL] and [URL="http://www.primenumbers.net/prptop/searchform.php?form=%286%5En-11%29%2F5&action=Search"]http://www.primenumbers.net/prptop/searchform.php?form=%286%5En-11%29%2F5&action=Search[/URL], Paul Bourdelais searched generalized repunit number family {1} in bases 2<=b<=101 and generalized Wagstaff number family {z0}z1 (which is equivalent to generalized repunit number in negative bases), but he also search families {1}0z and {y}z in base b=6 (but not in other bases), for {y}z in base b=6, see [URL="https://oeis.org/A248613"]https://oeis.org/A248613[/URL] and [URL="http://www.primenumbers.net/prptop/searchform.php?form=%284*6%5En%2B1%29%2F5&action=Search"]http://www.primenumbers.net/prptop/searchform.php?form=%284*6%5En%2B1%29%2F5&action=Search[/URL]

2{1} and {1}0z are dual families (see [URL="https://mersenneforum.org/showpost.php?p=593715&postcount=213"]this post[/URL] for the definition and references of "dual" families, "dual" families have the same [URL="https://www.rieselprime.de/ziki/Nash_weight"]Nash weight[/URL] (or [URL="https://stdkmd.net/nrr/prime/primedifficulty.txt"]difficulty[/URL]), thus if one can be ruled out as only contain composites (only count the numbers > base) by (covering congruence, algebraic factorization, or combine of them), then the other can also be ruled out as only contain composites (only count the numbers > base) by the same reason.

the dual family of {1}2 (for odd bases) is {y}z, which is already in [URL="https://docs.google.com/spreadsheets/d/e/2PACX-1vTKkSNKGVQkUINlp1B3cXe90FWPwiegdA07EE7-U7sqXntKAEQrynoI1sbFvvKriieda3LfkqRwmKME/pubhtml"]this list[/URL]

sweety439 2021-12-12 09:40

Note: If b and 2*b-1 are both squares (i.e. b is in [URL="https://oeis.org/A008844"]A008844[/URL]), then there cannot be a prime of the form 2{1} or {1}0z, since they can be factored as difference of squares (their formulas are ((2*b-1)*b^n-1)/(b-1) and (b^n-(2*b-1))/(b-1), thus if b and 2*b-1 are both squares, then b^n and 2*b-1 are both squares for all n, and ((2*b-1)*b^n-1)/(b-1) and (b^n-(2*b-1))/(b-1) can be factored as difference of squares: x^2-y^2 = (x-y) * (x+y))

(thus, for the files "21111" and "1110z" in last post, b=25 and b=841 should be "RC" instead of "unknown")

(also, since there is no digit "2" in base 2, thus, families 2{1} and {1}2 are not interpretable in base 2, and for the files "21111" and "11112" in last post, b=2 should be "NB")

This is like the GRU / GFN families:

{1}:

If b is of the form m^r with r>1 (i.e. b is in [URL="https://oeis.org/A001597"]A001597[/URL]), then {1} can be factored as sum of r-th powers (the only possible prime is b = m^(p^r) with prime p, 111...111 with p 1's may be prime, such number is prime for b = 4, 8, 27, 36, 100, 128, 196, 400, 576, 676, 1331, 1600, 2916, 3136, 4356, 5476, 7056, 8000, 8100, 8836, 9261, 12100, 13456, 14400, 15376, 15876, 16900, 17576, 17956, 21316, 22500, 24336, 25600, 27000, 28900, ..., but not for b = 9, 16, 25, 32, 49, 81, 121, 125, 144, 169, 216, 225, 243, 256, 289, 324, 343, 361, 441, 484, 512, 529, 625, 784, 841, 900, 961, 1000, 1089, 1156, 1225, 1296, 1369, 1444, 1521, 1681, 1728, 1764, 1849, 1936, 2025, 2048, 2116, 2187, 2197, 2209, 2304, 2401, 2500, 2601, 2704, 2744, 2809, 3025, 3125, 3249, 3364, 3375, 3481, 3600, 3721, 3844, 3969, 4225, 4489, 4624, 4761, 4900, 4913, 5041, 5184, 5329, 5625, 5776, 5832, 5929, 6084, 6241, 6400, 6561, 6724, 6859, 6889, 7225, 7396, 7569, 7744, 7776, 7921, 8192, 8281, 8464, 8649, 9025, 9216, 9409, 9604, 9801, 10000, ..., also, for base b = m^c where c is not prime power, there is no possible prime, such b are 64, 729, 1024, 4096, 15625, 16384, 32768, 46656, 59049, ...)

1{0}1 (for even b):

If b is of the form m^r with odd r>1 (i.e. b is in [URL="https://oeis.org/A070265"]A070265[/URL]), then 1{0}1 can be factored as sum of r-th powers

{#}$ (#=(b-1)/2, $=(b+1)/2, for odd b):

If b is of the form m^r with odd r>1 (i.e. b is in [URL="https://oeis.org/A070265"]A070265[/URL]), then {#}$ can be factored as sum of r-th powers

{z0}z1:

If b is of the form m^r with odd r>1 (i.e. b is in [URL="https://oeis.org/A070265"]A070265[/URL]), then {z0}z1 can be factored as sum of r-th powers (the only possible prime is b = m^(2^r*p^s) with prime p, z0z0z0...z0z0z1 with (p-3)/2 (z0)'s may be prime, such number is prime for b = 128, 216, 343, 729, 7776, 16807, ..., but not for b = 8, 27, 32, 64, 125, 243, 512, 1000, 1024, 1331, 1728, 2048, 2187, 2197, 2744, 3125, 3375, 4096, 4913, 5832, 6859, 8000, 8192, 9261, ..., also, for base b = m^c where c is not of the form 2^r*p^s with prime p, there is no possible prime, the smallest such b is 32768 = 2^15)

If b is of the form 4*m^4 (i.e. b is in [URL="https://oeis.org/A141046"]A141046[/URL]), then {z0}z1 can be factored as Aurifeuillian factorization of x^4+4*y^4 (the only possible prime is b = 4, z1 = 13 in decimal is prime)

sweety439 2021-12-16 00:44

The minimal prime (start with b+1) problem covers [URL="http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm"]CRUS Riesel problem[/URL] for the same base if the CK for the CRUS Riesel problem is < base, such bases 2<=b<=1024 are: (i.e. for these bases b, CRUS Riesel problem is a part of the minimal prime (start with b+1) problem in the same base b, and hence if the minimal prime (start with b+1) problem is solved, then CRUS Riesel problem in the same base b is also solved)

{14, 20, 29, 32, 34, 38, 41, 44, 47, 50, 54, 56, 59, 62, 64, 65, 68, 69, 74, 77, 81, 83, 84, 86, 89, 90, 92, 94, 98, 104, 110, 113, 114, 116, 118, 119, 122, 125, 128, 129, 131, 132, 134, 137, 139, 140, 142, 144, 146, 149, 152, 153, 155, 158, 164, 167, 170, 173, 174, 176, 178, 179, 182, 184, 185, 186, 188, 189, 194, 197, 200, 202, 203, 204, 206, 208, 209, 212, 214, 216, 218, 219, 221, 224, 227, 229, 230, 233, 234, 236, 237, 239, 242, 244, 245, 246, 248, 251, 252, 254, 257, 258, 259, 260, 263, 264, 265, 266, 269, 272, 274, 275, 278, 279, 284, 285, 286, 289, 290, 293, 294, 296, 298, 299, 300, 302, 305, 307, 308, 309, 311, 314, 317, 318, 320, 321, 322, 324, 326, 328, 329, 332, 334, 335, 338, 339, 340, 341, 344, 347, 349, 350, 353, 354, 356, 359, 362, 363, 364, 365, 368, 369, 371, 373, 374, 376, 377, 379, 380, 383, 384, 386, 387, 389, 390, 392, 394, 395, 398, 401, 402, 404, 405, 407, 410, 412, 413, 414, 416, 417, 419, 422, 424, 425, 426, 428, 429, 433, 434, 437, 439, 440, 441, 443, 444, 446, 447, 449, 450, 452, 454, 455, 458, 459, 461, 464, 467, 468, 470, 472, 473, 474, 475, 476, 479, 480, 482, 484, 488, 489, 491, 492, 493, 494, 497, 500, 503, 504, 506, 509, 510, 512, 514, 515, 516, 517, 518, 519, 521, 523, 524, 527, 528, 529, 530, 531, 533, 534, 536, 538, 539, 542, 544, 545, 548, 549, 551, 552, 554, 557, 558, 559, 560, 562, 563, 564, 566, 569, 571, 572, 573, 574, 577, 578, 579, 580, 581, 582, 584, 587, 588, 590, 593, 594, 596, 597, 599, 602, 604, 605, 608, 609, 611, 614, 615, 617, 619, 620, 622, 623, 626, 628, 629, 632, 634, 635, 636, 637, 638, 641, 643, 644, 645, 648, 649, 650, 653, 654, 656, 657, 659, 662, 664, 665, 668, 669, 670, 671, 674, 676, 677, 678, 679, 680, 681, 683, 684, 686, 688, 689, 692, 694, 695, 696, 698, 699, 701, 702, 704, 706, 707, 710, 712, 713, 714, 716, 719, 720, 722, 724, 725, 727, 729, 730, 731, 734, 737, 739, 740, 741, 743, 744, 746, 747, 749, 752, 753, 755, 758, 759, 761, 762, 764, 767, 769, 770, 773, 774, 776, 778, 779, 780, 781, 782, 783, 784, 785, 788, 789, 790, 791, 794, 797, 798, 800, 802, 803, 804, 805, 806, 809, 811, 812, 813, 814, 815, 816, 818, 819, 824, 825, 827, 828, 829, 830, 832, 833, 834, 835, 836, 839, 842, 844, 845, 846, 848, 849, 850, 851, 853, 854, 857, 859, 860, 863, 864, 866, 867, 868, 869, 870, 872, 873, 874, 875, 878, 879, 881, 883, 884, 887, 888, 889, 890, 892, 893, 894, 896, 898, 899, 900, 901, 902, 905, 908, 909, 911, 912, 914, 916, 917, 919, 920, 922, 923, 924, 926, 929, 930, 932, 934, 935, 938, 939, 941, 942, 944, 945, 947, 948, 949, 950, 951, 953, 954, 956, 958, 959, 961, 962, 964, 965, 967, 968, 969, 972, 974, 977, 978, 979, 980, 983, 984, 985, 986, 987, 988, 989, 992, 993, 994, 995, 998, 1000, 1002, 1003, 1004, 1007, 1010, 1011, 1013, 1014, 1016, 1017, 1019, 1021, 1022, 1024}

The minimal prime (start with b+1) problem covers [URL="http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm"]CRUS Sierpinski problem[/URL] for the same base if the CK for the CRUS Sierpinski problem is < base, such bases 2<=b<=1024 are: (i.e. for these bases b, CRUS Sierpinski problem is a part of the minimal prime (start with b+1) problem in the same base b, and hence if the minimal prime (start with b+1) problem is solved, then CRUS Sierpinski problem in the same base b is also solved)

{14, 20, 29, 32, 34, 38, 41, 44, 47, 50, 54, 56, 59, 62, 64, 65, 68, 69, 74, 76, 77, 83, 84, 86, 89, 90, 92, 94, 98, 101, 104, 109, 110, 113, 114, 116, 118, 119, 122, 125, 128, 129, 131, 132, 134, 137, 139, 140, 142, 144, 146, 149, 152, 153, 154, 155, 158, 159, 160, 164, 167, 169, 170, 172, 173, 174, 176, 179, 181, 182, 184, 185, 186, 188, 189, 194, 197, 200, 202, 203, 204, 206, 208, 209, 212, 214, 216, 218, 219, 220, 221, 224, 227, 229, 230, 233, 234, 236, 237, 239, 242, 244, 245, 246, 248, 251, 252, 254, 257, 258, 259, 260, 263, 264, 265, 266, 269, 272, 274, 275, 278, 279, 281, 284, 285, 289, 290, 293, 294, 296, 298, 299, 300, 302, 304, 305, 307, 308, 309, 311, 314, 317, 318, 320, 321, 322, 324, 326, 328, 329, 332, 334, 335, 338, 339, 340, 341, 344, 347, 349, 350, 353, 354, 356, 359, 362, 363, 364, 365, 368, 369, 370, 371, 373, 374, 377, 379, 380, 384, 386, 389, 390, 392, 394, 395, 398, 401, 402, 404, 405, 406, 407, 409, 410, 412, 413, 414, 416, 417, 419, 422, 424, 425, 426, 428, 429, 433, 434, 436, 437, 439, 440, 441, 443, 444, 446, 447, 449, 450, 452, 454, 455, 458, 459, 461, 464, 467, 468, 469, 470, 472, 473, 474, 475, 476, 479, 480, 482, 483, 484, 488, 489, 491, 492, 493, 494, 496, 497, 500, 501, 503, 504, 505, 506, 509, 510, 512, 514, 515, 516, 517, 518, 519, 521, 524, 526, 527, 528, 530, 531, 532, 533, 534, 536, 538, 539, 542, 544, 545, 548, 549, 550, 551, 552, 554, 557, 558, 559, 560, 562, 563, 564, 566, 569, 571, 572, 573, 574, 578, 579, 580, 581, 582, 584, 587, 588, 589, 590, 593, 594, 596, 597, 599, 601, 602, 604, 605, 608, 609, 610, 611, 614, 615, 617, 619, 620, 622, 623, 626, 629, 632, 634, 635, 636, 637, 638, 641, 643, 644, 645, 647, 648, 649, 650, 653, 654, 656, 657, 659, 662, 664, 665, 666, 668, 669, 670, 671, 674, 677, 678, 679, 680, 681, 683, 684, 686, 688, 689, 692, 695, 696, 698, 699, 701, 702, 703, 704, 706, 707, 709, 710, 712, 713, 714, 716, 718, 719, 720, 722, 724, 725, 727, 729, 730, 731, 734, 736, 737, 739, 740, 741, 743, 744, 746, 747, 748, 749, 752, 753, 754, 755, 758, 759, 761, 762, 764, 766, 767, 769, 770, 773, 774, 776, 778, 779, 780, 781, 782, 783, 784, 785, 788, 789, 790, 791, 792, 794, 797, 798, 800, 802, 803, 804, 805, 806, 809, 811, 812, 813, 814, 815, 816, 818, 819, 821, 824, 825, 827, 828, 829, 830, 832, 833, 834, 835, 836, 839, 842, 844, 845, 846, 848, 849, 850, 851, 853, 854, 857, 859, 860, 863, 864, 866, 867, 868, 869, 870, 872, 873, 874, 875, 878, 879, 881, 883, 884, 887, 888, 889, 890, 892, 893, 894, 896, 898, 899, 900, 901, 902, 903, 904, 905, 908, 909, 911, 912, 914, 916, 917, 919, 920, 922, 923, 924, 926, 929, 930, 932, 934, 935, 937, 938, 939, 941, 942, 944, 945, 947, 948, 949, 950, 951, 953, 954, 956, 958, 959, 962, 964, 965, 967, 968, 969, 972, 974, 977, 978, 979, 980, 983, 984, 985, 986, 987, 989, 992, 993, 994, 995, 998, 1000, 1001, 1004, 1006, 1007, 1009, 1010, 1011, 1013, 1014, 1016, 1019, 1022, 1024}

Families convert:

Riesel: (k-1):{(b-1)}
Sierpinski: (k):{0}:1

For these bases b, all primes in the CRUS Riesel/Sierpinski problems base b are minimal primes (start with b+1) base b, and the families are unsolved families in base b if and only if the corresponding k is remaining k (including GFNs) without known primes for the CRUS Riesel/Sierpinski problems base b.

sweety439 2021-12-18 19:32

The numbers in simple families are of the form for some fixed integers a, b, c where a≥1, b≥2 (b is the base), c≠0, gcd(a,c)=1, gcd(b,c)=1 (thus, all large minimal primes base b (but possible not all minimal primes base b if b is large, e.g. b = 25, 29, 31, 35) have simple expression ([URL="https://en.wikipedia.org/wiki/Expression_(mathematics)"]expression[/URL] with ≤ 40 [URL="https://en.wikipedia.org/wiki/Character_(computing)"]characters[/URL], all taken from “0” “1” “2” “3” “4” “5” “6” “7” “8” “9” “+” “-” “*” “/” “^” “(“ “)”), factorial (!), double factorial (!!), and primorial (#) are not allowed since they can be used to ensure many small factors, see [URL="http://primerecords.dk/primegaps/gaps20.htm"]this page[/URL]).

Thus, the sequence of the numbers in simple families also satisfy the [URL="https://en.wikipedia.org/wiki/Recurrence_relation"]recurrence relation[/URL] a(0) = k, a(n) = b*a(n-1)+r ([URL="https://en.wikipedia.org/wiki/Linear_function"]linear function[/URL]) for all integers n>0, with integers k, b (b is the base), r, and we want to find the smallest (probable) prime > b in this sequence.

Reference of such recurrence relations:

[URL="https://oeis.org/A102006"]https://oeis.org/A102006[/URL] (family 1{0}3 in base 10 (10^n+3), which is a(0) = 13, a(n) = 10*a(n-1) - 27 for n > 0 (a shifted value of n: 10^(n+1)+3))

[URL="https://oeis.org/A101823"]https://oeis.org/A101823[/URL] (family 3{0}1 in base 10 (3*10^n+1), which is a(0) = 31, a(n) = 10*a(n-1) - 9 for n > 0 (a shifted value of n: 3*10^(n+1)+1))

[URL="https://oeis.org/A056250"]https://oeis.org/A056250[/URL] (family 1{9}1 in base 10 (2*10^(n+1)-9), which is a(0) = 11, a(n) = 10*a(n-1) + 81 for n > 0)

[URL="https://oeis.org/A056251"]https://oeis.org/A056251[/URL] (family 3{1}3 in base 10 ((28*10^(n+1)+17)/9), which is a(0) = 33, a(n) = 10*A(n-1) - 17 for n > 0)

[URL="https://oeis.org/A050412"]https://oeis.org/A050412[/URL] (family *{1} in base 2 (k*2^n-1), which is a(n) = 2*a(n-1)+1 for n>0)

[URL="https://oeis.org/A052333"]https://oeis.org/A052333[/URL] (family *{1} in base 2 (k*2^n-1), which is a(n) = 2*a(n-1)+1 for n>0)

[URL="https://oeis.org/A257495"]https://oeis.org/A257495[/URL] (family *{1} in base 2 (k*2^n-1), which is a(n) = 2*a(n-1)+1 for n>0)

[URL="https://oeis.org/A225721"]https://oeis.org/A225721[/URL] (family *{0}1 in base 2 (k*2^n+1), which is a(n) = 2*a(n-1)-1 for n>0)

[URL="https://mersenneforum.org/showthread.php?t=21832&page=6"]https://mersenneforum.org/showthread.php?t=21832&page=6[/URL] (more generalization of this, i.e. with more (k,b,r) value triples)

Also, the sequence a(0) = 1, a(n) = b*a(n-1) + 1 for n > 0 is exactly the [URL="https://en.wikipedia.org/wiki/Repunit"]repunit[/URL] (i.e. family {1}) in base b, and the sequence a(0) = 1, a(n) = b*a(n-1) - 1 for n > 0 is exactly the family {y}z in base b. ([URL="https://docs.google.com/spreadsheets/d/e/2PACX-1vTKkSNKGVQkUINlp1B3cXe90FWPwiegdA07EE7-U7sqXntKAEQrynoI1sbFvvKriieda3LfkqRwmKME/pubhtml"]this table[/URL] already has the smallest indices of primes in these two sequences, i.e. the smallest primes in these two families)

sweety439 2021-12-18 21:24

[QUOTE=sweety439;592123]* Case (3,1):

** Since 32, 41, 61 are primes, we only need to consider the family 3{0,1,3,5}1 (since any digits 2, 4, 6 between them will produce smaller primes)

*** Since 551 is prime, we only need to consider the family 3{0,1,3}1 and 3{0,1,3}5{0,1,3}1 (since any digits combo 55 between (3,1) will produce smaller primes)

**** For the 3{0,1,3}1 family, since [B]3031[/B] and 131 are primes, we only need to consider the families 3{0,1}1 and 3{3}3{0,1}1 (since any digits combo 03, 13 between (3,1) will produce smaller primes, thus for the digits between (3,1), all 3's must be before all 0's and 1's, and thus we can let the red [COLOR="Red"]3[/COLOR] in 3{3}[COLOR="red"]3[/COLOR]{0,1}1 be the rightmost 3 between (3,1), all digits before this 3 must be 3's, and all digits after this 3 must be either 0's or 1's)

***** For the 3{0,1}1 family:

****** If there are >=2 0's and >=1 1's between (3,1), then at least one of [B]30011[/B], [B]30101[/B], [B]31001[/B] will be a subsequence.

****** If there are no 1's between (3,1), then the form will be 3{0}1

******* All numbers of the form 3{0}1 are divisible by 2, thus cannot be prime.

****** If there are no 0's between (3,1), then the form will be 3{1}1

******* The smallest prime of the form 3{1}1 is [B]31111[/B]

****** If there are exactly 1 0's between (3,1), then there must be <3 1's between (3,1), or [B]31111[/B] will be a subsequence.

******* If there are 2 1's between (3,1), then the digit sum is 6, thus the number is divisible by 6 and cannot be prime.

******* If there are 1 1's between (3,1), then the number can only be either 3101 or 3011

******** Neither 3101 nor 3011 is prime.

******* If there are no 1's between (3,1), then the number must be 301

******** 301 is not prime.

***** For the 3{3}3{0,1}1 family:

****** If there are at least one 3 between (3,3{0,1}1) and at least one 1 between (3{3}3,1), then [B]33311[/B] will be a subsequence.

****** If there are no 3 between (3,3{0,1}1), then the form will be 33{0,1}1

******* If there are at least 3 1's between (33,1), then 31111 will be a subsequence.

******* If there are exactly 2 1's between (33,1), then the digit sum is 12, thus the number is divisible by 3 and cannot be prime.

******* If there are exactly 1 1's between (33,1), then the digit sum is 11, thus the number is divisible by 2 and cannot be prime.

******* If there are no 1's between (33,1), then the form will be 33{0}1

******** The smallest prime of the form 33{0}1 is [B]33001[/B]

****** If there are no 1 between (3{3}3,1), then the form will be 3{3}3{0}1

******* If there are at least 2 0's between (3{3}3,1), then 33001 will be a subsequence.

******* If there are exactly 1 0's between (3{3}3,1), then the form is 3{3}301

******** The smallest prime of the form 3{3}301 is [B]33333301[/B]

******* If there are no 0's between (3{3}3,1), then the form is 3{3}31

******** The smallest prime of the form 3{3}31 is [B]33333333333333331[/B]

**** For the 3{0,1,3}5{0,1,3}1 family, since 335 is prime, we only need to consider the family 3{0,1}5{0,1,3}1

***** Numbers containing 3 between (3{0,1}5,1):

****** The form is 3{0,1}5{0,1,3}3{0,1,3}1

******* Since 3031 and 131 are primes, we only need to consider the family 35{3}3{0,1,3}1 (since any digits combo 03, 13 between (3,1) will produce smaller primes)

******** Since 533 is prime, we only need to consider the family 353{0,1}1 (since any digits combo 33 between (35,1) will produce smaller primes)

********* Since 5011 is prime, we only need to consider the family 353{1}{0}1 (since any digits combo 01 between (353,1) will produce smaller primes)

********** If there are at least 3 1's between (353,{0}1), then 31111 will be a subsequence.

********** If there are exactly 2 1's between (353,{0}1), then the digit sum is 20, thus the number is divisible by 2 and cannot be prime.

********** If there are exactly 1 1's between (353,{0}1), then the form is 3531{0}1

*********** The smallest prime of the form 3531{0}1 is 3531001, but it is not minimal prime since 31001 is prime.

********** If there are no 1's between (353,{0}1), then the digit sum is 15, thus the number is divisible by 6 and cannot be prime.

***** Numbers not containing 3 between (3{0,1}5,1):

****** The form is 3{0,1}5{0,1}1

******* If there are >=2 0's and >=1 1's between (3,1), then at least one of 30011, 30101, 31001 will be a subsequence.

******* If there are no 1's between (3,1), then the form will be 3{0}5{0}1

******** All numbers of the form 3{0}5{0}1 are divisible by 3, thus cannot be prime.

******* If there are no 0's between (3,1), then the form will be 3{1}5{1}1

******** If there are >=3 1's between (3,1), then 31111 will be a subsequence.

******** If there are exactly 2 1's between (3,1), then the number can only be 31151, 31511, 35111

********* None of 31151, 31511, 35111 are primes.

******** If there are exactly 1 1's between (3,1), then the digit sum is 13, thus the number is divisible by 2 and cannot be prime.

******** If there are no 1's between (3,1), then the number is 351

********* 351 is not prime.

******* If there are exactly 1 0's between (3,1), then the form will be 3{1}0{1}5{1}1 or 3{1}5{1}0{1}1

******** No matter 3{1}0{1}5{1}1 or 3{1}5{1}0{1}1, if there are >=3 1's between (3,1), then 31111 will be a subsequence.

******** If there are exactly 2 1's between (3,1), then the number can only be 311051, 310151, 310511, 301151, 301511, 305111, 311501, 315101, 315011, 351101, 351011, 350111

********* Of these numbers, 311051, 301151, 311501, 351101, 350111 are primes.

********** However, 311051, 301151, 311501 have 115 as subsequence, and 350111 has 5011 as subsequence, thus only [B]351101[/B] is minimal prime.

******** No matter 3{1}0{1}5{1}1 or 3{1}5{1}0{1}1, if there are exactly 1 1's between (3,1), then the digit sum is 13, thus the number is divisible by 2 and cannot be prime.

******** If there are no 1's between (3,1), then the number is 3051 for 3{1}0{1}5{1}1 or 3501 for 3{1}5{1}0{1}1

********* Neither 3051 nor 3501 is prime.[/QUOTE]

* Case (3,2):

** [B]32[/B] is prime, and thus the only minimal prime in this family.

* Case (3,3):

** Since 32, 23, 43, [B]313[/B] are primes, we only need to consider the family 3{0,3,5,6}3 (since any digits 1, 2, 4 between them will produce smaller primes)

*** If there are >=2 5's in {}, then 553 will be a subsequence.

*** If there are no 5's in {}, then the family will be 3{0,3,6}3

**** All numbers of the form 3{0,3,6}3 are divisible by 3, thus cannot be prime.

*** If there are exactly 1 5's in {}, then the family will be 3{0,3,6}5{0,3,6}3

**** Since 335, 65, [B]3503[/B], 533, 56 are primes, we only need to consider the family 3{0}53 (since any digit 3, 6 between (3,5{0,3,6}3) and any digit 0, 3, 6 between (3{0,3,6}5,3) will produce smaller primes)

***** The smallest prime of the form 3{0}53 is [B]300053[/B]

* Case (3,4):

** Since 32, 14, [B]304[/B], [B]344[/B], [B]364[/B] are primes, we only need to consider the family 3{3,5}4 (since any digits 0, 1, 2, 4, 6 between them will produce smaller primes)

*** Since [B]3334[/B] and 335 are primes, we only need to consider the family 3{5}4 and 3{5}34 (since any digits combo 33, 35 between them will produce smaller primes)

**** The smallest prime of the form 3{5}4 is [URL="http://factordb.com/index.php?id=1100000002766595757"]3(5^9234)4[/URL] (not minimal prime, since 35555 and 5554 are primes)

**** The smallest prime of the form 3{5}34 is 355555555555555555555555555555555555555555555555555555555555555534 (not minimal prime, since 35555, 553, and 5554 are primes)

* Case (3,5):

** Since 32, 25, 65, [B]335[/B] are primes, we only need to consider the family 3{0,1,4,5}5 (since any digits 2, 3, 6 between them will produce smaller primes)

*** If there are at least one 1's and at least one 5's in {}, then either 155 or 515 will be a subsequence.

*** If there are at least one 1's and at least one 4's in {}, then either 14 or 41 will be a subsequence.

*** If there are at least two 1's in {}, then 115 will be a subsequence.

*** If there are exactly one 1's and no 4's or 5's in {}, then the family will be 3{0}1{0}5

**** All numbers of the form 3{0}1{0}5 are divisible by 3, thus cannot be prime.

*** If there is no 1's in {}, then the family will be 3{0,4,5}5

**** If there are at least to 4's in {}, then 344 and 445 will be subsequences.

**** If there is no 4's in {}, then the family will be 3{0,5}5

***** Since [B]3055[/B] and [B]3505[/B] are primes, we only need to consider the families 3{0}5 and 3{5}5

****** All numbers of the form 3{0}5 are divisible by 2, thus cannot be prime.

****** The smallest prime of the form 3{5}5 is [B]35555[/B]

**** If there is exactly one 4's in {}, then the family will be 3{0,5}4{0,5}5

***** Since 304, [B]3545[/B] are primes, we only need to consider the families 34{0,5}5 (since any digits 0 or 5 between (3,4{0,5}5) will produce small primes)

****** All numbers of the form 34{0,5}5 are divisible by 5, thus cannot be prime.

* Case (3,6):

** Since 32, 16, 56, [B]346[/B] are primes, we only need to consider the family 3{0,3,6}6 (since any digits 1, 2, 4, 5 between them will produce smaller primes)

*** All numbers of the form 3{0,3,6}6 are divisible by 3, thus cannot be prime.

sweety439 2021-12-18 21:47

* Case (4,1):

** [B]41[/B] is prime, and thus the only minimal prime in this family.

* Case (4,2):

** Since 41, 43, 32, 52 are primes, we only need to consider the family 4{0,2,4,6}2 (since any digits 1, 3, 5 between them will produce smaller primes)

*** All numbers of the form 4{0,2,4,6}2 are divisible by 2, thus cannot be prime.

* Case (4,3):

** [B]43[/B] is prime, and thus the only minimal prime in this family.

* Case (4,4):

** Since 41, 43, 14 are primes, we only need to consider the family 4{0,2,4,5,6}4 (since any digits 1, 3 between them will produce smaller primes)

*** If there is no 5's in {}, then the family will be 4{0,2,4,6}4

**** All numbers of the form 4{0,2,4,6}4 are divisible by 2, thus cannot be prime.

*** If there is at least one 5's in {}, then there cannot be 2 in {} (since if so, then either 25 or 52 will be a subsequence) and there cannot be 6 in {} (since if so, then either 65 or 56 will be a subsequence), thus the family is 4{0,4,5}5{0,4,5}4

**** Since 445, [B]4504[/B], 544 are primes, we only need to consider the family 4{0,5}5{5}4 (since any digit 4 between (4,5{0,4,5}4) and any digit 0, 4 between (4{0,4,5}5,4) will produce smaller primes)

***** If there are at least two 0's between (4,5{0,4,5}4), then [B]40054[/B] will be a subsequence.

***** If there is no 0's between (4,5{0,4,5}4), then the family will be 4{5}5{5}4, which is equivalent to 4{5}4

****** The smallest prime of the form 4{5}4 is 45555555555555554 (not minimal prime, since 4555 and 5554 are primes)

***** If there is exactly one 0's between (4,5{0,4,5}4), then the family will be 4{5}0{5}5{5}4

****** Since 4504 is prime, we only need to consider the family 40{5}5{5}4 (since any digit 5 between (4,0{5}5{5}4) will produce small primes), which is equivalent to 40{5}4

******* The smallest prime of the form 40{5}4 is 405555555555555554 (not minimal prime, since 4555 and 5554 are primes)

* Case (4,5):

** Since 41, 43, 25, 65, [B]445[/B] are primes, we only need to consider the family 4{0,5}5 (since any digits 1, 2, 3, 4, 6 between them will produce smaller primes)

*** If there are at least two 5's in {}, then [B]4555[/B] will be a subsequence.

*** If there is exactly one 5's in {}, then the digit sum is 20, and the number will be divisible by 2 and cannot be prime.

*** If there is no 5's in {}, then the family will be 4{0}5

**** All numbers of the form 4{0}5 are divisible by 3, thus cannot be prime.

* Case (4,6):

** Since 41, 43, 16, 56 are primes, we only need to consider the family 4{0,2,4,6}6 (since any digits 1, 3, 5 between them will produce smaller primes)

*** All numbers of the form 4{0,2,4,6}6 are divisible by 2, thus cannot be prime.

sweety439 2021-12-19 19:52

Base 7 is really an interesting bases (although these properties may not related to minimal primes (start with b+1) in base b=7):

* There are no known (probable) primes of the form 12121...12121 ([URL="http://www.worldofnumbers.com/undulat.htm"]Smoothly undulating palindromic primes[/URL]) (reference: [URL="https://github.com/RaymondDevillers/primes/blob/master/left49"]https://github.com/RaymondDevillers/primes/blob/master/left49[/URL], this family is the same as the unsolved family 1{F} in base 49)

* There are no known (probable) primes of the form {1}21 (I found this when searched the minimal primes (start with b+1) in base b=7, although this family will not produce minimal primes (start with b+1), since 11111 and 1112 are primes) (reference: [URL="https://github.com/RaymondDevillers/primes/blob/master/left49"]https://github.com/RaymondDevillers/primes/blob/master/left49[/URL], this family is the same as the unsolved family {8}F in base 49 when the number of 1's in {} is even, note that this number cannot be prime when the number of 1's in {} is odd, since if so, then the number will be an even number)

sweety439 2021-12-20 04:42

The smallest primes of the form {1} in base b (always minimal prime (start with b+1) base b) for 2<=b<=100:

3, 13, 5, 31, 7, 2801, 73, (not exist), 11, 50544702849929377, 13, 30941, 211, 241, 17, 307, 19, 109912203092239643840221, 421, 463, 23, 292561, 601, (not exist), 321272407, 757, 29, 732541, 31, 917087137, (not exist), 1123, 2458736461986831391, (35^313-1)/34, 37, 6765811783780036261, 1483, (39^349-1)/38, 41, 1723, 43, 3500201, 3835261, 585578449280908796570517800071, 47, 4939353696332137648660158610486273245800498531219046056285398249895046060595791007616253627660064463584012737427605759732894439061580553419678353685587762357233722998146101218334328347614340561470069315963989297, 1868467947605686541562499217713, (not exist), 2551, (51^4229-1)/50, 53, 178250690949465223, 2971, 7141212583461249612878870081, 31401724537, 3307, 59, 3541, 61, 52379047267, 3907, 16007041, (not exist), 435686197988821897112429141998291, 67, 751410597400064602523400427092397, 21700501, 4831, 71, 5113, 73, 28792661, 30397351, 5701, 1730995823890967242587808530196681934696717844461757939646025856921040336541, 6007, 79, 39449441, 6481, (not exist), 83, 48037081, 6218272796370530483675222621221, 52822061, 22390512687494871811, 438668366137, 89, 8011, 8191, (91^4421-1)/90, (92^439-1)/91, 654022685443, 78914411, 742912017121, 97, 62065212901958868055012327674641, 792806586866086631668831, 9901, 101

(lengths: 2, 3, 2, 3, 2, 5, 3, (not exist), 2, 17, 2, 5, 3, 3, 2, 3, 2, 19, 3, 3, 2, 5, 3, (not exist), 7, 3, 2, 5, 2, 7, (not exist), 3, 13, 313, 2, 13, 3, 349, 2, 3, 2, 5, 5, 19, 2, 127, 19, (not exist), 3, 4229, 2, 11, 3, 17, 7, 3, 2, 3, 2, 7, 3, 5, (not exist), 19, 2, 19, 5, 3, 2, 3, 2, 5, 5, 3, 41, 3, 2, 5, 3, (not exist), 2, 5, 17, 5, 11, 7, 2, 3, 3, 4421, 439, 7, 5, 7, 2, 17, 13, 3, 2)

The smallest primes of the form {y}z in base b (always minimal prime (start with b+1) base b) for 3<=b<=100 (since this family has leading digit "y", thus base b=2 is not available):

5, 11, 19, 29, 41, 439, 71, 89, 109, 131, (11*13^564+1)/12, 181, 535461077009, 239, 271, 98801, 754617425461612781, 379, 419, 461, 74751395041, 317351, 599, 11406121, 701, 21139, 811, 703862069, 929, 991, (31*33^252+1)/32, 38113, 7385772222129586256251615636489, 1259, 2494456231, 2028781, 1481, 1559, 2755117, 1721, 1558753710276070629972594251713686824401, 83203, 1979, 2069, 2161, 5195471, 2351, 122449, 2549, 145699033217806348921408781051, 236694247338220787957359344916201671880441, 2861, 2969, 3079, 3191, 191689, 11908441, 3539, 3659, 14534101, 15498881, 72615754448888393663, 4159, 4289, 4421, 1432233067, 4691, (68*70^555+1)/69, 4969, 1907665271, 795259257284441, 4856538474104427001, 31248293212942174963049, 433123, 5851, 468389, 239970949681, 4662589691947244641370734177215189873417721519, 42508637, 300253502641, (81*83^680+1)/82, 6971, 372659640439, 7309, 7481, 673639, 206308659677878787942238728669477849139002326971125920277494982244624742112469994669963447472945487047748925382258769209352782944085048415241443200073603004312410186589399084763844457936246598248431316944992075024254288100854204343884044113084729872468008781, 8009, 18236539101242584939773724671515281275916872322822196569481086274001352684603849641802753638243049517747444391767358741697136702377284369289870004752365894524863040193875279847118108550688639161244660867997023347326607966379705505251891378728107735869696193530297505392306761, 70852051, 73992101, 8741, 8929, 32076562714376628748804112036298983524468388012134426965970127012090218136861237329876966780347068031249130531700233282691952665764901560395220654002360958652776602527, 9311, 621543986037312561989074843437923599751689820022203926184214847126107201853363777344400467841110613254083527536032858130161348209726494439079405240789126468313805891021946318821044254078655554233276554461261763230238431382098112789156212297664967204632934288754351041323937, 931873209101, 9898989899

(lengths: 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 564, 2, 10, 2, 2, 4, 14, 2, 2, 2, 8, 4, 2, 5, 2, 3, 2, 6, 2, 2, 252, 3, 20, 2, 6, 4, 2, 2, 4, 2, 24, 3, 2, 2, 2, 4, 2, 3, 2, 17, 24, 2, 2, 2, 2, 3, 4, 2, 2, 4, 4, 11, 2, 2, 2, 5, 2, 555, 2, 5, 8, 10, 12, 3, 2, 3, 6, 24, 4, 6, 680, 2, 6, 2, 2, 3, 132, 2, 140, 4, 4, 2, 2, 84, 2, 137, 6, 5)

sweety439 2021-12-20 04:49

The smallest primes of the form 1{0}z in base b (always minimal prime (start with b+1) base b) for 2<=b<=100:

3, 5, 7, 29, 11, 13, 71, 17, 19, 131, 23, 181, 2177953337809371149, 29, 31, 83537, 5849, 37, 419, 41, 43, 279863, 47, 15649, 701, 53, 811, 420707233300229, 59, 61, 3599131035634557106248430806148785487095757694641533306480604458089470064537190296255232548883112685719936728506816716098566612844395439751206812144692131084107807, 35969, 67, 1259, 71, 73, 1481, 1559, 79, 1721, 83, 79549, 1979, 89, 2161, 2766668711962335809450748011342447, 2351, 97, 2549, 101, 103, 2861, 107, 109, 3191, 113, 11316553, 3539, 3659, 3142742836081, 218340105584957, 250109, 127, 4289, 131, 66956888672235945457062019127709902451882787, 4691, 137, 139, 16409682740640811134311, 26873927, 389089, 899194740203849, 149, 151, 35153117, 474629, 157, 1582914569427869017987216134525742016224840634247755775450003558994542240681803949890024966791955487933425097126503291771848736219563207743792295392652116274666283341751674470400000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000079, 531521, 163, 6971, 167, 7309, 7481, 173, 3596345248055383, 8009, 179, 181, 71639387, 8741, 8929, 62672163268978486152220736113427031521574282073283717377501840800586629942614536048495210707187652587890719, 191, 193, 92236913, 197, 199

(lengths: 2, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 3, 17, 2, 2, 5, 4, 2, 3, 2, 2, 5, 2, 4, 3, 2, 3, 11, 2, 2, 109, 4, 2, 3, 2, 2, 3, 3, 2, 3, 2, 4, 3, 2, 3, 21, 3, 2, 3, 2, 2, 3, 2, 2, 3, 2, 5, 3, 3, 8, 9, 4, 2, 3, 2, 25, 3, 2, 2, 13, 5, 4, 9, 2, 2, 5, 4, 2, 195, 4, 2, 3, 2, 3, 3, 2, 9, 3, 2, 2, 5, 3, 3, 55, 2, 2, 5, 2, 2)

The smallest primes of the form z{0}1 in base b (always minimal prime (start with b+1) base b) for 2<=b<=100:

3, 7, 13, 101, 31, 43, 449, 73, 9001, 259374246011, 19009, 157, 2549, 211, 241, 1336337, 307, 218336795902605993201009018384568383223, 31129600000000000001, 421, 463, 255042399139852495799, 13249, 601, 16901, 13817467, 757, 23549, 23490001, 858874531, 35740566642812256257, 34849, 1123, 41651, 45361, 4678622632622773, 699421969744001971270254593, 1483, 62401, 42147180671470348388835886625344411346196083191529631288482561146240326026998221506440783978133517939838164995885825751477859968041, 1723, 77659, 161168129, 89101, 4380121, 26080959134473636132102571567, 108289, 115249, 122501, 2551, 2802982140528952023258759169, 52*53^960+1, 5130766694717659087768092673, 2971, 540897281, 10370809, 3307, 359216400347725176472840139, 3541, 3091222461661, 42969828958366879401068146141598580737, 3907, 16515073, 64*65^946+1, 5372506751041, 4423, 64749322924298179419277948572663809, 166341196124523666469, 4831, 1778817671, 5113, 149261154697, 234993088291936566889001874832943568280082096415477544968258287961251922992693249, 2341406251, 5701, 76*77^828+1, 6007, 6163, 505601, 6481, 7486024557894291753209857, 564899, 29157736624129, 4384852501, 921239313498782686653014814287960958346849550024859147304961, 56631259, 87*88^3022+1, 697049, 8011, 8191, 467029174555181057, 795709, 7260965329, 7656358751, 875521, 686689346964557196168289, 825240131904332033, 960499, 9901

(lengths: 2, 2, 2, 3, 2, 2, 3, 2, 4, 11, 4, 2, 3, 2, 2, 5, 2, 30, 15, 2, 2, 15, 3, 2, 3, 5, 2, 3, 5, 6, 13, 3, 2, 3, 3, 10, 17, 2, 3, 81, 2, 3, 5, 3, 4, 17, 3, 3, 3, 2, 16, 961, 16, 2, 5, 4, 2, 15, 2, 7, 21, 2, 4, 947, 7, 2, 19, 11, 2, 5, 2, 6, 43, 5, 2, 829, 2, 2, 3, 2, 13, 3, 7, 5, 31, 4, 3023, 3, 2, 2, 9, 3, 5, 5, 3, 12, 9, 3, 2)

sweety439 2021-12-20 04:58

The smallest primes of the form {z}1 in base b (always minimal prime (start with b+1) base b) for 2<=b<=100:

3, 7, 13, 3121, 31, 43, 549755813881, 73, 991, 1321, 248821, 157, 2731, 211, 241, 34271896307617, 307, 6841, 13107199999999999999981, 421, 463, 141050039560662968926081, 331753, 601, 17551, 7625597484961, 757, 1816075630094014572464024421543167816955354437761, 21869999971, 29761, 34359738337, 1185889, 1123, 42841, 60466141, 1173587600912967505181585220815870451386152316472799938266409866089889961869797411886878993830039201370297, 79235131, 1483, 262143999999961, 68881, 1723, 3418759, 121987944123281928470243645070631579418581, 91081, 4477411, 229344961, 254803921, 36703368217294125441230211032033660188753, 124951, 2551, 140557, 1621038246414954860589967996431649201, 157411, 2971, 5416169448144841, 185137, 3307, 30155888444737842601, 3541, 844596241, 238267, 3907, 16777153, 27883916671284601415195465087890561, 254602111920194024074608072436520292411608321458245205951, 4423, 138917982831632221022646640257006246528083468188909501, 22667053, 4831, 71^3019-70, 5113, 1348279907365869037210940254745047725601, 214199131839040635233999278781065638639145954194193262334275941174952043636376856413982450253751, 421801, 5701, 456457, 6007, 6163, 134217727999999921, 6481, 271982557273863678841753073605626002507679137645543667495471311053616993453470564708294248779498865580099754980028625286424287001013541608441630559920314975786687829835617478796177988454578148883323659802024345006968986457411371581654584985642656490840054833369455010687173903712282140558186224074122542984523434130283691951, 83^965-82, 592621, 614041, 126024529890394348379345391831458407434589541727066914731, 658417, 88^2848-87, 350356403707485121, 8011, 8191, 778597, (unknown, >93^60000-92), 78074803, 857281, 75144747810721, 832972004833, 9039207871, 970201, 9901

(lengths: 2, 2, 2, 5, 2, 2, 13, 2, 3, 3, 5, 2, 3, 2, 2, 11, 2, 3, 17, 2, 2, 17, 4, 2, 3, 9, 2, 33, 7, 3, 7, 4, 2, 3, 5, 67, 5, 2, 9, 3, 2, 4, 25, 3, 4, 5, 5, 24, 3, 2, 3, 21, 3, 2, 9, 3, 2, 11, 2, 5, 3, 2, 4, 19, 31, 2, 29, 4, 2, 3019, 2, 21, 51, 3, 2, 3, 2, 2, 9, 2, 169, 965, 3, 3, 29, 3, 2848, 9, 2, 2, 3, (unknown, >60000), 4, 3, 7, 6, 5, 3, 2)

The smallest primes of the form y{z} in base b (always minimal prime (start with b+1) base b) for 3<=b<=100 (since this family has leading digit "y", thus base b=2 is not available):

5, 11, 19, 29, 41, 3583, 71, 89, 109, 131, 2027, 181, 408700964355468749, 239, 271, 5507, 846825857, 379, 419, 461, 17276416353328819798072137388863592892072278184923153720493777138850572564953, 839967991029301247, 599, 3885038158778096269468893991882380063764065770433606110283149695964997245520484669311748838825973451239771955518933348332721403496018696846203290707966794803507099534240007184258836096614399, 701, 2368778164222232774191928573951, 811, 26099, 929, 991, 34847, 3095263992211830248865791, 1457749, 1259, 3417547576787, 37*38^136211-1, 1481, 1559, 7790170955239, 1721, 11416381666493, 13728945815551, 1979, 2069, 2161, 108287, 2351, 146031379699707031249999999999999999999999999, 2549, 19390405631, 7741603, 2861, 2969, 3079, 3191, 191747, 11911981, 3539, 3659, 61*62^899-1, 15502913, 4648579506574807007231, 4159, 4289, 4421, 1879906732393711733925510475137857760503062460875743807798664124070281660123652821731052897176373035753185153797586943, 4691, 66853417180829999999999999, 4969, 1925620760471083622891481383291847136824810012671, 58065126616373831, 29581351, 416249, 1098389592883199, 5851, 13761102938344700291819041271278618982733762179772125443309654334452663746650714406911, 486797, 67860483276799999999999, 42515279, 8460714920405607680979411179674012039906592349943469819895302427737964209997000306003319158242216832837056676965039854908398751686697181608337645440589627391, 82*83^21495-1, 6971, 606899, 7309, 7481, 59288063, 477785360038488838329543538347169179362263139957911, 8009, 90*91^519-1, 1481508330753453728491805383593670105823471153485518018370135116052371415571240807373643012628916031447449255358158844479345054906912548026050220478431231, 92*93^476-1, 8741, 8929, 875519, 9311, 97*98^4983-1, 960497, 989999

(lengths: 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 3, 2, 15, 2, 2, 3, 7, 2, 2, 2, 56, 13, 2, 134, 2, 21, 2, 3, 2, 2, 3, 16, 4, 2, 8, 136212, 2, 2, 8, 2, 8, 8, 2, 2, 2, 3, 2, 26, 2, 6, 4, 2, 2, 2, 2, 3, 4, 2, 2, 900, 4, 12, 2, 2, 2, 64, 2, 14, 2, 26, 9, 4, 3, 8, 2, 45, 3, 12, 4, 82, 21496, 2, 3, 2, 2, 4, 26, 2, 520, 78, 477, 2, 2, 3, 2, 4984, 3, 3)

sweety439 2021-12-20 05:26

[QUOTE=sweety439;595602]* Case (4,1):

** [B]41[/B] is prime, and thus the only minimal prime in this family.

* Case (4,2):

** Since 41, 43, 32, 52 are primes, we only need to consider the family 4{0,2,4,6}2 (since any digits 1, 3, 5 between them will produce smaller primes)

*** All numbers of the form 4{0,2,4,6}2 are divisible by 2, thus cannot be prime.

* Case (4,3):

** [B]43[/B] is prime, and thus the only minimal prime in this family.

* Case (4,4):

** Since 41, 43, 14 are primes, we only need to consider the family 4{0,2,4,5,6}4 (since any digits 1, 3 between them will produce smaller primes)

*** If there is no 5's in {}, then the family will be 4{0,2,4,6}4

**** All numbers of the form 4{0,2,4,6}4 are divisible by 2, thus cannot be prime.

*** If there is at least one 5's in {}, then there cannot be 2 in {} (since if so, then either 25 or 52 will be a subsequence) and there cannot be 6 in {} (since if so, then either 65 or 56 will be a subsequence), thus the family is 4{0,4,5}5{0,4,5}4

**** Since 445, [B]4504[/B], 544 are primes, we only need to consider the family 4{0,5}5{5}4 (since any digit 4 between (4,5{0,4,5}4) and any digit 0, 4 between (4{0,4,5}5,4) will produce smaller primes)

***** If there are at least two 0's between (4,5{0,4,5}4), then [B]40054[/B] will be a subsequence.

***** If there is no 0's between (4,5{0,4,5}4), then the family will be 4{5}5{5}4, which is equivalent to 4{5}4

****** The smallest prime of the form 4{5}4 is 45555555555555554 (not minimal prime, since 4555 and 5554 are primes)

***** If there is exactly one 0's between (4,5{0,4,5}4), then the family will be 4{5}0{5}5{5}4

****** Since 4504 is prime, we only need to consider the family 40{5}5{5}4 (since any digit 5 between (4,0{5}5{5}4) will produce small primes), which is equivalent to 40{5}4

******* The smallest prime of the form 40{5}4 is 405555555555555554 (not minimal prime, since 4555 and 5554 are primes)

* Case (4,5):

** Since 41, 43, 25, 65, [B]445[/B] are primes, we only need to consider the family 4{0,5}5 (since any digits 1, 2, 3, 4, 6 between them will produce smaller primes)

*** If there are at least two 5's in {}, then [B]4555[/B] will be a subsequence.

*** If there is exactly one 5's in {}, then the digit sum is 20, and the number will be divisible by 2 and cannot be prime.

*** If there is no 5's in {}, then the family will be 4{0}5

**** All numbers of the form 4{0}5 are divisible by 3, thus cannot be prime.

* Case (4,6):

** Since 41, 43, 16, 56 are primes, we only need to consider the family 4{0,2,4,6}6 (since any digits 1, 3, 5 between them will produce smaller primes)

*** All numbers of the form 4{0,2,4,6}6 are divisible by 2, thus cannot be prime.[/QUOTE]

* Case (5,1):

** Since 52, 56, 41, 61, [B]551[/B] are primes, we only need to consider the family 5{0,1,3}1 (since any digits 2, 4, 5, 6 between them will produce smaller primes)

*** If there are at least two 3's in {}, then 533 will be a subsequence.

*** If there is no 3's in {}, then the family will be 5{0,1}1

**** Since [B]5011[/B] is prime, we only need to consider the family 5{1}{0}1

***** Since 11111 is prime, we only need to consider the families 5{0}1, 51{0}1, 511{0}1, 5111{0}1 (since any digits combo 1111 between (5,1) will produce small primes)

****** All numbers of the form 5{0}1 are divisible by 6, thus cannot be prime.

****** The smallest prime of the form 51{0}1 is [B]5100000001[/B]

****** All numbers of the form 511{0}1 are divisible by 2, thus cannot be prime.

****** All numbers of the form 5111{0}1 are divisible by 3, thus cannot be prime.

*** If there is exactly one 3's in {}, then the family will be 5{0,1}3{0,1}1

**** If there is at least one 1's between (5,3{0,1}1), then 131 will be a subsequence.

***** Thus we only need to consider the family 5{0}3{0,1}1

****** If there are no 1's between (5{0}3,1), then the digit sum is 12, and the number will be divisible by 3 and cannot be prime.

****** If there are exactly one 1's between (5{0}3,1), then the digit sum is 13, and the number will be divisible by 2 and cannot be prime.

****** If there are exactly three 1's between (5{0}3,1), then the digit sum is 15, and the number will be divisible by 6 and cannot be prime.

****** If there are at least four 1's between (5{0}3,1), then 11111 will be a subsequence.

****** If there are exactly two 1's between (5{0}3,1), then the family will be 5{0}3{0}1{0}1{0}1

******* Since 5011 is prime, we only need to consider the family 5311{0}1 (since any digit 0 between (5,1{0}1) will produce small primes, this includes the leftmost three {} in 5{0}3{0}1{0}1{0}1, and thus only the rightmost {} can contain 0)

******** The smallest prime of the form 5311{0}1 is [B]531101[/B]

* Case (5,2):

** [B]52[/B] is prime, and thus the only minimal prime in this family.

* Case (5,3):

** Since 52, 56, 23, 43, [B]533[/B], [B]553[/B] are primes, we only need to consider the family 5{0,1}3 (since any digits 2, 3, 4, 5, 6 between them will produce smaller primes)

*** If there are at least two 1's in {}, then 113 will be a subsequence.

*** If there is exactly one 1's in {}, then the digit sum is 12, and the number will be divisible by 3 and cannot be prime.

*** If there is no 1's in {}, then the digit sum is 11, and the number will be divisible by 2 and cannot be prime.

* Case (5,4):

** Since 52, 56, 14, [B]544[/B] are primes, we only need to consider the family 5{0,3,5}4 (since any digits 1, 2, 4, 6 between them will produce smaller primes)

*** If there are no 5's in {}, then the family will be 5{0,3}4

**** All numbers of the form 5{0,3}4 are divisible by 3, thus cannot be prime.

*** If there are at least one 5's and at least one 3's in {}, then either 535 or 553 will be a subsequence.

*** If there are exactly one 5's and no 3's in {}, then the digit sum is 20, and the number will be divisible by 2 and cannot be prime.

*** If there are at least two 5's in {}, then [B]5554[/B] will be a subsequence.

* Case (5,5):

** Since 52, 56, 25, 65, [B]515[/B], [B]535[/B] are primes, we only need to consider the family 5{0,4,5}5 (since any digits 1, 2, 3, 6 between them will produce smaller primes)

*** If there are no 4's in {}, then the family will be 5{0,5}5

**** All numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime.

*** If there are no 5's in {}, then the family will be 5{0,4}5

**** All numbers of the form 5{0,4}5 are divisible by 2, thus cannot be prime.

*** If there are at least one 4's and at least one 5's in {}, then either [B]5455[/B] or [B]5545[/B] will be a subsequence.

* Case (5,6):

** [B]56[/B] is prime, and thus the only minimal prime in this family.

sweety439 2021-12-20 05:53

* Case (6,1):

** [B]61[/B] is prime, and thus the only minimal prime in this family.

* Case (6,2):

** Since 61, 65, 32, 52 are primes, we only need to consider the family 6{0,2,4,6}2 (since any digits 1, 3, 5 between them will produce smaller primes)

*** All numbers of the form 6{0,2,4,6}2 are divisible by 2, thus cannot be prime.

* Case (6,3):

** Since 61, 65, 23, 43 are primes, we only need to consider the family 6{0,3,6}3 (since any digits 1, 2, 4, 5 between them will produce smaller primes)

*** All numbers of the form 6{0,3,6}3 are divisible by 3, thus cannot be prime.

* Case (6,4):

** Since 61, 65, 14 are primes, we only need to consider the family 6{0,2,3,4,6}4 (since any digits 1, 5 between them will produce smaller primes)

*** If there is no 3's in {}, then the family will be 6{0,2,4,6}4

**** All numbers of the form 6{0,2,4,6}4 are divisible by 2, thus cannot be prime.

*** If there are exactly two 3's in {}, then the family will be 6{0,2,4,6}3{0,2,4,6}3{0,2,4,6}4

**** All numbers of the form 6{0,2,4,6}3{0,2,4,6}3{0,2,4,6}4 are divisible by 2, thus cannot be prime.

*** If there are at least three 3's in {}, then 3334 will be a subsequence.

*** If there is exactly one 3's in {}, then the family will be 6{0,2,4,6}3{0,2,4,6}4

**** If there is 0 between (6,3{0,2,4,6}4), then [B]6034[/B] will be a subsequence.

**** If there is 2 between (6,3{0,2,4,6}4), then 23 will be a subsequence.

**** If there is 4 between (6,3{0,2,4,6}4), then 43 will be a subsequence.

**** If there is 6 between (6,3{0,2,4,6}4), then [B]6634[/B] will be a subsequence.

**** If there is 0 between (6{0,2,4,6}3,4), then 304 will be a subsequence.

**** If there is 2 between (6{0,2,4,6}3,4), then 32 will be a subsequence.

**** If there is 4 between (6{0,2,4,6}3,4), then 344 will be a subsequence.

**** If there is 6 between (6{0,2,4,6}3,4), then 364 will be a subsequence.

**** Thus the number can only be 634

***** 634 is not prime.

* Case (6,5):

** [B]65[/B] is prime, and thus the only minimal prime in this family.

* Case (6,6):

** Since 61, 65, 16, 56 are primes, we only need to consider the family 6{0,2,3,4,6}6 (since any digits 1, 5 between them will produce smaller primes)

*** If there is no 3's in {}, then the family will be 6{0,2,4,6}6

**** All numbers of the form 6{0,2,4,6}6 are divisible by 2, thus cannot be prime.

*** If there is no 2's and no 4's in {}, then the family will be 6{0,3,6}6

**** All numbers of the form 6{0,3,6}6 are divisible by 3, thus cannot be prime.

*** If there is at least one 3's and at least one 2's in {}, then either 32 or 23 will be a subsequence.

*** If there is at least one 3's and at least one 4's in {}, then either 346 or 43 will be a subsequence.

sweety439 2021-12-20 05:55

Base 7 is now completely solved!!! The minimal set is {14, 16, 23, 25, 32, 41, 43, 52, 56, 61, 65, 113, 115, 131, 133, 155, 212, 221, 304, 313, 335, 344, 346, 364, 445, 515, 533, 535, 544, 551, 553, 1022, 1051, 1112, 1202, 1211, 1222, 2111, 3031, 3055, 3334, 3503, 3505, 3545, 4504, 4555, 5011, 5455, 5545, 5554, 6034, 6634, 11111, 11201, 30011, 30101, 31001, 31111, 33001, 33311, 35555, 40054, 100121, 150001, 300053, 351101, 531101, 1100021, 33333301, 5100000001, 33333333333333331}, with 71 elements.

sweety439 2021-12-20 21:23

The smallest primes of the form 1{z} in base b (always minimal prime (start with b+1) base b) for 2<=b<=100:

3, 5, 7, 1249, 11, 13, 127, 17, 19, 241, 23, 337, 76831, 29, 31, 577, 647, 37, 20479999999999, 41, 43, 296071777, 47, 1249, 617831551, 53, 1567, 15387133080032326246081223292828787411221911122916017220126284227825703776392672467768318856009763825207593900596158761682711294895921233392537083406917227083982402321012446032594528728383203531755841, 59, 61, 2147483647, 2582935937, 67, 3676531249, 71, 73, 2887, 3041, 79, 3361, 83, 3697, 7496191, 89, 4231, 9759361, 10616831, 97, 4999, 101, 103, 14762783749438524018088313240622157671545425891033638774020213131211643094561, 107, 109, 6271, 113, 390223, 6961, 1555199999, 453961, 7687, 7937, 127, 35701249, 131, 2*67^768-1, 42762751, 137, 139, 3685226021389754364081406998013620614177373233840954223262516393648197304721188581672950092102081, 2820556958550470572987049795649933350494568280745031706857177087, 10657, 10955116495297186033389799920407521119823438170983067411284733410338720021136605880400331969750325727283776539553179715828948746553389246803325612720511513248619960935886421747050162442794804812613833601971860122867866007685442586844128834278653951, 149, 151, 515110198093444174897047217, 35131137709823, 157, 12799, 13121, 163, 4504584464278081, 167, 14449, 44260315777606141951, 173, 93028094906919615583719418453514602923973148671, 1838359291054664377057535487394089069295354282527628876589216679708146054555978759346341266203765093789368997103003201, 179, 181, 16927, 1608713, 14678080447, 18049, 191, 193, 19207, 197, 199

(lengths: 2, 2, 2, 5, 2, 2, 3, 2, 2, 3, 2, 3, 5, 2, 2, 3, 3, 2, 11, 2, 2, 7, 2, 3, 7, 2, 3, 137, 2, 2, 7, 7, 2, 7, 2, 2, 3, 3, 2, 3, 2, 3, 5, 2, 3, 5, 5, 2, 3, 2, 2, 45, 2, 2, 3, 2, 4, 3, 6, 4, 3, 3, 2, 5, 2, 769, 5, 2, 2, 53, 35, 3, 133, 2, 2, 15, 8, 2, 3, 3, 2, 9, 2, 3, 11, 2, 25, 61, 2, 2, 3, 4, 6, 3, 2, 2, 3, 2, 2)

The smallest primes of the form {z}y in base b (always minimal prime (start with b+1) base b) for 3<=b<=100 (since this family has ending digit "y", thus base b=2 is not available):

7, (not exist), 23, (not exist), 47, (not exist), 79, (not exist), 14639, (not exist), 167, (not exist), 223, (not exist), 24137567, (not exist), 359, (not exist), 439, (not exist), 480250763996501976790165756943039, (not exist), 6103515623, (not exist), 727, (not exist), 839, (not exist), 29789, (not exist), 1087, (not exist), 1223, (not exist), 1367, (not exist), 2313439, (not exist), 2825759, (not exist), 1847, (not exist), 1532278301220703123, (not exist), 2207, (not exist), 2399, (not exist), 45767944570399, (not exist), 7890479, (not exist), 3023, (not exist), 1176246293903439667999, (not exist), 12117359, (not exist), 3719, (not exist), 3967, (not exist), 318644812890623, (not exist), 300761, (not exist), 4759, (not exist), 5039, (not exist), 28398239, (not exist), 5623, (not exist), 5927, (not exist), 1287743804278744050410620426954739687963064854495168753870500853746064159, (not exist), 126780498249647572006884144885170471304489016861879438617760751383229084323232338257117700598826138594153330162501779530073312173213845425140567243580938604038739769427385468006506866897314769910412165745223337576718744062067606378331982456148394399, (not exist), 47458319, (not exist), 52200623, (not exist), 57289759, (not exist), 7919, (not exist), 753569, (not exist), 8647, (not exist), 81450623, (not exist), 97^747-2, (not exist), 970297, (not exist)

(lengths: 2, (not exist), 2, (not exist), 2, (not exist), 2, (not exist), 4, (not exist), 2, (not exist), 2, (not exist), 6, (not exist), 2, (not exist), 2, (not exist), 24, (not exist), 7, (not exist), 2, (not exist), 2, (not exist), 3, (not exist), 2, (not exist), 2, (not exist), 2, (not exist), 4, (not exist), 4, (not exist), 2, (not exist), 11, (not exist), 2, (not exist), 2, (not exist), 8, (not exist), 4, (not exist), 2, (not exist), 12, (not exist), 4, (not exist), 2, (not exist), 2, (not exist), 8, (not exist), 3, (not exist), 2, (not exist), 2, (not exist), 4, (not exist), 2, (not exist), 2, (not exist), 38, (not exist), 130, (not exist), 4, (not exist), 4, (not exist), 4, (not exist), 2, (not exist), 3, (not exist), 2, (not exist), 4, (not exist), 747, (not exist), 3, (not exist))

sweety439 2021-12-20 21:32

The smallest primes of the form 1{0}1 in base b (always minimal prime (start with b+1) base b) for 2<=b<=100:

3, (not exist), 5, (not exist), 7, (not exist), (not exist), (not exist), 11, (not exist), 13, (not exist), 197, (not exist), 17, (not exist), 19, (not exist), 401, (not exist), 23, (not exist), 577, (not exist), 677, (not exist), 29, (not exist), 31, (not exist), (not exist), (not exist), 1336337, (not exist), 37, (not exist), (unknown, >=38^8388608+1), (not exist), 41, (not exist), 43, (not exist), 197352587024076973231046657, (not exist), 47, (not exist), 5308417, (not exist), (unknown, >=50^8388608+1), (not exist), 53, (not exist), 2917, (not exist), 3137, (not exist), 59, (not exist), 61, (not exist), (unknown, >=62^8388608+1), (not exist), (not exist), (not exist), 67, (not exist), (unknown, >=68^8388608+1), (not exist), 71, (not exist), 73, (not exist), 5477, (not exist), 1238846438084943599707227160577, (not exist), 79, (not exist), 40960001, (not exist), 83, (not exist), 7057, (not exist), (unknown, >=86^8388608+1), (not exist), 89, (not exist), 8101, (not exist), (unknown, >=92^8388608+1), (not exist), 8837, (not exist), 97, (not exist), (unknown, >=98^8388608+1), (not exist), 101

(lengths: 2, (not exist), 2, (not exist), 2, (not exist), (not exist), (not exist), 2, (not exist), 2, (not exist), 3, (not exist), 2, (not exist), 2, (not exist), 3, (not exist), 2, (not exist), 3, (not exist), 3, (not exist), 2, (not exist), 2, (not exist), (not exist), (not exist), 5, (not exist), 2, (not exist), (unknown, >=8388609), (not exist), 2, (not exist), 2, (not exist), 17, (not exist), 2, (not exist), 5, (not exist), (unknown, >=8388609), (not exist), 2, (not exist), 3, (not exist), 3, (not exist), 2, (not exist), 2, (not exist), (unknown, >=8388609), (not exist), (not exist), (not exist), 2, (not exist), (unknown, >=8388609), (not exist), 2, (not exist), 2, (not exist), 3, (not exist), 17, (not exist), 2, (not exist), 5, (not exist), 2, (not exist), 3, (not exist), (unknown, >=8388609), (not exist), 2, (not exist), 3, (not exist), (unknown, >=8388609), (not exist), 3, (not exist), 2, (not exist), (unknown, >=8388609), (not exist), 2)

The smallest primes of the form {#}$ (#=(b-1)/2, $=(b+1)/2) in base b (always minimal prime (start with b+1) base b) for 2<=b<=100 (only available for odd b):

(not exist), 5, (not exist), 13, (not exist), 1201, (not exist), 41, (not exist), 61, (not exist), 14281, (not exist), 113, (not exist), 41761, (not exist), 181, (not exist), 97241, (not exist), 139921, (not exist), 313, (not exist), (not exist), (not exist), 421, (not exist), (unknown, >=(31^524288+1)/2), (not exist), 703204309121, (not exist), 613, (not exist), (unknown, >=(37^524288+1)/2), (not exist), 761, (not exist), 31879515457326527173216321, (not exist), 5844100138801, (not exist), 1013, (not exist), 11905643330881, (not exist), 1201, (not exist), 1301, (not exist), 31129845205681, (not exist), (unknown, >=(55^524288+1)/2), (not exist), 5278001, (not exist), 1741, (not exist), 1861, (not exist), (unknown, >=(63^524288+1)/2), (not exist), 2113, (not exist), (unknown, >=(67^524288+1)/2), (not exist), 2381, (not exist), 2521, (not exist), 14199121, (not exist), 502262128603166429097887091259622138750273734331130981445313, (not exist), (unknown, >=(77^524288+1)/2), (not exist), 3121, (not exist), 21523361, (not exist), (unknown, >=(83^524288+1)/2), (not exist), 3613, (not exist), 5386145066875877753173052384321, (not exist), (unknown, >=(89^524288+1)/2), (not exist), (unknown, >=(91^524288+1)/2), (not exist), (unknown, >=(93^524288+1)/2), (not exist), 4513, (not exist), (unknown, >=(97^524288+1)/2), (not exist), (unknown, >=(99^524288+1)/2), (not exist)

(lengths: (not exist), 2, (not exist), 2, (not exist), 4, (not exist), 2, (not exist), 2, (not exist), 4, (not exist), 2, (not exist), 4, (not exist), 2, (not exist), 4, (not exist), 4, (not exist), 2, (not exist), (not exist), (not exist), 2, (not exist), (unknown, >=524288), (not exist), 8, (not exist), 2, (not exist), (unknown, >=524288), (not exist), 2, (not exist), 16, (not exist), 8, (not exist), 2, (not exist), 8, (not exist), 2, (not exist), 2, (not exist), 8, (not exist), (unknown, >=524288), (not exist), 4, (not exist), 2, (not exist), 2, (not exist), (unknown, >=524288), (not exist), 2, (not exist), (unknown, >=524288), (not exist), 2, (not exist), 2, (not exist), 4, (not exist), 32, (not exist), (unknown, >=524288), (not exist), 2, (not exist), 4, (not exist), (unknown, >=524288), (not exist), 2, (not exist), 16, (not exist), (unknown, >=524288), (not exist), (unknown, >=524288), (not exist), (unknown, >=524288), (not exist), 2, (not exist), (unknown, >=524288), (not exist), (unknown, >=524288), (not exist))

sweety439 2021-12-20 21:35

The smallest primes of the form 1{0}2 in base b (always minimal prime (start with b+1) base b) for 3<=b<=100 (since this family has digit "2", thus base b=2 is not available):

5, (not exist), 7, (not exist), (not exist), (not exist), 11, (not exist), 13, (not exist), (not exist), (not exist), 17, (not exist), 19, (not exist), (not exist), (not exist), 23, (not exist), 952809757913929, (not exist), (not exist), (not exist), 29, (not exist), 31, (not exist), (not exist), (not exist), 1091, (not exist), 37, (not exist), (not exist), (not exist), 41, (not exist), 43, (not exist), (not exist), (not exist), 47, (not exist), 885233716287722386108568808645559198522547790058305212262181780420828956357982973084581935827930464156048602918053397761948271781610736426217362565287242033121579185919812362859356307201329, (not exist), (not exist), (not exist), 53, (not exist), 1174711139839, (not exist), (not exist), (not exist), 59, (not exist), 61, (not exist), (not exist), (not exist), 250049, (not exist), 67, (not exist), (not exist), (not exist), 71, (not exist), 73, (not exist), (not exist), (not exist), 31676352024078369140627, (not exist), 79, (not exist), (not exist), (not exist), 83, (not exist), 571789, (not exist), (not exist), (not exist), 89, (not exist), 89^255+2, (not exist), (not exist), (not exist), 5595818096650403, (not exist), 97, (not exist), (not exist), (not exist), 101, (not exist)

(lengths: (not exist), 2, (not exist), 2, (not exist), (not exist), (not exist), 2, (not exist), 2, (not exist), (not exist), (not exist), 2, (not exist), 2, (not exist), (not exist), (not exist), 2, (not exist), 12, (not exist), (not exist), (not exist), 2, (not exist), 2, (not exist), (not exist), (not exist), 3, (not exist), 2, (not exist), (not exist), (not exist), 2, (not exist), 2, (not exist), (not exist), (not exist), 2, (not exist), 114, (not exist), (not exist), (not exist), 2, (not exist), 8, (not exist), (not exist), (not exist), 2, (not exist), 2, (not exist), (not exist), (not exist), 4, (not exist), 2, (not exist), (not exist), (not exist), 2, (not exist), 2, (not exist), (not exist), (not exist), 13, (not exist), 2, (not exist), (not exist), (not exist), 2, (not exist), 4, (not exist), (not exist), (not exist), 2, (not exist), 256, (not exist), (not exist), (not exist), 9, (not exist), 2, (not exist), (not exist), (not exist), 2, (not exist))

The smallest primes of the form 2{0}1 in base b (always minimal prime (start with b+1) base b) for 3<=b<=100 (since this family has digit "2", thus base b=2 is not available):

5, 7, (not exist), 11, 13, (not exist), 17, 19, (not exist), 23, 3457, (not exist), 29, 31, (not exist), 13555929465559461990942712143872578804076607708197374744547, 37, (not exist), 41, 43, (not exist), 47, 1153, (not exist), 53, 1459, (not exist), 59, 61, (not exist), 65537, 67, (not exist), 71, 73, (not exist), 2*38^2729+1, 79, (not exist), 83, 3529, (not exist), 89, 4051, (not exist), 82823796591884729837907950243851987042491027688029791782033968173988787397927431168748344242980462637086843228831225333542602440512725127029105275975234384910715377295392116427292929375082823988662090607733781357479215392846048752706418227733688234263166843856633793191822664770551012658601887, 97, (not exist), 101, 103, (not exist), 107, 109, (not exist), 113, 370387, (not exist), 410759, 432001, (not exist), 236522599840432068647134316649762315445236710001482847056204302486382634336257, 127, (not exist), 131, 8713, (not exist), 137, 139, (not exist), 715823, 10369, (not exist), 149, 151, (not exist), 913067, 157, (not exist), 1717986918400000000001, 163, (not exist), 167, 99574273, (not exist), 173, 15139, (not exist), 179, 181, (not exist), 1557377, 17299, (not exist), 191, 193, (not exist), 197, 199, (not exist)

(lengths: 2, 2, (not exist), 2, 2, (not exist), 2, 2, (not exist), 2, 4, (not exist), 2, 2, (not exist), 48, 2, (not exist), 2, 2, (not exist), 2, 3, (not exist), 2, 3, (not exist), 2, 2, (not exist), 4, 2, (not exist), 2, 2, (not exist), 2730, 2, (not exist), 2, 3, (not exist), 2, 3, (not exist), 176, 2, (not exist), 2, 2, (not exist), 2, 2, (not exist), 2, 4, (not exist), 4, 4, (not exist), 44, 2, (not exist), 2, 3, (not exist), 2, 2, (not exist), 4, 3, (not exist), 2, 2, (not exist), 4, 2, (not exist), 12, 2, (not exist), 2, 5, (not exist), 2, 3, (not exist), 2, 2, (not exist), 4, 3, (not exist), 2, 2, (not exist), 2, 2, (not exist))

sweety439 2021-12-20 21:42

[QUOTE=sweety439;595709]Base 7 is now completely solved!!! The minimal set is {14, 16, 23, 25, 32, 41, 43, 52, 56, 61, 65, 113, 115, 131, 133, 155, 212, 221, 304, 313, 335, 344, 346, 364, 445, 515, 533, 535, 544, 551, 553, 1022, 1051, 1112, 1202, 1211, 1222, 2111, 3031, 3055, 3334, 3503, 3505, 3545, 4504, 4555, 5011, 5455, 5545, 5554, 6034, 6634, 11111, 11201, 30011, 30101, 31001, 31111, 33001, 33311, 35555, 40054, 100121, 150001, 300053, 351101, 531101, 1100021, 33333301, 5100000001, 33333333333333331}, with 71 elements.[/QUOTE]

and now we have these solved minimal sets:

[CODE]
base: minimal sets
2: {11}
3: {12, 21, 111}
4: {11, 13, 23, 31, 221}
5: {12, 21, 23, 32, 34, 43, 104, 111, 131, 133, 313, 401, 414, 3101, 10103, 14444, 30301, 33001, 33331, 44441, 300031, 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000013}
6: {11, 15, 21, 25, 31, 35, 45, 51, 4401, 4441, 40041}
7: {14, 16, 23, 25, 32, 41, 43, 52, 56, 61, 65, 113, 115, 131, 133, 155, 212, 221, 304, 313, 335, 344, 346, 364, 445, 515, 533, 535, 544, 551, 553, 1022, 1051, 1112, 1202, 1211, 1222, 2111, 3031, 3055, 3334, 3503, 3505, 3545, 4504, 4555, 5011, 5455, 5545, 5554, 6034, 6634, 11111, 11201, 30011, 30101, 31001, 31111, 33001, 33311, 35555, 40054, 100121, 150001, 300053, 351101, 531101, 1100021, 33333301, 5100000001, 33333333333333331}
8: {13, 15, 21, 23, 27, 35, 37, 45, 51, 53, 57, 65, 73, 75, 107, 111, 117, 141, 147, 161, 177, 225, 255, 301, 343, 361, 401, 407, 417, 431, 433, 463, 467, 471, 631, 643, 661, 667, 701, 711, 717, 747, 767, 3331, 3411, 4043, 4443, 4611, 5205, 6007, 6101, 6441, 6477, 6707, 6777, 7461, 7641, 47777, 60171, 60411, 60741, 444641, 500025, 505525, 3344441, 4444477, 5500525, 5550525, 55555025, 444444441, 744444441, 77774444441, 7777777777771, 555555555555525, 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447}
10: {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027}
12: {11, 15, 17, 1B, 25, 27, 31, 35, 37, 3B, 45, 4B, 51, 57, 5B, 61, 67, 6B, 75, 81, 85, 87, 8B, 91, 95, A7, AB, B5, B7, 221, 241, 2A1, 2B1, 2BB, 401, 421, 447, 471, 497, 565, 655, 665, 701, 70B, 721, 747, 771, 77B, 797, 7A1, 7BB, 907, 90B, 9BB, A41, B21, B2B, 2001, 200B, 202B, 222B, 229B, 292B, 299B, 4441, 4707, 4777, 6A05, 6AA5, 729B, 7441, 7B41, 929B, 9777, 992B, 9947, 997B, 9997, A0A1, A201, A605, A6A5, AA65, B001, B0B1, BB01, BB41, 600A5, 7999B, 9999B, AAAA1, B04A1, B0B9B, BAA01, BAAA1, BB09B, BBBB1, 44AAA1, A00065, BBBAA1, AAA0001, B00099B, AA000001, BBBBBB99B, B0000000000000000000000000009B, 400000000000000000000000000000000000000077}
[/CODE]

and we have this condensed table:

[CODE]
b number of minimal primes base b base-b form of largest known minimal prime base b length of largest known minimal prime base b algebraic ((a×b^n+c)/d) form of largest known minimal prime base b
2 1 11 2 3
3 3 111 3 13
4 5 221 3 41
5 22 1(0^93)13 96 5^95+8
6 11 40041 5 5209
7 71 (3^16)1 17 (7^17−5)/2
8 75 (4^220)7 221 (4×8^221+17)/7
10 77 5(0^28)27 31 5×10^30+27
12 106 4(0^39)77 42 4×12^41+91
[/CODE]

Now I try to prove base 9 (in base 9 I known 149 minimal primes, I doubt that the current set may not be complete) ....

sweety439 2021-12-20 22:36

[QUOTE=sweety439;595339]The minimal prime (start with b+1) problem covers [URL="http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm"]CRUS Riesel problem[/URL] for the same base if the CK for the CRUS Riesel problem is < base, such bases 2<=b<=1024 are: (i.e. for these bases b, CRUS Riesel problem is a part of the minimal prime (start with b+1) problem in the same base b, and hence if the minimal prime (start with b+1) problem is solved, then CRUS Riesel problem in the same base b is also solved)

{14, 20, 29, 32, 34, 38, 41, 44, 47, 50, 54, 56, 59, 62, 64, 65, 68, 69, 74, 77, 81, 83, 84, 86, 89, 90, 92, 94, 98, 104, 110, 113, 114, 116, 118, 119, 122, 125, 128, 129, 131, 132, 134, 137, 139, 140, 142, 144, 146, 149, 152, 153, 155, 158, 164, 167, 170, 173, 174, 176, 178, 179, 182, 184, 185, 186, 188, 189, 194, 197, 200, 202, 203, 204, 206, 208, 209, 212, 214, 216, 218, 219, 221, 224, 227, 229, 230, 233, 234, 236, 237, 239, 242, 244, 245, 246, 248, 251, 252, 254, 257, 258, 259, 260, 263, 264, 265, 266, 269, 272, 274, 275, 278, 279, 284, 285, 286, 289, 290, 293, 294, 296, 298, 299, 300, 302, 305, 307, 308, 309, 311, 314, 317, 318, 320, 321, 322, 324, 326, 328, 329, 332, 334, 335, 338, 339, 340, 341, 344, 347, 349, 350, 353, 354, 356, 359, 362, 363, 364, 365, 368, 369, 371, 373, 374, 376, 377, 379, 380, 383, 384, 386, 387, 389, 390, 392, 394, 395, 398, 401, 402, 404, 405, 407, 410, 412, 413, 414, 416, 417, 419, 422, 424, 425, 426, 428, 429, 433, 434, 437, 439, 440, 441, 443, 444, 446, 447, 449, 450, 452, 454, 455, 458, 459, 461, 464, 467, 468, 470, 472, 473, 474, 475, 476, 479, 480, 482, 484, 488, 489, 491, 492, 493, 494, 497, 500, 503, 504, 506, 509, 510, 512, 514, 515, 516, 517, 518, 519, 521, 523, 524, 527, 528, 529, 530, 531, 533, 534, 536, 538, 539, 542, 544, 545, 548, 549, 551, 552, 554, 557, 558, 559, 560, 562, 563, 564, 566, 569, 571, 572, 573, 574, 577, 578, 579, 580, 581, 582, 584, 587, 588, 590, 593, 594, 596, 597, 599, 602, 604, 605, 608, 609, 611, 614, 615, 617, 619, 620, 622, 623, 626, 628, 629, 632, 634, 635, 636, 637, 638, 641, 643, 644, 645, 648, 649, 650, 653, 654, 656, 657, 659, 662, 664, 665, 668, 669, 670, 671, 674, 676, 677, 678, 679, 680, 681, 683, 684, 686, 688, 689, 692, 694, 695, 696, 698, 699, 701, 702, 704, 706, 707, 710, 712, 713, 714, 716, 719, 720, 722, 724, 725, 727, 729, 730, 731, 734, 737, 739, 740, 741, 743, 744, 746, 747, 749, 752, 753, 755, 758, 759, 761, 762, 764, 767, 769, 770, 773, 774, 776, 778, 779, 780, 781, 782, 783, 784, 785, 788, 789, 790, 791, 794, 797, 798, 800, 802, 803, 804, 805, 806, 809, 811, 812, 813, 814, 815, 816, 818, 819, 824, 825, 827, 828, 829, 830, 832, 833, 834, 835, 836, 839, 842, 844, 845, 846, 848, 849, 850, 851, 853, 854, 857, 859, 860, 863, 864, 866, 867, 868, 869, 870, 872, 873, 874, 875, 878, 879, 881, 883, 884, 887, 888, 889, 890, 892, 893, 894, 896, 898, 899, 900, 901, 902, 905, 908, 909, 911, 912, 914, 916, 917, 919, 920, 922, 923, 924, 926, 929, 930, 932, 934, 935, 938, 939, 941, 942, 944, 945, 947, 948, 949, 950, 951, 953, 954, 956, 958, 959, 961, 962, 964, 965, 967, 968, 969, 972, 974, 977, 978, 979, 980, 983, 984, 985, 986, 987, 988, 989, 992, 993, 994, 995, 998, 1000, 1002, 1003, 1004, 1007, 1010, 1011, 1013, 1014, 1016, 1017, 1019, 1021, 1022, 1024}

The minimal prime (start with b+1) problem covers [URL="http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm"]CRUS Sierpinski problem[/URL] for the same base if the CK for the CRUS Sierpinski problem is < base, such bases 2<=b<=1024 are: (i.e. for these bases b, CRUS Sierpinski problem is a part of the minimal prime (start with b+1) problem in the same base b, and hence if the minimal prime (start with b+1) problem is solved, then CRUS Sierpinski problem in the same base b is also solved)

{14, 20, 29, 32, 34, 38, 41, 44, 47, 50, 54, 56, 59, 62, 64, 65, 68, 69, 74, 76, 77, 83, 84, 86, 89, 90, 92, 94, 98, 101, 104, 109, 110, 113, 114, 116, 118, 119, 122, 125, 128, 129, 131, 132, 134, 137, 139, 140, 142, 144, 146, 149, 152, 153, 154, 155, 158, 159, 160, 164, 167, 169, 170, 172, 173, 174, 176, 179, 181, 182, 184, 185, 186, 188, 189, 194, 197, 200, 202, 203, 204, 206, 208, 209, 212, 214, 216, 218, 219, 220, 221, 224, 227, 229, 230, 233, 234, 236, 237, 239, 242, 244, 245, 246, 248, 251, 252, 254, 257, 258, 259, 260, 263, 264, 265, 266, 269, 272, 274, 275, 278, 279, 281, 284, 285, 289, 290, 293, 294, 296, 298, 299, 300, 302, 304, 305, 307, 308, 309, 311, 314, 317, 318, 320, 321, 322, 324, 326, 328, 329, 332, 334, 335, 338, 339, 340, 341, 344, 347, 349, 350, 353, 354, 356, 359, 362, 363, 364, 365, 368, 369, 370, 371, 373, 374, 377, 379, 380, 384, 386, 389, 390, 392, 394, 395, 398, 401, 402, 404, 405, 406, 407, 409, 410, 412, 413, 414, 416, 417, 419, 422, 424, 425, 426, 428, 429, 433, 434, 436, 437, 439, 440, 441, 443, 444, 446, 447, 449, 450, 452, 454, 455, 458, 459, 461, 464, 467, 468, 469, 470, 472, 473, 474, 475, 476, 479, 480, 482, 483, 484, 488, 489, 491, 492, 493, 494, 496, 497, 500, 501, 503, 504, 505, 506, 509, 510, 512, 514, 515, 516, 517, 518, 519, 521, 524, 526, 527, 528, 530, 531, 532, 533, 534, 536, 538, 539, 542, 544, 545, 548, 549, 550, 551, 552, 554, 557, 558, 559, 560, 562, 563, 564, 566, 569, 571, 572, 573, 574, 578, 579, 580, 581, 582, 584, 587, 588, 589, 590, 593, 594, 596, 597, 599, 601, 602, 604, 605, 608, 609, 610, 611, 614, 615, 617, 619, 620, 622, 623, 626, 629, 632, 634, 635, 636, 637, 638, 641, 643, 644, 645, 647, 648, 649, 650, 653, 654, 656, 657, 659, 662, 664, 665, 666, 668, 669, 670, 671, 674, 677, 678, 679, 680, 681, 683, 684, 686, 688, 689, 692, 695, 696, 698, 699, 701, 702, 703, 704, 706, 707, 709, 710, 712, 713, 714, 716, 718, 719, 720, 722, 724, 725, 727, 729, 730, 731, 734, 736, 737, 739, 740, 741, 743, 744, 746, 747, 748, 749, 752, 753, 754, 755, 758, 759, 761, 762, 764, 766, 767, 769, 770, 773, 774, 776, 778, 779, 780, 781, 782, 783, 784, 785, 788, 789, 790, 791, 792, 794, 797, 798, 800, 802, 803, 804, 805, 806, 809, 811, 812, 813, 814, 815, 816, 818, 819, 821, 824, 825, 827, 828, 829, 830, 832, 833, 834, 835, 836, 839, 842, 844, 845, 846, 848, 849, 850, 851, 853, 854, 857, 859, 860, 863, 864, 866, 867, 868, 869, 870, 872, 873, 874, 875, 878, 879, 881, 883, 884, 887, 888, 889, 890, 892, 893, 894, 896, 898, 899, 900, 901, 902, 903, 904, 905, 908, 909, 911, 912, 914, 916, 917, 919, 920, 922, 923, 924, 926, 929, 930, 932, 934, 935, 937, 938, 939, 941, 942, 944, 945, 947, 948, 949, 950, 951, 953, 954, 956, 958, 959, 962, 964, 965, 967, 968, 969, 972, 974, 977, 978, 979, 980, 983, 984, 985, 986, 987, 989, 992, 993, 994, 995, 998, 1000, 1001, 1004, 1006, 1007, 1009, 1010, 1011, 1013, 1014, 1016, 1019, 1022, 1024}

Families convert:

Riesel: (k-1):{(b-1)}
Sierpinski: (k):{0}:1

For these bases b, all primes in the CRUS Riesel/Sierpinski problems base b are minimal primes (start with b+1) base b, and the families are unsolved families in base b if and only if the corresponding k is remaining k (including GFNs) without known primes for the CRUS Riesel/Sierpinski problems base b.[/QUOTE]

If the 2nd/3rd/4th/... CK for base b is also < b, then the minimal prime (start with b+1) problem base b also covers the 2nd/3rd/4th/... Riesel/Sierpinski problems base b, e.g. for base 14 the 2nd CK for both sides are both 11 (<14) and for base 20 the 2nd CK for both sides are both 13 (<20), thus the minimal prime (start with b+1) problem in bases b=14 and b=20 cover the 2nd Riesel problems and the 2nd Sierpinski problems, for the same base b.

If we require k<b instead of k<CK (thus, many of the k are >CK, see [URL="https://mersenneforum.org/showthread.php?t=15188"]https://mersenneforum.org/showthread.php?t=15188[/URL]), then [I]all[/I] primes for the problem are minimal primes (start with b+1)

Sometimes (but not always), the primes for k > b can also be minimal primes (start with b+1), for b<k<b^2, let the digits of k is (x,y):

the Riesel form k*b^n-1 will be x:y-1:{b-1} in base b, thus if k-1 (=x:y-1) is not prime and both x:(b-1) and (y-1):(b-1) have no possible primes (e.g. when gcd(x,b-1) and gcd(y-1,b-1) are both >1), then the prime for this k for the Riesel problem base b is minimal prime (start with b+1)

the Sierpinski form k*b^n+1 will be x:y:{0}:1 in base b, thus if k (=x:y) is not prime and both x:{0}:1 and y:{0}:1 have no possible primes (e.g. when gcd(x+1,b-1) and gcd(y+1,b-1) are both >1), then the prime for this k for the Sierpinski problem base b is minimal prime (start with b+1)

e.g. the prime 262*17^186768+1 (found by Sierpinski problem base 17) is minimal prime (start with b+1) in base b=17, since it is F7(0^186767)1, and both F{0}1 and 7{0}1 have no possible primes, since all numbers of the form F{0}1 and 7{0}1 are even, thus cannot be primes.

e.g. the prime 178*20^176+1 (the Sierpinski problem, with k > CK) is minimal prime (start with b+1) in base b=20, since it is 8I(0^175)1, and both 8{0}1 and I{0}1 have no possible primes, since all numbers of the form 8{0}1 are divisible either by 3 or by 7, and all numbers of the form I{0}1 are divisible by 19, thus cannot be primes.

e.g. the prime 118*21^19849+1 (found by Sierpinski problem base 21) is minimal prime (start with b+1) in base b=21, since it is 5D(0^19848)1, and both 5{0}1 and D{0}1 have no possible primes, since all numbers of the form 5{0}1 and D{0}1 are even, thus cannot be primes.

e.g. the prime 590*37^22021-1 (found by Riesel problem base 37) is minimal prime (start with b+1) in base b=37, since it is R8(a^22021), and both R{a} and 8{a} have no possible primes, since all numbers of the form R{a} are divisible by 3, and all numbers of the form 8{a} are even, thus cannot be primes.

sweety439 2021-12-21 22:14

The number of possible (first digit,last digit) combo of a minimal prime (start with b+1) in base b is (b−1)*[URL="https://en.wikipedia.org/wiki/Euler%27s_totient_function"]eulerphi[/URL](b), since the first digit has b−1 choices (all digits except 0 can be the first digit), and the last digit has eulerphi(b) choices (only digits coprime to b (i.e. the digits in the reduced residue system mod b) can be the last digit), by the [URL="https://en.wikipedia.org/wiki/Rule_of_product"]rule of product[/URL], there are (b−1)*eulerphi(b) choices of the (first digit,last digit) combo.

Thus, (b−1)*eulerphi(b) is the relative hardness for (solving the minimal prime (start with b+1) problem in) base b, and both "the number of minimal primes (start with b+1) base b" and "the length of the largest minimal prime (start with b+1) base b" are [URL="https://en.wikipedia.org/wiki/Asymptotic_analysis"]roughly[/URL] e^(gamma*(b−1)*eulerphi(b)), where e is [URL="https://en.wikipedia.org/wiki/E_(mathematical_constant)"]the base of the natural logarithm[/URL] and gamma is [URL="https://en.wikipedia.org/wiki/Euler%27s_constant"]the Euler–Mascheroni constant[/URL].

Start with b+1 makes the formula of the number of possible (first digit,last digit) combo of a minimal prime in base b more simple and [URL="https://en.wikipedia.org/wiki/Smooth_number"]smooth number[/URL], since if start with b, then when b is prime, there is an additional possible (first digit,last digit) combo: (1,0), and hence the formula will be (b−1)*eulerphi(b)+1 if b is prime, or (b−1)*eulerphi(b) if b is composite (the fully formula will be (b−1)*eulerphi(b)+isprime(b) or (b−1)*eulerphi(b)+floor((b-eulerphi(b)) / (b-1))), which is more complex, and if start with 1 (i.e. the [URL="https://cs.uwaterloo.ca/~cbright/reports/mepn.pdf"]original minimal prime problem[/URL]), the formula is much more complex, since the prime digits (i.e. the single-digit primes) should be excluded, and (for such prime >b) the first digit has b−1−[URL="https://oeis.org/A000720"]pi[/URL](b) choices, and the last digit has [URL="https://oeis.org/A048864"]A048864[/URL](b) choices, by the [URL="https://en.wikipedia.org/wiki/Rule_of_product"]rule of product[/URL], there are (b−1−[URL="https://oeis.org/A000720"]pi[/URL](b))*([URL="https://oeis.org/A048864"]A048864[/URL](b)) choices of the (first digit,last digit) combo (if for such prime >=b instead of >b, then the formula will be (b−1−[URL="https://oeis.org/A000720"]pi[/URL](b))*([URL="https://oeis.org/A048864"]A048864[/URL](b))+1 if b is prime, or (b−1−[URL="https://oeis.org/A000720"]pi[/URL](b))*([URL="https://oeis.org/A048864"]A048864[/URL](b)) if b is composite), which is much more complex, (also, the possible (first digit,last digit) combo for a prime >b in base b are exactly the (first digit,last digit) combos which there are infinitely many primes have, while this is not true when the requiring is prime>=b or prime>=2 instead of prime>b, since this will contain the prime factors of b, which are not coprime to b and hence there is only this prime (and not infinitely many primes) have this (first digit,last digit) combo) thus the problem in this forum (i.e. the minimal prime (start with b+1) problem) is much better than the original minimal prime problem.

Another reason is that this problem is regardless [URL="https://primes.utm.edu/notes/faq/one.html"]whether 1 is considered as prime or not[/URL], i.e. no matter 1 is considered as prime or not prime (in the beginning of the 20th century, 1 is regarded as prime, e.g. in base 10, if 1 is considered as prime, then the set in the original minimal prime problem is {1, 2, 3, 5, 7, 89, 409, 449, 499, 6469, 6949, 9049, 9649, 9949, 60649, 666649, 946669, 60000049, 66000049, 66600049}, while if 1 is not considered as prime, then the set in the original minimal prime problem is {2, 3, 5, 7, 11, 19, 41, 61, 89, 409, 449, 499, 881, 991, 6469, 6949, 9001, 9049, 9649, 9949, 60649, 666649, 946669, 60000049, 66000049, 66600049}, however, in base 10, the set in this new problem is always {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027}, no matter 1 is considered as prime or not prime.

Another reason is that if we include the prime = b (i.e. the prime "10") or the primes < b (i.e. the single-digit primes), then some properties in [URL="https://mersenneforum.org/showpost.php?p=593116&postcount=208"]this post[/URL] will be incorrect.

Thus, start with b+1 (instead of b, 2, 1, b^2, b^2+1, b+2, 2*b, 2*b+1, ...) makes this minimal prime problem most beautiful (prime = b (i.e. the prime "10") and primes < b (i.e. single-digit primes) need to be excluded, while the prime = b+1 (i.e. the prime "11") and other two-digit primes and other repunit primes do not need).

Reference: [URL="https://mersenneforum.org/showpost.php?p=562832&postcount=52"]https://mersenneforum.org/showpost.php?p=562832&postcount=52[/URL]

sweety439 2021-12-23 09:54

2 Attachment(s)
Update new files, both PDF and WORD (zipped)

sweety439 2021-12-26 05:39

Some sequences from this problem:

* Number of minimal primes (start with b+1) base b for b = 2, 3, 4, ...:

1, 3, 5, 22, 11, 71, 75, 149 (conjectured), 77, >=914, 106, >=2497, >=606, >=1212, >=2045, ...

* Largest minimal prime (start with b+1) base b (written in base 10) for b = 2, 3, 4, ...:

3, 13, 41, 2524354896707237777317531408904915934954260592348873615264892578133, 5209, 116315256993601, 21870014779720278736374332149114462520188534743847615898363462279537144492484599310778624146468224150373895489844303219383829573677353011540369291867378470695590964880740521967077028064041941947533607, 2502834912130941228981594838568047007685391589426558809617400213418282527350716547775328604270549538213046909875482264691462374583011445603072337046349820356657444760155517460988693569161900376763260063042633631135244220815105656622524951200259007718495917864117405547147590222419375058379574594080658297207604131854610535599489346198821817431117091965632520151514541808501416706046842760217146877727723295324835109217998918911955471784007676341431748342770193933000722184682405269827878905265116459080907918793395867167367499638489767391992537709059524884652896769102415537839348416960850467681983060002162390800572925622177581991626917601755614769750667555366972378636352064082870607863678474941101871860712524059504055163796718520779508406291839451362511818447413705902997975107357845560785637900417278099493046212539395848504694452701430507487110023093092162881186398359015719438275294036102483885424712035725605375511622913992739195207856408178305713981491162803603900452533739457555869699509477310280520560512672265632708352842398727598199181486638532882325131901716020795309902315550951482539354571213, 5000000000000000000000000000027, ?, 705490352625161496279722666407220454094798939, ...

* Largest minimal prime (start with b+1) base b (written in base b) for b = 2, 3, 4, ...:

11, 111, 221, 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000013, 40041, 33333333333333331, 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447, 300000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000011, 5000000000000000000000000000027, ?, 400000000000000000000000000000000000000077, ...

* Length of largest minimal prime (start with b+1) base b for b = 2, 3, 4, ...:

2, 3, 3, 96, 5, 17, 221, 1161, 31, >=1013, 42, >=32021, >=19699, >=157, >=32235, ...

* The two digit number in base b from the first and last digit of the largest minimal prime (start with b+1) base b for b = 2, 3, 4, ...:

11, 11, 21, 13, 41, 31, 47, 31, 57, ? (likely 57, from the unsolved family 5777...777), 47, ...

The same sequence written in base 10 (will be a number between b and b^2 which is coprime with b):

3, 4, 9, 8, 25, 22, 39, 28, 57, ? (likely 62, from the unsolved family 5777...777), 55, ...

* The two digit number in base b from the first and last digit such that there are the most minimal primes (start with b+1) base b having this (first digit,last digit) combo (using & if there is a tie):

11 (the prime 11)
11 (the prime 111) & 12 (the prime 12) & 21 (the prime 21)
11 (the prime 11) & 13 (the prime 13) & 21 (the prime 221) & 23 (the prime 23) & 31 (the prime 31)
31 (the primes 3101, 30301, 33001, 33331, 300031)
41 (the primes 4401, 4441, 40041)
31 (the primes 3031, 30011, 30101, 31001, 31111, 33001, 33311, 351101, 33333301, 33333333333333331)
55 (the primes 5205, 500025, 505525, 5500525, 5550525, 55555025, 555555555555525) & 61 (the primes 631, 661, 6101, 6441, 60171, 60411, 60741) & 71 (the primes 701, 711, 7461, 7641, 744444441, 77774444441, 7777777777771)
51 (the primes 531, 5051, 5071, 5101, 5501, 50161, 50611, 55111, 55551, 57061, 555611, 51116111, 5161111111, 5700000000001, 5111111111111161, 56111111111111111111111111111111111111)
51 (the primes 521, 5051, 5081, 5501, 5581, 5801, 5851, 555555555551)
?
B1 (the primes B21, B001, B0B1, BB01, BB41, B04A1, BAA01, BAAA1, BBBB1, BBBAA1)

The same sequence written in base 10 (will be numbers between b and b^2 which is coprime with b):

3, 4&5&7, 5&7&9&11&13, 16, 25, 22, 45&49&57, 46, 51, ?, 133

sweety439 2021-12-26 14:52

For a minimal prime (start with b+1), we can let its first digit be d1 and its last digit be d2, and consider the two-digit integer (d1,d2) in base b....

If two-digit integer (d1,d2) is not coprime to b (equivalently, d2 is not coprime to b), then there cannot be prime >b with first digit d1 and last digit d2, thus there is no minimal prime (start with b+1) with first digit d1 and last digit d2 (thus, d2 can only be a digit coprime to b, of course d1 cannot be 0)

If two-digit integer (d1,d2) is prime number, since (d1,d2) is a subsequence of any prime with first digit d1 and last digit d2, the prime (d1,d2) is the only minimal prime (start with b+1) with first digit d1 and last digit d2

Thus, for any minimal prime (start with b+1) >=3 digits with first digit d1 and last digit d2, (d1,d2) must be composite numbers coprime to b (note that the range of (d1,d2) is >b and <=b^2)

However, not all (d1,d2) (>b and <=b^2) coprime to b have a minimal prime (start with b+1) with first digit d1 and last digit d2 (if (d1,d2) is itself prime, then (d1,d2) is the only minimal prime (start with b+1) with first digit d1 and last digit d2), e.g.

* In base 3, there are no minimal primes (start with b+1) with first digit 2 and last digit 2
* In base 4, there are no minimal primes (start with b+1) with first digit 3 and last digit 3
* In base 5, there are no minimal primes (start with b+1) with first digit 2 and last digit 2
* In base 5, there are no minimal primes (start with b+1) with first digit 2 and last digit 4
* In base 5, there are no minimal primes (start with b+1) with first digit 4 and last digit 2
* In base 5, there are no minimal primes (start with b+1) with first digit 4 and last digit 4
* In base 6, there are no minimal primes (start with b+1) with first digit 5 and last digit 5
* In base 7, there are no minimal primes (start with b+1) with first digit 2 and last digit 4
* In base 7, there are no minimal primes (start with b+1) with first digit 2 and last digit 6
* In base 7, there are no minimal primes (start with b+1) with first digit 4 and last digit 2
* In base 7, there are no minimal primes (start with b+1) with first digit 4 and last digit 6
* In base 7, there are no minimal primes (start with b+1) with first digit 6 and last digit 2
* In base 7, there are no minimal primes (start with b+1) with first digit 6 and last digit 3
* In base 7, there are no minimal primes (start with b+1) with first digit 6 and last digit 6
* In base 9, there are no minimal primes (start with b+1) with first digit 2 and last digit 4
* In base 9, there are no minimal primes (start with b+1) with first digit 4 and last digit 2
* In base 9, there are no minimal primes (start with b+1) with first digit 4 and last digit 4
* In base 9, there are no minimal primes (start with b+1) with first digit 6 and last digit 2
* In base 9, there are no minimal primes (start with b+1) with first digit 6 and last digit 4
* In base 9, there are no minimal primes (start with b+1) with first digit 8 and last digit 2
* In base 10, there are no minimal primes (start with b+1) with first digit 3 and last digit 3
* In base 10, there are no minimal primes (start with b+1) with first digit 6 and last digit 3
* In base 10, there are no minimal primes (start with b+1) with first digit 9 and last digit 3

(there are all such examples for bases b<=10)

Conjectures:

* There are only finitely many such examples (i.e. for every [I]enough large[/I] base b, for any (d1,d2) digit pair such that d1 is not 0 and d2 is coprime to b, there is a minimal prime (start with b+1) with first digit d1 and last digit d2)
* All such examples have gcd(d1,d2,b-1) > 1 (i.e. there is a prime number which divides d1, d2, and b-1, simultaneously)

sweety439 2021-12-29 07:20

[QUOTE=sweety439;595887]The number of possible (first digit,last digit) combo of a minimal prime (start with b+1) in base b is (b−1)*[URL="https://en.wikipedia.org/wiki/Euler%27s_totient_function"]eulerphi[/URL](b), since the first digit has b−1 choices (all digits except 0 can be the first digit), and the last digit has eulerphi(b) choices (only digits coprime to b (i.e. the digits in the reduced residue system mod b) can be the last digit), by the [URL="https://en.wikipedia.org/wiki/Rule_of_product"]rule of product[/URL], there are (b−1)*eulerphi(b) choices of the (first digit,last digit) combo.

Thus, (b−1)*eulerphi(b) is the relative hardness for (solving the minimal prime (start with b+1) problem in) base b, and both "the number of minimal primes (start with b+1) base b" and "the length of the largest minimal prime (start with b+1) base b" are [URL="https://en.wikipedia.org/wiki/Asymptotic_analysis"]roughly[/URL] e^(gamma*(b−1)*eulerphi(b)), where e is [URL="https://en.wikipedia.org/wiki/E_(mathematical_constant)"]the base of the natural logarithm[/URL] and gamma is [URL="https://en.wikipedia.org/wiki/Euler%27s_constant"]the Euler–Mascheroni constant[/URL].[/QUOTE]

However, in general, odd bases b have lower [URL="https://www.rieselprime.de/ziki/Nash_weight"]weight[/URL] than even bases, since they have many families which 2 divides half of the numbers in the family, similarly, bases b == 2 mod 3 have lower weight than other bases, since they have many families which 3 divides half of the numbers in the family (thus, bases b == 5 mod 6 have extremely low weight, and the original minimal prime (i.e. prime > base is not need) for base b = 23 is solved is very unusual, since this base is low weight), also, bases b == 1 mod 3 have lower weight than bases b == 0 mod 3, since they have many families which 3 divides a third of the numbers in the family, thus in these bases, the estimation value e^(gamma*(b−1)*eulerphi(b)) will be much smaller than the actual values, in fact, bases b with b+1 having a lot of prime factors (such as b = 29, 41, 59) will be much lower weight, and bases b with b+1 prime power (especially bases b with b+1 prime) have high weight, thus in general, if [URL="https://oeis.org/A001221"]omega[/URL](b+1) is larger, then base b have smaller weight, and if [URL="https://oeis.org/A020639"]lpf[/URL](b+1) is larger, then b have larger weight (however, these results should be combined the formula e^(gamma*(b−1)*eulerphi(b)), the better formula of the relative hardness is (e^(gamma*(b−1)*eulerphi(b)))/W, where W is the weight of base b, W [URL="https://en.wikipedia.org/wiki/Approximation"]≈[/URL] lpf(b+1)/omega(b+1), thus this formula of the relative hardness is (e^(gamma*(b−1)*eulerphi(b)))*omega(b+1)/lpf(b+1)

(of course, primes p dividing the base (b) cannot divide any number in a family, or p will divide all numbers in this family and thus this family can be ruled out as only contain composite numbers (only count numbers > base (b)))

e.g. only consider the primes p = 2 and p = 3:

[CODE]
b == 0 mod 6 weight = 1
b == 3 mod 6 weight = 3/4, since 1/2 of the families have 2 dividing 1/2 of the numbers
b == 2 mod 6 weight = 5/6, since 1/3 of the families have 3 dividing 1/2 of the numbers
b == 4 mod 6 weight = 8/9, since 1/3 of the families have 3 dividing 1/3 of the numbers
b == 5 mod 6 weight = 5/8 (3/4 * 5/6)
b == 1 mod 6 weight = 2/3 (3/4 * 8/9)
[/CODE]

weight for base b (in the [URL="https://en.wikipedia.org/wiki/Ring_(mathematics)"]ring[/URL] Z6 (integers mod 6)): 0 > 4 > 2 > 3 > 1 > 5

[CODE]
base (b) for fixed prime factor p
== 1 mod 2 2 divides 1/2 of numbers in a family
== 1 mod 3 3 divides 1/3 of numbers in a family
== 2 mod 3 3 divides 1/2 of numbers in a family
== 1 mod 5 5 divides 1/5 of numbers in a family
== 2, 3 mod 5 5 divides 1/4 of numbers in a family
== 4 mod 5 5 divides 1/2 of numbers in a family
== 1 mod 7 7 divides 1/7 of numbers in a family
== 2, 4 mod 7 7 divides 1/3 of numbers in a family
== 3, 5 mod 7 7 divides 1/6 of numbers in a family
== 6 mod 7 7 divides 1/2 of numbers in a family
[/CODE]

The estimate of the weight is similar to [URL="https://en.wikipedia.org/wiki/Artin%27s_conjecture_on_primitive_roots"]the formula of the density of primes p which have primitive root b[/URL] when b is not in [URL="https://oeis.org/A085397"]https://oeis.org/A085397[/URL] (when b is in [URL="https://oeis.org/A085397"]https://oeis.org/A085397[/URL], then the density is [URL="https://oeis.org/A005596"]C_Artin (the Artin constant)[/URL]) and [URL="https://en.wikipedia.org/wiki/Chebyshev%27s_bias"]Chebyshev's bias[/URL] extension to higher power residue (related to [URL="https://en.wikipedia.org/wiki/Dirichlet_character"]Dirichlet character[/URL] and [URL="https://en.wikipedia.org/wiki/Power_residue_symbol"]power residue symbol[/URL]).

sweety439 2021-12-30 19:17

[QUOTE=sweety439;593784]they are the families (generalized octagonal number){1}, and since base b families can be converted to base b^n families for n>1 (the repeating digit (i.e. the digit in {}) will be multiple of Rn(b), where Rn(b) is the base b [URL="https://en.wikipedia.org/wiki/Repunit"]repunit[/URL] with length n), these base 4 families can be converted to base 16 (=4^2) families, {1} in base 4 = 1{5} in base 16, 20{1} in base 4 = 8{5} in base 16, 100{1} in base 4 = 10{5} in base 16, 220{1} in base 4 = A1{5} in base 16, etc. ([URL="https://www.rose-hulman.edu/~rickert/Compositeseq/#b9d4"]reference for base b families converted to base b^n families for n>1, the case base 3 families *{1} converted to base 9=3^2 families *{4}[/URL])[/QUOTE]

Exactly, all minimal primes (start with b+1) base b are also minimal primes (start with b'+1) base b' = b^r with integer r>1, if they are > b', since all [URL="https://en.wikipedia.org/wiki/Subsequence"]subsequences[/URL] of [URL="https://en.wikipedia.org/wiki/Radix"]base[/URL] b^r representation are also subsequences of base b representation, of the same number.

e.g. the prime 5^95+8 is minimal prime (start with b+1) in base b = 5, thus this prime is also minimal primes (start with b+1) in base b = 5^r for r = 2, 3, 4, ..., 95 (not for r>95, since in such bases, this prime is not > base), similarly, the prime 5*10^30+27 is minimal prime (start with b+1) in base b = 10, thus this prime is also minimal primes (start with b+1) in base b = 10^r for r = 2, 3, 4, ..., 30, also, the prime (2^665+17)/7 is minimal prime (start with b+1) in base b = 8, thus this prime is also minimal primes (start with b+1) in base b = 8^r for r = 2, 3, 4, ..., 220 (this prime has 221 digits in base 8), but possible not for base b = 2^r with r not divisible by 3, since such bases are not [I]integer[/I] powers of 8, but only [I]rational[/I] powers of 8, and subsequences of representations in these bases need not to be subsequences of base 8 representations.

sweety439 2021-12-30 19:45

[QUOTE=sweety439;582061]We can use the sense of [URL="http://www.iakovlev.org/zip/riesel2.pdf"]http://www.iakovlev.org/zip/riesel2.pdf[/URL], [URL="https://scholar.google.com/scholar_url?url=https://www.sciencedirect.com/science/article/pii/S0022314X08000462/pdf%3Fmd5%3Dcb11465f6eb6873d749c67b2b31dbb1d%26pid%3D1-s2.0-S0022314X08000462-main.pdf%26_valck%3D1&hl=zh-TW&sa=T&oi=ucasa&ct=ufr&ei=TnR0YYi5IoP2yASq-aXQCA&scisig=AAGBfm1x9DSu578ydrXxfMnrRUPp1l8rcA"]https://scholar.google.com/scholar_url?url=https://www.sciencedirect.com/science/article/pii/S0022314X08000462/pdf%3Fmd5%3Dcb11465f6eb6873d749c67b2b31dbb1d%26pid%3D1-s2.0-S0022314X08000462-main.pdf%26_valck%3D1&hl=zh-TW&sa=T&oi=ucasa&ct=ufr&ei=TnR0YYi5IoP2yASq-aXQCA&scisig=AAGBfm1x9DSu578ydrXxfMnrRUPp1l8rcA[/URL], [URL="https://people.math.sc.edu/filaseta/papers/SierpinskiEtCoPapNew.pdf"]https://people.math.sc.edu/filaseta/papers/SierpinskiEtCoPapNew.pdf[/URL], [URL="https://mersenneforum.org/showpost.php?p=138737&postcount=24"]https://mersenneforum.org/showpost.php?p=138737&postcount=24[/URL], [URL="https://mersenneforum.org/showpost.php?p=153508&postcount=147"]https://mersenneforum.org/showpost.php?p=153508&postcount=147[/URL], [URL="https://mersenneforum.org/showpost.php?p=155243&postcount=176"]https://mersenneforum.org/showpost.php?p=155243&postcount=176[/URL], [URL="https://mersenneforum.org/showpost.php?p=549958&postcount=867"]https://mersenneforum.org/showpost.php?p=549958&postcount=867[/URL], [URL="https://mersenneforum.org/showpost.php?p=550208&postcount=883"]https://mersenneforum.org/showpost.php?p=550208&postcount=883[/URL], [URL="https://mersenneforum.org/showpost.php?p=550364&postcount=891"]https://mersenneforum.org/showpost.php?p=550364&postcount=891[/URL], [URL="https://mersenneforum.org/showpost.php?p=550372&postcount=893"]https://mersenneforum.org/showpost.php?p=550372&postcount=893[/URL], [URL="https://stdkmd.net/nrr/1/11113.htm#prime_period"]https://stdkmd.net/nrr/1/11113.htm#prime_period[/URL], [URL="https://stdkmd.net/nrr/1/13333.htm#prime_period"]https://stdkmd.net/nrr/1/13333.htm#prime_period[/URL], [URL="https://stdkmd.net/nrr/1/10003.htm#prime_period"]https://stdkmd.net/nrr/1/10003.htm#prime_period[/URL], [URL="https://mersenneforum.org/showpost.php?p=452132&postcount=66"]https://mersenneforum.org/showpost.php?p=452132&postcount=66[/URL] ("Mersenne number" can be generated to generalized repunit number (i.e. (a*b^n+c)/gcd(a+c,b-1) can be written as (x^(y*n+z)-1)/(x-1) where x is a root of the base (b)) with x>2 (the "Mersenne number" is the x=2 case) and generalized Wagstaff number (i.e. (a*b^n+c)/gcd(a+c,b-1) can be written as (x^(2*(y*n+z)+1)+1)/(x+1) where x is a root of the base (b)), and "GFN" can be generated to generalized half Fermat number (i.e. (a*b^n+c)/gcd(a+c,b-1) can be written as (x^(y*n+z)+1)/2 where x is a root of the base (b)) with odd x (the "GFN" is x^(y*n+z)+1 with even x)), to conclude that the unsolved families (unsolved families are families which are neither primes (>base) found nor can be proven to contain no primes > base) eventually should yield a prime, this can be calculated for the [URL="https://www.rieselprime.de/ziki/Nash_weight"]Nash weight[/URL] (or the [URL="https://stdkmd.net/nrr/prime/primedifficulty.txt"]difficulty[/URL]), families which can be proven to contain no primes > base have Nash weight (or difficulty) 0, e.g. for the base 11 unsolved family 5{7}:

5(7^n) = (57*11^n-7)/10, but there is no n satisfying that 57*11^n and 7 are both r-th powers for some r>1 (since 7 is not perfect power), nor there is n satisfying that 57*11^n and -7 are (one is 4th power, another is of the form 4*m^4) (since -7 is neither 4th power nor of the form 4*m^4), thus, 5(7^n) has no algebra factors for any n, thus 5(7^n) eventually should yield a prime unless it can be proven to contain no primes > base using covering congruence, and we have:

5(7^n) is divisible by 2 for n == 1 mod 2
5(7^n) is divisible by 13 for n == 2 mod 12
5(7^n) is divisible by 17 for n == 4 mod 16
5(7^n) is divisible by 5 for n == 0 mod 5
5(7^n) is divisible by 23 for n == 6 mod 22
5(7^n) is divisible by 601 for n == 8 mod 600
5(7^n) is divisible by 97 for n == 12 mod 48
5(7^n) is divisible by 1279 for n == 16 mod 426
...

and it does not appear to be any covering set of primes (and its Nash weight (or difficulty) is positive, and it has prime candidate), so there must be a prime at some point.

(see post [URL="https://mersenneforum.org/showpost.php?p=568675&postcount=103"]#103[/URL] for examples of families which can be proven to contain no primes > base)[/QUOTE]

Also, for the largest minimal prime (start with b+1) in bases 5, 8, 10, 12 (their corresponding families are hence the "last unsolved family", i.e. the family which is the last (for the corresponding base) to be eliminated, if we test the primality of larger and larger numbers in these families), we can also use this sense to show they have Nash weight (or difficulty) > 0

The largest minimal prime (start with b+1) in base b = 5 is 1(0^93)13, and its family is 1{0}13, and we have:

1(0^n)13 has sum-of-cubes algebraic factorization for n == 1 mod 3
1(0^n)13 is divisible by 3 for n == 0 mod 2
1(0^n)13 is divisible by 7 for n == 1 mod 6
1(0^n)13 is divisible by 13 for n == 3 mod 4
1(0^n)13 is divisible by 11 for n == 0 mod 5
1(0^n)13 is divisible by 691 for n == 9 mod 115
1(0^n)13 is divisible by 41 for n == 17 mod 20

and it is not appear to be a full covering set, thus there must be a prime at some point, exactly, 1(0^93)13 is prime.

The largest minimal prime (start with b+1) in base b = 8 is (4^220)7, and its family is {4}7, and we have:

(4^n)7 is divisible by 3 for n == 1 mod 2
(4^n)7 is divisible by 5 for n == 2 mod 4
(4^n)7 is divisible by 61 for n == 4 mod 20
(4^n)7 is divisible by 7 for n == 0 mod 7
(4^n)7 is divisible by 13 for n == 1 mod 4
(4^n)7 is divisible by 7333 for n == 8 mod 2444
(4^n)7 is divisible by 563 for n == 12 mod 562
(4^n)7 is divisible by 23 for n == 5 mod 11
(4^n)7 is divisible by 137 for n == 20 mod 68
(4^n)7 is divisible by 1747 for n == 32 mod 582

and it is not appear to be a full covering set, thus there must be a prime at some point, exactly, (4^220)7 is prime.

The largest minimal prime (start with b+1) in base b = 10 is 5(0^28)27, and its family is 5{0}27, and we have:

5(0^n)27 is divisible by 11 if n == 1 mod 2
5(0^n)27 is divisible by 17 if n == 0 mod 16
5(0^n)27 is divisible by 31 if n == 0 mod 15
5(0^n)27 is divisible by 19 if n == 2 mod 18
5(0^n)27 is divisible by 7 if n == 5 mod 6
5(0^n)27 is divisible by 241 if n == 4 mod 30
5(0^n)27 is divisible by 113 if n == 6 mod 112
5(0^n)27 is divisible by 3511 if n == 8 mod 1755
5(0^n)27 is divisible by 23 if n == 10 mod 22

and it is not appear to be a full covering set, thus there must be a prime at some point, exactly, 5(0^28)27 is prime.

The largest minimal prime (start with b+1) in base b = 12 is 4(0^39)77, and its family is 4{0}77, and we have:

4(0^n)77 is divisible by 5 if n == 2 mod 4
4(0^n)77 is divisible by 29 if n == 0 mod 4
4(0^n)77 is divisible by 23 if n == 0 mod 11
4(0^n)77 is divisible by 47 if n == 1 mod 23
4(0^n)77 is divisible by 67 if n == 3 mod 66
4(0^n)77 is divisible by 19 if n == 4 mod 6
4(0^n)77 is divisible by 17 if n == 5 mod 16
4(0^n)77 is divisible by 4759 if n == 7 mod 4758
4(0^n)77 is divisible by 443629 if n == 9 mod 221814

and it is not appear to be a full covering set, thus there must be a prime at some point, exactly, 4(0^39)77 is prime.

sweety439 2022-01-02 10:18

The possible values of the divisor (i.e. the "d" in (a*b^n+c)/d) are exactly the [URL="https://en.wikipedia.org/wiki/Table_of_divisors"]divisors of b-1[/URL], in fact, the divisor is (b-1)/gcd(repeating digit,b-1), where "repeating digit" is the "y" in x{y}z, thus, there is no divisor (i.e. the divisor is 1, the formula is a*b^n+c) if and only if the "repeating digit" is either 0 or b-1

sweety439 2022-01-02 19:45

I try to prove base 9 (this base is much harder than bases 7 and 12, see [URL="https://mersenneforum.org/showpost.php?p=595887&postcount=250"]this post[/URL]: the relative hardness of base b is (b−1)*eulerphi(b) = [URL="https://oeis.org/A062955"]A062955[/URL](b), the value of it for bases 2<=b<=64 are:

[CODE]
2,1
3,4
4,6
5,16
6,10
7,36
8,28
9,48
10,36
11,100
12,44
13,144
14,78
15,112
16,120
17,256
18,102
19,324
20,152
21,240
22,210
23,484
24,184
25,480
26,300
27,468
28,324
29,784
30,232
31,900
32,496
33,640
34,528
35,816
36,420
37,1296
38,666
39,912
40,624
41,1600
42,492
43,1764
44,860
45,1056
46,990
47,2116
48,752
49,2016
50,980
51,1600
52,1224
53,2704
54,954
55,2160
56,1320
57,2016
58,1596
59,3364
60,944
61,3600
62,1830
63,2232
64,2016
65,3072
66,1300
67,4356
68,2144
69,2992
70,1656
71,4900
72,1704
73,5184
74,2628
75,2960
76,2700
77,4560
78,1848
79,6084
80,2528
81,4320
82,3240
83,6724
84,1992
85,5376
86,3570
87,4816
88,3480
89,7744
90,2136
91,6480
92,4004
93,5520
94,4278
95,6768
96,3040
97,9216
98,4074
99,5880
100,3960
101,10000
102,3232
103,10404
104,4944
105,4992
106,5460
107,11236
108,3852
109,11664
110,4360
111,7920
112,5328
113,12544
114,4068
115,10032
116,6440
117,8352
118,6786
119,11328
120,3808
121,13200
122,7260
123,9760
124,7380
125,12400
126,4500
127,15876
128,8128
129,10752
130,6192
131,16900
132,5240
133,14256
134,8778
135,9648
136,8640
137,18496
138,6028
139,19044
140,6672
141,12880
142,9870
143,17040
144,6864
145,16128
146,10440
147,12264
148,10584
149,21904
150,5960
151,22500
152,10872
153,14592
154,9180
155,18480
156,7440
157,24336
158,12246
159,16432
160,10176
161,21120
162,8694
163,26244
164,13040
165,13120
166,13530
167,27556
168,8016
169,26208
170,10816
171,18360
172,14364
173,29584
174,9688
175,20880
176,14000
177,20416
178,15576
179,31684
180,8592
181,32400
182,13032
183,21840
184,16104
185,26496
186,11100
187,29760
188,17204
189,20304
190,13608
191,36100
192,12224
193,36864
194,18528
195,18624
196,16380
197,38416
198,11820
199,39204
200,15920
[/CODE]

The bases b sorted by relative hardness (i.e. (b−1)*eulerphi(b), which is the number of possible (first digit,last digit) combo for prime > b in base b) are:

[CODE]
2,1
3,4
4,6
6,10
5,16
8,28
7,36
10,36
12,44
9,48
14,78
11,100
18,102
15,112
16,120
13,144
20,152
24,184
22,210
30,232
21,240
17,256
26,300
19,324
28,324
36,420
27,468
25,480
23,484
42,492
32,496
34,528
40,624
33,640
38,666
48,752
29,784
35,816
44,860
31,900
39,912
60,944
54,954
50,980
46,990
45,1056
52,1224
37,1296
66,1300
56,1320
58,1596
41,1600
51,1600
70,1656
72,1704
43,1764
62,1830
78,1848
84,1992
49,2016
57,2016
64,2016
47,2116
90,2136
68,2144
55,2160
63,2232
80,2528
74,2628
76,2700
53,2704
75,2960
69,2992
96,3040
65,3072
102,3232
82,3240
59,3364
88,3480
86,3570
61,3600
120,3808
108,3852
100,3960
92,4004
114,4068
98,4074
94,4278
81,4320
67,4356
110,4360
126,4500
77,4560
87,4816
71,4900
104,4944
105,4992
73,5184
132,5240
112,5328
85,5376
106,5460
93,5520
99,5880
150,5960
138,6028
79,6084
130,6192
116,6440
91,6480
140,6672
83,6724
95,6768
118,6786
144,6864
122,7260
124,7380
156,7440
89,7744
111,7920
168,8016
128,8128
117,8352
180,8592
136,8640
162,8694
134,8778
154,9180
97,9216
135,9648
174,9688
123,9760
142,9870
101,10000
[/CODE]

sweety439 2022-01-02 20:17

The possible (first digit,last digit) for an element with >=3 digits in the minimal set of the strings for primes with at least two digits are

(1,1), (1,2), (1,4), (1,5), (1,7), (1,8), (2,1), (2,2), (2,4), (2,5), (2,7), (2,8), (3,1), (3,2), (3,4), (3,5), (3,7), (3,8), (4,1), (4,2), (4,4), (4,5), (4,7), (4,8), (5,1), (5,2), (5,4), (5,5), (5,7), (5,8), (6,1), (6,2), (6,4), (6,5), (6,7), (6,8), (7,1), (7,2), (7,4), (7,5), (7,7), (7,8), (8,1), (8,2), (8,4), (8,5), (8,7), (8,8)

* Case (1,1):

** Since 12, 14, 18, 21, 41, 81, [B]131[/B], [B]151[/B] are primes, we only need to consider the family 1{0,1,6,7}1 (since any digits 2, 3, 4, 5, 8 between them will produce smaller primes)

*** If there are at least one 0's and at least one 1's in {}, then either [B]1011[/B] or [B]1101[/B] will be a subsequence.

*** If there are at least one 1's and at least one 7's in {}, then either 117 or 711 will be a subsequence.

*** If there are at least one 1's and no 0's or 7's in {}, then the family will be 1{1,6}1{1,6}1

**** If there are at least two 6's, then 661 will be a subsequence.

**** If there are exactly one 6's, then the family will be {1}6{1}

***** All numbers of the form {1}6{1} are divisible by 2 (if the total number of 1's is even) or 5 (if the total number of 1's is odd), thus cannot be prime.

**** If there are no 6's, then the family will be {1}

***** All numbers of the form {1} factored as (10^n-1)/8 = (3^n-1)/2 * (3^n+1)/4 (if n is odd) or (3^n-1)/4 * (3^n+1)/2 (if n is even), thus cannot be prime.

*** If there are no 1's in {}, then the family will be 1{0,6,7}1

**** Since 601, 661, 67 are primes, we only need to consider the families 1{0,7}1 and 1{0,7}61 (since any digits combo 60, 66, 67 between them will produce small primes)

***** For the 1{0,7}1 family, since 177 and 771 are primes, we only need to consider the families 1{0}1 and 1{0}7{0}1 (since any digits combo 77 between them will produce small primes)

****** All numbers of the form 1{0}1 are divisible by 2, thus cannot be prime.

****** For the 1{0}7{0}1 family, since [B]1701[/B] is prime, we only need to consider the family 1{0}71 (since any digits 0 between (1{0}7,1) will produce small primes)

******* The smallest prime of the form 1{0}71 is [B]100071[/B]

***** For the 1{0,7}61 family, since 177 and 771 are primes, we only need to consider the families 1{0}61 and 1{0}7{0}61 (since any digits combo 77 between them will produce small primes)

****** All numbers of the form 1{0}61 are divisible by 2, thus cannot be prime.

****** For the 1{0}7{0}61 family, since 1701 is prime, we only need to consider the family 1{0}761 (since any digits 0 between (1{0}7,61) will produce small primes)

******* The smallest prime of the form 1{0}761 is [B]100761[/B]

sweety439 2022-01-02 20:58

* Case (1,2):

** [B]12[/B] is prime, and thus the only minimal prime in this family.

* Case (1,4):

** [B]14[/B] is prime, and thus the only minimal prime in this family.

* Case (1,5):

** Since 12, 14, 18, 25, 45, 65, [B]135[/B], [B]155[/B], [B]175[/B] are primes, we only need to consider the family 1{0,1}5 (since any digits 2, 3, 4, 5, 6, 7, 8 between them will produce smaller primes)

*** Since [B]1015[/B] is prime, we only need to consider the family 1{1}{0}5

**** All numbers of the form 1{1}{0}5 are divisible by 2 (if the total number of 1's is odd) or 5 (if the total number of 1's is even), thus cannot be prime.

* Case (1,7):

** Since 12, 14, 18, 47, 67, 87, [B]117[/B], [B]177[/B] are primes, we only need to consider the family 1{0,3,5}7 (since any digits 1, 2, 4, 6, 7, 8 between them will produce smaller primes)

*** If there are at least two 3's in {}, then 337 will be a subsequence.

*** If there are at least two 5's in {}, then both 155 and 557 will be a subsequence.

*** If there are at least one 3's and at least one 5's in {}, then either 135 or 537 will be a subsequence.

*** If there are no 3's and no 5's in {}, then the family will be 1{0}7

**** All numbers of the form 1{0}7 are divisible by 8, thus cannot be prime

*** If there are exactly one 3's and no 5's in {}, then the family will be 1{0}3{0}7

**** Since [B]100037[/B] is prime, we only need to consider the families 13{0}7, 103{0}7, 1003{0}7

***** The smallest prime of the form 13{0}7 is [B]1300000007[/B]

***** All numbers of the form 103{0}7 are divisible by 7, thus cannot be prime

***** The smallest prime of the form 1003{0}7 is 100300000007 (not minimal prime, since 1300000007 is prime)

*** If there are no 3's and exactly one 5's in {}, then the family will be 1{0}5{0}7

**** Since [B]105007[/B] is prime, we only need to consider the families 15{0}7, 1{0}57, 1{0}507

***** All numbers of the form 15{0}7 are divisible by 7, thus cannot be prime

***** The smallest prime of the form 1{0}57 is [B]1000000000000000000000000057[/B]

***** The smallest prime of the form 1{0}507 is [B]100000000000507[/B]

* Case (1,8):

** [B]18[/B] is prime, and thus the only minimal prime in this family.

sweety439 2022-01-04 00:33

1 Attachment(s)
* Case (2,1):

** [B]21[/B] is prime, and thus the only minimal prime in this family.

* Case (2,2):

** Since 21, 25, 12, 32, 52, [B]272[/B] are primes, we only need to consider the family 2{0,2,4,6,8}2 (since any digits 1, 3, 5, 7 between them will produce smaller primes)

*** All numbers of the form 2{0,2,4,6,8}2 are divisible by 2, thus cannot be prime

* Case (2,4):

** Since 21, 25, 14, 34, 74 are primes, we only need to consider the family 2{0,2,4,6,8}4 (since any digits 1, 3, 5, 7 between them will produce smaller primes)

*** All numbers of the form 2{0,2,4,6,8}4 are divisible by 2, thus cannot be prime

* Case (2,5):

** [B]25[/B] is prime, and thus the only minimal prime in this family.

* Case (2,7):

** Since 21, 25, 47, 67, 87 are primes, we only need to consider the family 2{0,2,3,7}7 (since any digits 1, 4, 5, 6, 8 between them will produce smaller primes)

*** If there are at least two 3's in {}, then 337 will be a subsequence.

*** If there are exactly one 3's in {}, then there cannot be 7's in {}, otherwise, either 377 or 737 will be a subsequence, thus the family is 2{0,2}3{0,2}7

**** All numbers of the form 2{0,2}3{0,2}7 are divisible by 2, thus cannot be prime

*** If there are no 3's in {}, then the family will be 2{0,2,7}7

**** Since [B]2027[/B] and 272 are primes, we only need to consider the family 2{2}{0,7}7 (since any digits combo 02, 72 between them will produce smaller primes, thus let "d" be the rightmost digit 2 in {}, then all digits before "d" are 2 (cannot be 0 or 7, otherwise 02 or 72 will be in {}, and hence either [B]2027[/B] or 272 will be a subsequence); also, all digits after "d" are 0 or 7, since "d" is the rightmost digit 2, thus the family is 2{2}{0,7}7)

***** Since [B]22227[/B] is prime, we only need to consider the families 2{0,7}7, 22{0,7}7, 222{0,7}7

****** Since [B]2207[/B] is prime, we only need to consider the families 2{0,7}7, 22{7}7, 222{7}7

******* For the 2{0,7}7 family, since the digit sum of primes must be odd (otherwise the number will be divisible by 2, thus cannot be prime), there is an odd total number of 7

******** If there are only 1 7's, then the form is 2{0}7

********* The smallest prime of the form 2{0}7 is [B]2000000000007[/B]

******** If there are at least 3 7's, then there cannot be any 0 before the 3rd rightmost 7, or [B]20777[/B] will be a subsequence, thus the family is 2{7}7{0}7{0}7

********* Since [B]270707[/B] is prime, we only need to consider the families 2{7}7{0}77 and 2{7}77{0}7

********** All numbers of the form 2{7}7{0}77 are divisible by 2 (if the total number of 7's is even) or 5 (if the total number of 7's is odd), thus cannot be prime.

********** For the 2{7}77{0}7 family, since [B]2770007[/B] is prime, we only need to consider the families 2{7}777, 2{7}7707, 2{7}77007

*********** All numbers of the form 2{7}777 are divisible by 2 (if the total number of 7's is even) or 5 (if the total number of 7's is odd), thus cannot be prime.

*********** The smallest prime of the form 2{7}7707 is [B]27777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777707[/B] with 687 7's, which can be written as 2(7^686)07 and equal the prime (23*9^688-511)/8 ([URL="http://factordb.com/cert.php?id=1100000002495467486"]primality certificate of this prime[/URL])

*********** All numbers of the form 2{7}77007 are divisible by 2 (if the total number of 7's is even) or 5 (if the total number of 7's is odd), thus cannot be prime.

******* The smallest prime of the form 22{7}7 is [B]22777[/B]

******* All numbers of the form 222{7}7 are divisible by 2 (if the total number of 7's is even) or 5 (if the total number of 7's is odd), thus cannot be prime.

* Case (2,8):

** Since 21, 25, 18, 58, 78, [B]238[/B] are primes, we only need to consider the family 2{0,2,4,6,8}8 (since any digits 1, 3, 5, 7 between them will produce smaller primes)

*** All numbers of the form 2{0,2,4,6,8}8 are divisible by 2, thus cannot be prime

sweety439 2022-01-04 17:08

This problem (i.e. the minimal primes (start with b+1) problem) covers these problems in the same base b:

* Find the smallest prime of the form (b^n-1)/(b-1) with n>=2 ([URL="https://oeis.org/A084740"]A084740[/URL], [URL="https://oeis.org/A084738"]A084738[/URL], [URL="https://oeis.org/A065854"]A065854[/URL] for prime b, [URL="https://oeis.org/A279068"]A279068[/URL] for prime b) ([URL="https://archive.fo/tf7jx"]https://archive.fo/tf7jx[/URL], [URL="http://www.fermatquotient.com/PrimSerien/GenRepu.txt"]http://www.fermatquotient.com/PrimSerien/GenRepu.txt[/URL])
* Find the smallest prime of the form b^(2^n)+1 with n>=0 ([URL="https://oeis.org/A228101"]A228101[/URL], [URL="https://oeis.org/A079706"]A079706[/URL], [URL="https://oeis.org/A084712"]A084712[/URL]) ([URL="http://jeppesn.dk/generalized-fermat.html"]http://jeppesn.dk/generalized-fermat.html[/URL], [URL="http://www.noprimeleftbehind.net/crus/GFN-primes.htm"]http://www.noprimeleftbehind.net/crus/GFN-primes.htm[/URL])
* Find the smallest prime of the form b^n+2 with n>=1 ([URL="https://oeis.org/A138066"]A138066[/URL], [URL="https://oeis.org/A084713"]A084713[/URL])
* Find the smallest prime of the form b^n-2 with n>=2 ([URL="https://oeis.org/A250200"]A250200[/URL]) ([URL="http://www.primepuzzles.net/puzzles/puzz_887.htm"]http://www.primepuzzles.net/puzzles/puzz_887.htm[/URL] for prime b)
* Find the smallest prime of the form 2*b^n+1 with n>=1 ([URL="https://oeis.org/A119624"]A119624[/URL]) ([URL="https://mersenneforum.org/showthread.php?t=19725"]https://mersenneforum.org/showthread.php?t=19725[/URL] for b == 11 mod 12)
* Find the smallest prime of the form 2*b^n-1 with n>=1 ([URL="https://oeis.org/A119591"]A119591[/URL]) ([URL="https://mersenneforum.org/showthread.php?t=24576"]https://mersenneforum.org/showthread.php?t=24576[/URL])
* Find the smallest prime of the form b^n+(b-1) with n>=1 ([URL="https://oeis.org/A076845"]A076845[/URL], [URL="https://oeis.org/A076846"]A076846[/URL])
* Find the smallest prime of the form b^n-(b-1) with n>=2 ([URL="https://oeis.org/A113516"]A113516[/URL], [URL="https://oeis.org/A343589"]A343589[/URL]) ([URL="https://cs.uwaterloo.ca/journals/JIS/VOL3/mccranie.html"]https://cs.uwaterloo.ca/journals/JIS/VOL3/mccranie.html[/URL] for prime b)
* Find the smallest prime of the form (b-1)*b^n+1 with n>=1 ([URL="https://oeis.org/A305531"]A305531[/URL], [URL="https://oeis.org/A087139"]A087139[/URL] for prime b)
* Find the smallest prime of the form (b-1)*b^n-1 with n>=1 ([URL="https://oeis.org/A122396"]A122396[/URL] for prime b) ([URL="https://harvey563.tripod.com/wills.txt"]https://harvey563.tripod.com/wills.txt[/URL])

In fact, this problem (i.e. the minimal primes (start with b+1) problem) also covers these problems in the same base b:

* Find the smallest prime of the form b^n+k with n>=1, for all k<b
* Find the smallest prime of the form b^n-k with n>=2, for all k<b
* Find the smallest prime of the form k*b^n+1 with n>=1, for all k<b ([URL="http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm"]http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm[/URL])
* Find the smallest prime of the form k*b^n-1 with n>=1, for all k<b ([URL="http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm"]http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm[/URL])

sweety439 2022-01-05 20:00

All families containing no minimal primes (start with b+1) as subsequences are ruled out as only contain composites (only count numbers > b) (may by covering congruence, algebraic factorization, or combine of them), furthermore, all families containing neither known minimal primes (start with b+1) (i.e. the minimal primes (start with b+1) in the current list, not including the smallest primes in the unsolved families) as subsequences nor unsolved families as sub-families are ruled out as only contain composites (only count numbers > b)

e.g. for base 10, families containing no minimal primes (start with b+1) as subsequences are the sub-families of these families (including these families themselves):

* {0,3,6,9}{0,2,4,5,6,8}
* {0,7}{0,2,4,5,6,8}
* {0}2{0}1{0,2,4,5,6,8}
* {0}5{0}1{0,2,4,5,6,8}
* {0}5{0}7{0,2,4,5,6,8}
* {0}8{0}1{0,2,4,5,6,8}
* {0}4{6}9{0,2,4,5,6,8}
* {0}28{0}7{0,2,4,5,6,8}

and for base 2, families containing no minimal primes (start with b+1) as subsequences are the sub-families of these families (including these families themselves):

* {0}1{0}

and for base 3, families containing no minimal primes (start with b+1) as subsequences are the sub-families of these families (including these families themselves):

* {0,2}
* {0}1{0}1{0}

and for base 4, families containing no minimal primes (start with b+1) as subsequences are the sub-families of these families (including these families themselves):

* {0,3}{0,2}
* {0}2{0}1{0,2}

and for base 6, families containing no minimal primes (start with b+1) as subsequences are the sub-families of these families (including these families themselves):

* {0,5}{0,2,3,4}
* {0}4{0}1{0,2,3,4}

Also, for base 10, assume the largest minimal prime (start with b+1) (i.e. 5(0^28)27) has not been found, i.e. we have a list of 76 smaller minimal primes (start with b+1) and one unsolved family 5{0}27 in base b=10, then we have completely the same set of families containing no minimal primes (start with b+1) as subsequences, since they are the only families containing neither the 76 primes as subsequences nor the family 5{0}27 as sub-family.

sweety439 2022-01-05 20:19

Note: Family a{b}c means the family of strings starting with a and ending with c, and all remaining characters are b, and family ab{c,d}ef means the family of strings starting with ab and ending with ef, and all remaining characters are either c or d (thus, abdcdccdef belongs to family ab{c,d}ef), and family a{b}c{d}e means the family of strings starting with a and ending with e and there is c in this string and all characters (may be empty) between a and c are b and all characters (may be empty) between c and e are d

Note: Family X is a sub-family of family Y if and only if all numbers in family X are subsequence of some number in family Y

e.g. (if a, b, c, ... are digits)

family ab{c}de is sub-family of these families:

* ab{c}def
* ab{c}dfe
* ab{c}fde
* abf{c}de
* afb{c}de
* fab{c}de
* ab{c,f}de

but not of these families:

* ab{c}edf
* fab{c}ed
* ba{c}def
* fba{c}de
* a{b}cdef
* fa{b}cde
* abc{d}ef
* fabc{d}e
* a{b,f}cde
* abc{d,f}e

Also, all sub-families of family a{b,c}d are: (note: a{}d = ad is only a number (or a string), not a family, thus not listed, so is {} = empty string, a{} = a, {}d = d)

* {b}
* {c}
* {b,c}
* a{b}
* a{c}
* a{b,c}
* {b}d
* {c}d
* {b,c}d
* a{b}d
* a{c}d
* a{b,c}d itself

Like subsequence and substring, sub-family is [URL="https://en.wikipedia.org/wiki/Partial_order"]partial order[/URL] relation.

sweety439 2022-01-10 02:27

5 Attachment(s)
I ran a program to compute the minimal primes (start with b+1) <= 2^32 in bases 17<=b<=36, now done to b=23

sweety439 2022-01-10 03:17

Proving the set of the minimal primes (start with b+1) in base b is S, is equivalent to:

* Prove that all elements in S, when read as base b representation, are primes > b.

* Prove that all proper subsequence of all elements in S, when read as base b representation, which are > b, are composite.

* Prove that all primes > b, when written in base b, contain at least one element in S as subsequence (equivalently, prove that all strings not containing any element in S as subsequence, when read as base b representation, which are > b, are composite).

e.g. proving the set of the minimal primes (start with b+1) in base b = 10 is {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027}, is equivalent to:

* Prove that all of 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027 are primes > 10.

* Prove that all proper subsequence of all elements in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} which are > 10 are composite.

* Prove that all primes > 10 contain at least one element in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} as subsequence (equivalently, prove that all numbers > 10 not containing any element in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} as subsequence are composite).

sweety439 2022-01-10 03:33

This is the prime game:

Write down a prime number > 10, then you can always strike 0 or more digits to get a prime in this set: {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027}

(the same game when the restriction of prime>base is not required: [URL="http://www.wiskundemeisjes.nl/wp-content/uploads/2007/02/primes2.pdf"]http://www.wiskundemeisjes.nl/wp-content/uploads/2007/02/primes2.pdf[/URL])

Also, for known minimal primes (start with b+1) base b and known unsolved families, it is proved that all primes not in these unsolved families have a subsequence in the current set of minimal primes (start with b+1) base b.

e.g. (for the case that the restriction of prime>base is not required, i.e. [URL="https://cs.uwaterloo.ca/~cbright/reports/mepn.pdf"]the original minimal prime problem[/URL]):

* In base 23, it is proved that all primes contain at least one element in [URL="https://raw.githubusercontent.com/curtisbright/mepn-data/master/data/minimal.23.txt"]https://raw.githubusercontent.com/curtisbright/mepn-data/master/data/minimal.23.txt[/URL] as subsequence, and it is proved that all proper subsequence of all numbers in [URL="https://raw.githubusercontent.com/curtisbright/mepn-data/master/data/minimal.23.txt"]https://raw.githubusercontent.com/curtisbright/mepn-data/master/data/minimal.23.txt[/URL] are composite (or 0 or 1), and it is proved that all numbers in [URL="https://raw.githubusercontent.com/curtisbright/mepn-data/master/data/minimal.23.txt"]https://raw.githubusercontent.com/curtisbright/mepn-data/master/data/minimal.23.txt[/URL] other than the 6013th, the 6014th, the 6015th, the 6016th, the 6017th, the 6018th, the 6019th, the 6021st are primes (the 6012th and the 6020th are proven to be primes using the [URL="https://primes.utm.edu/prove/prove3_1.html"]N-1 primality test[/URL]), also, these 9 numbers (the 6013th, the 6014th, the 6015th, the 6016th, the 6017th, the 6018th, the 6019th, the 6021st element in [URL="https://raw.githubusercontent.com/curtisbright/mepn-data/master/data/minimal.23.txt"]https://raw.githubusercontent.com/curtisbright/mepn-data/master/data/minimal.23.txt[/URL] passes the [URL="https://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test"]Miller-Rabin test[/URL] to many bases, and the property that they are in fact composite is less than one googolth ([URL="https://primes.utm.edu/notes/prp_prob.html"]reference[/URL]), thus the property that the set of the base 23 minimal primes is [URL="https://raw.githubusercontent.com/curtisbright/mepn-data/master/data/minimal.23.txt"]https://raw.githubusercontent.com/curtisbright/mepn-data/master/data/minimal.23.txt[/URL] is larger than (googol-1)/googol

* In base 25, it is proved that all primes not in any family in [URL="https://raw.githubusercontent.com/curtisbright/mepn-data/master/data/unsolved.25.txt"]https://raw.githubusercontent.com/curtisbright/mepn-data/master/data/unsolved.25.txt[/URL] contain at least one element in [URL="https://raw.githubusercontent.com/curtisbright/mepn-data/master/data/minimal.25.txt"]https://raw.githubusercontent.com/curtisbright/mepn-data/master/data/minimal.25.txt[/URL] as subsequence, and it is proved that all proper subsequence of all numbers in [URL="https://raw.githubusercontent.com/curtisbright/mepn-data/master/data/minimal.25.txt"]https://raw.githubusercontent.com/curtisbright/mepn-data/master/data/minimal.25.txt[/URL] are composite (or 0 or 1), and most (>95%) of the numbers in [URL="https://raw.githubusercontent.com/curtisbright/mepn-data/master/data/minimal.25.txt"]https://raw.githubusercontent.com/curtisbright/mepn-data/master/data/minimal.25.txt[/URL] are proven to be primes

sweety439 2022-01-10 23:24

5 Attachment(s)
[QUOTE=sweety439;597525]I ran a program to compute the minimal primes (start with b+1) <= 2^32 in bases 17<=b<=36, now done to b=23[/QUOTE]

done to 26

chalsall 2022-01-10 23:56

[QUOTE=sweety439;597609]done to 26[/QUOTE]

This is just an empirical experiment...

Does this 'bot read? Or only radiate?

sweety439 2022-01-12 02:17

[URL="https://docs.google.com/spreadsheets/d/e/2PACX-1vTKkSNKGVQkUINlp1B3cXe90FWPwiegdA07EE7-U7sqXntKAEQrynoI1sbFvvKriieda3LfkqRwmKME/pubhtml"]https://docs.google.com/spreadsheets/d/e/2PACX-1vTKkSNKGVQkUINlp1B3cXe90FWPwiegdA07EE7-U7sqXntKAEQrynoI1sbFvvKriieda3LfkqRwmKME/pubhtml[/URL]

e.g.

for base b=311:

minimal prime (start with b+1) 10:(0^314805):1 ([URL="http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm#S311"]generalized Sierpinski problem base 311[/URL], k=10)

minimal prime (start with b+1) 76:(0^135561):1 ([URL="http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm#S311"]generalized Sierpinski problem base 311[/URL], k=76)

minimal prime (start with b+1) 46:(0^8479):1 ([URL="http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm#S311"]generalized Sierpinski problem base 311[/URL], k=46)

unsolved family {155}:156 ([URL="http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt"]generalized half Fermat primes base 311[/URL])

minimal prime (start with b+1) (1^36497) ([URL="http://www.primenumbers.net/prptop/searchform.php?form=%28311%5En-1%29%2F310&action=Search"]generalized repunit primes base 311[/URL])

for base b=383:

unsolved families 2:{0}:1, 32:{0}:1, 44:{0}:1, 94:{0}:1, 98:{0}:1, ... [URL="http://www.noprimeleftbehind.net/crus/Sierp-conjecture-base383-reserve.htm"]generalized Sierpinski problem base 383[/URL]

minimal prime (start with b+1) 50:(0^463312):1 ([URL="http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm#S383"]generalized Sierpinski problem base 383[/URL], k=50)

minimal prime (start with b+1) 104:(0^408248):1 ([URL="http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm#S383"]generalized Sierpinski problem base 383[/URL], k=104)

minimal prime (start with b+1) 1:71:(0^354813):1 ([URL="http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm#S383"]generalized Sierpinski problem base 383[/URL], k=104)

minimal prime (start with b+1) 134:(0^225186):1 ([URL="http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm#S383"]generalized Sierpinski problem base 383[/URL], k=104)

unsolved families 115:{382}, 133:{382}, 135:{382}, ... ([URL="http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm#R383"]generalized Riesel problem base 383[/URL])

minimal prime (start with b+1) 69:(382^147947) ([URL="http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm#R383"]generalized Riesel problem base 383[/URL])

minimal prime (start with b+1) 43:(382^143148) ([URL="http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm#R383"]generalized Riesel problem base 383[/URL])

minimal prime (start with b+1) 81:(382^47643) ([URL="http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm#R383"]generalized Riesel problem base 383[/URL])

minimal prime (start with b+1) 1:(382^20956) ([URL="http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm#R383"]generalized Riesel problem base 383[/URL])

unsolved family 1:{0}:2 ([URL="https://oeis.org/A138066"]b^n+2[/URL])

unsolved family 381:{382} ([URL="https://harvey563.tripod.com/wills.txt"]Williams primes base 383 (listed as "base 382" in this site)[/URL])

unsolved family {191}:192 ([URL="http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt"]generalized half Fermat primes base 383[/URL])

for base b=401:

unsolved family 20:{0}:1 ([URL="http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm#S401"]generalized Sierpinski problem base 401[/URL], k=20)

minimal prime (start with b+1) 16:(0^4211):1 ([URL="http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm#S401"]generalized Sierpinski problem base 401[/URL], k=16)

unsolved family 37:{400} ([URL="http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm#R401"]generalized Riesel problem base 401[/URL], k=38)

minimal prime (start with b+1) 399:(400^103669) ([URL="https://harvey563.tripod.com/wills.txt"]Williams primes base 401 (listed as "base 400" in this site)[/URL])

unsolved family {200}:201 ([URL="http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt"]generalized half Fermat primes base 401[/URL])

minimal prime (start with b+1) (1^127) ([URL="https://archive.fo/tf7jx"]generalized repunit primes base 401[/URL])

for base b=422:

unsolved families 8:{0}:1, 13:{0}:1, 17:{0}:1 ([URL="http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm#S422"]generalized Sierpinski problem base 422[/URL])

minimal prime (start with b+1) 22:(0^268037):1 ([URL="http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm#S422"]generalized Sierpinski problem base 422[/URL])

minimal prime (start with b+1) 16:(0^176283):1 ([URL="http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm#S422"]generalized Sierpinski problem base 422[/URL])

minimal prime (start with b+1) 31:(0^33727):1 ([URL="http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm#S422"]generalized Sierpinski problem base 422[/URL])

minimal prime (start with b+1) 37:(0^13019):1 ([URL="http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm#S422"]generalized Sierpinski problem base 422[/URL])

unsolved families 10:{421}, 12:{421}, 13:{421}, 28:{421}, 36:{421} ([URL="http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm#R422"]generalized Riesel problem base 422[/URL])

minimal prime (start with b+1) 40:(421^22802) ([URL="http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm#R422"]generalized Riesel problem base 422[/URL])

minimal prime (start with b+1) 3:(421^21737) ([URL="http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm#R422"]generalized Riesel problem base 422[/URL])

unsolved family 1:{0}:1 ([URL="http://jeppesn.dk/generalized-fermat.html"]generalized Fermat primes base 422[/URL])

minimal prime (start with b+1) (1^983) ([URL="https://archive.fo/tf7jx"]generalized repunit primes base 422[/URL])

unsolved family 421:{0}:1 ([URL="https://www.rieselprime.de/ziki/Williams_prime_MP_table"]Williams primes of the second kind base 422[/URL])

for base b=452:

unsolved families 23:{0}:1, 37:{0}:1, 41:{0}:1, 68:{0}:1, 96:{0}:1, 101:{0}:1, 124:{0}:1, 136:{0}:1 ([URL="http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm#S452"]generalized Sierpinski problem base 452[/URL])

minimal prime (start with b+1) 151:(0^61687):1 ([URL="http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm#S452"]generalized Sierpinski problem base 452[/URL])

minimal prime (start with b+1) 4:(0^14153):1 ([URL="http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm#S452"]generalized Sierpinski problem base 452[/URL])

unsolved families 10:{451}, 42:{451}, 51:{451} ([URL="http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm#R452"]generalized Riesel problem base 452[/URL])

minimal prime (start with b+1) 45:(451^153285) ([URL="http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm#R452"]generalized Riesel problem base 452[/URL])

minimal prime (start with b+1) 451:(0^71939):1 ([URL="https://www.rieselprime.de/ziki/Williams_prime_MP_table"]Williams primes of the second kind base 452[/URL])

unsolved family {1} ([URL="https://oeis.org/A128164/a128164_7.txt"]generalized repunit primes base 452[/URL])

for base b=542:

unsolved families 2:{0}:1, 13:{0}:1 ([URL="http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm#S542"]generalized Sierpinski problem base 542[/URL])

minimal prime (start with b+1) 19:(0^18949):1 ([URL="http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm#S542"]generalized Sierpinski problem base 542[/URL])

minimal prime (start with b+1) 4:(0^15981):1 ([URL="http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm#S542"]generalized Sierpinski problem base 542[/URL])

unsolved families 10:{541}, 18:{541}, 36:{541}, 41:{541}, 70:{541}, 73:{541}, ... ([URL="http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm#R542"]generalized Riesel problem base 542[/URL])

minimal prime (start with b+1) 132:(541^83867) ([URL="http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm#R542"]generalized Riesel problem base 542[/URL])

minimal prime (start with b+1) 12:(541^70447) ([URL="http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm#R542"]generalized Riesel problem base 542[/URL])

minimal prime (start with b+1) 27:(541^66555) ([URL="http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm#R542"]generalized Riesel problem base 542[/URL])

minimal prime (start with b+1) 103:(541^56400) ([URL="http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm#R542"]generalized Riesel problem base 542[/URL])

minimal prime (start with b+1) 114:(541^41905) ([URL="http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm#R542"]generalized Riesel problem base 542[/URL])

unsolved family 541:{0}:1 ([URL="https://www.rieselprime.de/ziki/Williams_prime_MP_table"]Williams primes of the second kind base 542[/URL])

sweety439 2022-01-12 04:46

[QUOTE=chalsall;597614]This is just an empirical experiment...

Does this 'bot read? Or only radiate?[/QUOTE]

These are the data for "minimal primes (start with b+1)" <= 2^32 in bases b=22 to b=26, they are in my project, see [URL="https://docs.google.com/document/d/e/2PACX-1vQct6Hx-IkJd5-iIuDuOKkKdw2teGmmHW-P75MPaxqBXB37u0odFBml5rx0PoLa0odTyuW67N_vn96J/pub"]my article[/URL]

sweety439 2022-01-12 04:49

5 Attachment(s)
[QUOTE=sweety439;597609]done to 26[/QUOTE]

done to base b=31

the original text file for base 31 is too large (1546 KB) to upload, thus zipped it

sweety439 2022-01-12 04:57

look the size for the data files, my [URL="https://primes.utm.edu/glossary/xpage/Heuristic.html"]heuristic argument[/URL] is really true, i.e. the number of minimal primes (start with b+1) <= fixed limit (2^32 in these data files) in base b is C*[URL="https://en.wikipedia.org/wiki/E_(mathematical_constant)"]e[/URL]^([URL="https://en.wikipedia.org/wiki/Euler%27s_constant"]gamma[/URL]*(b-1)*[URL="https://en.wikipedia.org/wiki/Euler%27s_totient_function"]eulerphi[/URL](b)), i.e. C*[URL="https://en.wikipedia.org/wiki/E_(mathematical_constant)"]e[/URL]^([URL="https://en.wikipedia.org/wiki/Euler%27s_constant"]gamma[/URL]*[URL="https://oeis.org/A062955"]A062955[/URL](b)), where C is a [URL="https://en.wikipedia.org/wiki/Constant_(mathematics)"]constant[/URL]

(there is also a [URL="https://primes.utm.edu/glossary/xpage/Heuristic.html"]heuristic argument[/URL]: both the totally number of minimal primes (start with b+1) in base b (equivalently, the "fixed limit" in above section is [URL="https://en.wikipedia.org/wiki/Infinity"]infinity[/URL]) and the length of the largest minimal primes (start with b+1) in base b is C*[URL="https://en.wikipedia.org/wiki/E_(mathematical_constant)"]e[/URL]^([URL="https://en.wikipedia.org/wiki/Euler%27s_constant"]gamma[/URL]*(b-1)*[URL="https://en.wikipedia.org/wiki/Euler%27s_totient_function"]eulerphi[/URL](b)), i.e. C*[URL="https://en.wikipedia.org/wiki/E_(mathematical_constant)"]e[/URL]^([URL="https://en.wikipedia.org/wiki/Euler%27s_constant"]gamma[/URL]*[URL="https://oeis.org/A062955"]A062955[/URL](b)), where C is a [URL="https://en.wikipedia.org/wiki/Constant_(mathematics)"]constant[/URL])

VBCurtis 2022-01-12 07:06

Time for your periodic warning:
Posting too many attachments wastes forum resources. Your project does not need multiple update posts a day, ever. If you're finding so many things that you have multiple things to say in a week, your work is so trivial that it doesn't need to be said at all.

Store your files somewhere else. Post less, like once a week, on this useless project.

sweety439 2022-01-12 13:28

[QUOTE=VBCurtis;597732]Time for your periodic warning:
Posting too many attachments wastes forum resources. Your project does not need multiple update posts a day, ever. If you're finding so many things that you have multiple things to say in a week, your work is so trivial that it doesn't need to be said at all.

Store your files somewhere else. Post less, like once a week, on this useless project.[/QUOTE]

When I have a new researching result about my project (i.e. finding and proving the set of minimal primes (start with b+1) in bases 2<=b<=36), I either post on this thread or update [URL="https://docs.google.com/document/d/e/2PACX-1vQct6Hx-IkJd5-iIuDuOKkKdw2teGmmHW-P75MPaxqBXB37u0odFBml5rx0PoLa0odTyuW67N_vn96J/pub"]my article[/URL] (or both).

Uncwilly 2022-01-13 03:12

[QUOTE=sweety439;597751]I either post on this thread or update [URL="https://docs.google.com/document/d/e/2PACX-1vQct6Hx-IkJd5-iIuDuOKkKdw2teGmmHW-P75MPaxqBXB37u0odFBml5rx0PoLa0odTyuW67N_vn96J/pub"]my article[/URL] (or both).[/QUOTE]Maybe just over on your google doc. Then people can fetch it at their leisure. When you post here, you generate notifications for people that there is something new. While it is new, it is not worth noticing.

sweety439 2022-01-13 06:31

1 Attachment(s)
This is the smallest GFN primes and the smallest GRU primes in bases b<=64

Note 1: we do not include the case where the base of the GFNs is perfect odd power and the case where the base of the GRUs is either perfect power or of the form -4*m^4 with integer m, since such numbers have algebra factors and are composite for all n or are prime only for very small n, such families for bases 2<=b<=64 are:

[CODE]
base GFN family GRU family
4 {1}
8 1{0}1 {1}
9 {1}
16 {1}, 1{5}, {C}D
25 {1}
27 {D}E {1}
32 1{0}1 {1}
36 {1}
49 {1}
64 1{0}1 {1}, 1{L}, 5{L}, 1{9}, {u}v
[/CODE]

Such small primes are: 11 in base 4, 111 in base 8, 11 in base 16, 111 in base 27, 11 in base 36, 19 in base 64 (they are 5, 73, 17, 757, 37, 73, respectively, in decimal)

Note 2: All GFN base b and all GRU base b are strong-probable-primes (primes and [URL="https://en.wikipedia.org/wiki/Strong_pseudoprime"]strong pseudoprimes[/URL]) to base b, since they are over-probable-primes (primes and overpseudoprimes) to base b (references: [URL="https://oeis.org/A141232"]https://oeis.org/A141232[/URL] [URL="http://arxiv.org/abs/0806.3412"]http://arxiv.org/abs/0806.3412[/URL] [URL="http://arxiv.org/abs/0807.2332"]http://arxiv.org/abs/0807.2332[/URL] [URL="http://arxiv.org/abs/1412.5226"]http://arxiv.org/abs/1412.5226[/URL] [URL="https://cs.uwaterloo.ca/journals/JIS/VOL15/Shevelev/shevelev19.pdf"]https://cs.uwaterloo.ca/journals/JIS/VOL15/Shevelev/shevelev19.pdf[/URL]), and all overpseudoprimes are [URL="https://en.wikipedia.org/wiki/Strong_pseudoprime"]strong pseudoprimes[/URL] to the same base b, all strong pseudoprimes are [URL="https://en.wikipedia.org/wiki/Euler%E2%80%93Jacobi_pseudoprime"]Euler–Jacobi pseudoprimes[/URL] to the same base b, all Euler–Jacobi pseudoprimes are [URL="https://en.wikipedia.org/wiki/Euler_pseudoprime"]Euler pseudoprimes[/URL] to the same base b, all Euler pseudoprimes are [URL="https://en.wikipedia.org/wiki/Fermat_pseudoprime"]Fermat pseudoprimes[/URL] to the same base b, so don't test with this base (see [URL="https://mersenneforum.org/showthread.php?t=10476&page=2"]https://mersenneforum.org/showthread.php?t=10476&page=2[/URL], [URL="https://oeis.org/A171381"]https://oeis.org/A171381[/URL], [URL="https://oeis.org/A028491"]https://oeis.org/A028491[/URL], also see [URL="https://oeis.org/A210454"]https://oeis.org/A210454[/URL], [URL="https://oeis.org/A210461"]https://oeis.org/A210461[/URL], [URL="https://oeis.org/A216170"]https://oeis.org/A216170[/URL], [URL="https://oeis.org/A217841"]https://oeis.org/A217841[/URL], [URL="https://oeis.org/A243292"]https://oeis.org/A243292[/URL], [URL="https://oeis.org/A217853"]https://oeis.org/A217853[/URL], [URL="https://oeis.org/A293626"]https://oeis.org/A293626[/URL], [URL="https://oeis.org/A210454/a210454.pdf"]https://oeis.org/A210454/a210454.pdf[/URL], [URL="https://cs.uwaterloo.ca/journals/JIS/VOL10/Hamahata2/hamahata44.pdf"]https://cs.uwaterloo.ca/journals/JIS/VOL10/Hamahata2/hamahata44.pdf[/URL], all generalized repunits in base b^2 with length p (where p is prime not dividing b*(b^2-1)) are Fermat pseudoprimes to base b, thus there are infinitely many pseudoprimes to every base b), note that there are also (but very few) numbers in the simple families which are neither GFN families nor GRU families, which are pseudoprimes, e.g. for the family {5}25 in base 8 (which have the smallest prime 555555555555525, corresponding to the second-largest base 8 minimal prime (start with b+1)), a smaller number 525 is 341 in decimal, which is Fermat pseudoprime and Euler pseudoprime (although not strong pseudoprime, but there are many examples of strong pseudoprimes to base 2 and/or base 3, e.g. the smallest composite number which is strong pseudoprime to both base 2 and base 3 is 1373653 (see [URL="https://oeis.org/A072276"]https://oeis.org/A072276[/URL]), which has no proper subsequence which is prime > base (b) in bases b = 55, 58, 59, 65, 66, 70, 79, 82, 95, 103, 112, 113, 115, 116, 117, 121, 127, 130, 133, 134, 135, 136, 137, 138, 139, 141, 146, 147, 149, 151, 152, 155, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 177, 179, 183, 184, 185, 187, 188, 189, 191, 192, 193, 195, 196, 197, 199, 200, ..., and the second-smallest composite number which is strong pseudoprime to both base 2 and base 3 is 1530787, which has no proper subsequence which is prime > base (b) in bases b = 77, 91, 95, 98, 109, 113, 120, 123, 125, 127, 129, 131, 132, 135, 136, 139, 141, 142, 143, 144, 147, 151, 155, 159, 160, 161, 162, 169, 170, 173, 176, 177, 179, 181, 183, 184, 187, 188, 189, 190, 191, 192, 194, 197, 199, 200, ..., and if we assume a number which has passed the Miller–Rabin primality tests to both base 2 and base 3 is in fact prime, our data will be wrong for these bases b) to base 2 (and thus to base 8, since pseudoprimes to base b are always (the same type) pseudoprimes to base b^r for all r>1, and 8=2^3).

sweety439 2022-01-13 06:36

[QUOTE=Uncwilly;597815]it is not worth noticing.[/QUOTE]

but it is my newer researching result, and I think that it is important, sometimes I update my old posts, such as [URL="https://mersenneforum.org/showpost.php?p=593116&postcount=208"]#208[/URL] and [URL="https://mersenneforum.org/showpost.php?p=593783&postcount=215"]#215[/URL], also there are posts which are lists of references: [URL="https://mersenneforum.org/showpost.php?p=571731&postcount=140"]#140[/URL] and [URL="https://mersenneforum.org/showpost.php?p=582061&postcount=154"]#154[/URL]

sweety439 2022-01-13 11:42

5 Attachment(s)
[QUOTE=sweety439;597723]done to base b=31

the original text file for base 31 is too large (1546 KB) to upload, thus zipped it[/QUOTE]

all bases b<=36 are done.

the original text file for base 35 is too large (1743 KB) to upload, thus zipped it

sweety439 2022-01-15 07:11

Conjecture: There is no base b such that the largest minimal prime (start with b+1) and the second-largest minimal prime (start with b+1) have the same number of digits in base b, note that in the [URL="https://cs.uwaterloo.ca/~cbright/reports/mepn.pdf"]original minimal prime[/URL] (i.e. prime > base is not required), the three largest minimal primes in decimal (base 10) all have the same number of digits (60000049, 66000049, 66600049, all have 8 digits), and in base 2 the largest (and the only) two minimal primes are 10 and 11, both have 2 digits, also, in base 5, the largest two minimal primes are 14444 and 44441, both have 5 digits.

For the problem in this project (i.e. the minimal primes (start with b+1)), the largest and the second-largest minimal primes (start with b+1) have the numbers of digits: (combine with the third-largest and the fourth-largest minimal primes (start with b+1), see the table below):

[CODE]
base 1st largest 2nd largest 3rd largest 4th largest
2 2 N/A N/A N/A
3 3 2 2 N/A
4 3 2 2 2
5 96 6 5 5
6 5 4 4 2
7 17 10 8 7
8 221 15 13 11
9 1161 689 331 38 (conjectured)
10 31 12 8 8
12 42 30 9 8
[/CODE]

and (the number of digits of 1st largest) / (the number of digits of 2nd largest) getting large very quickly if (b-1)*eulerphi(b) gets large, thus I do not think such base can exist.

Another conjecture: For any number n>=2, there exists a minimal primes (start with b+1) with exactly n digits in base b, for every [I]enough large[/I] b

Clearly, all 2-digit primes (except "10" (i.e. = b) when b itself is prime) are minimal primes (start with b+1) base b, I conjectured that all bases b != 2, 6 have a 3-digit minimal prime (start with b+1), also all bases b>4 have a 4-digit minimal primes (start with b+1), all bases b>4 have a 5-digit minimal primes (start with b+1), etc. (note that in the [URL="https://cs.uwaterloo.ca/~cbright/reports/mepn.pdf"]original minimal prime[/URL] (i.e. prime > base is not required), all single-digit primes are minimal primes, and I conjectured that all bases b != 8 have a 2-digit minimal prime, all bases b != 2, 4, 6, 7 have a 3-digit minimal prime, all bases b != 2, 3, 4, 5, 7 have a 4-digit minimal prime, all bases b != 2, 3, 4, 9 have a 5-digit minimal prime, etc. (the bases with no n-digit minimal prime for given n is more complex, thus the problem in this project (i.e. the minimal primes (start with b+1)) is really better))

(for more data, see [URL="https://mersenneforum.org/showpost.php?p=571888&postcount=145"]post 145[/URL])

we can research:

* the possible length of the minimal primes (start with b+1)
* the possible (first,last) combo of the minimal primes (start with b+1)
* for these minimal primes (start with b+1), the digit which appears the most times in this minimal prime (start with b+1)
* the length such that there are the most minimal primes (start with b+1)
* the (first,last) combo such that there are the most minimal primes (start with b+1)

sweety439 2022-01-17 06:17

find the set of the minimal primes (start with b+1) base b for various bases b (2<=b<=36) is the target of the project in this thread.

minimal prime (start with b+1) base b is always minimal prime (start with b'+1) base b' = b^n, if it is > b', for any integer n>1

[URL="https://cs.uwaterloo.ca/~cbright/reports/mepn.pdf"]original minimal prime[/URL] (i.e. prime > b is not required) base b is always minimal prime (start with b+1) base b, if it is > b

sweety439 2022-01-21 15:44

1 Attachment(s)
zipped file for the minimal primes (start with b+1) in bases 17<=b<=36

sweety439 2022-01-21 15:52

[QUOTE=sweety439;597528]e.g. proving the set of the minimal primes (start with b+1) in base b = 10 is {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027}, is equivalent to:

* Prove that all of 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027 are primes > 10.

* Prove that all proper subsequence of all elements in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} which are > 10 are composite.

* Prove that all primes > 10 contain at least one element in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} as subsequence (equivalently, prove that all numbers > 10 not containing any element in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} as subsequence are composite).[/QUOTE]

the last (i.e. Proving that all primes > 10 contain at least one element in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} as subsequence) is equivalent to:

* If n is an integer which is > 10, and the base 10 representation of n contains none of {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} as subsequences, prove that n is composite.

Of course, this can be generalized to other bases, such as:

* If n is an integer which is > 7, and the base 7 representation of n contains none of {14, 16, 23, 25, 32, 41, 43, 52, 56, 61, 65, 113, 115, 131, 133, 155, 212, 221, 304, 313, 335, 344, 346, 364, 445, 515, 533, 535, 544, 551, 553, 1022, 1051, 1112, 1202, 1211, 1222, 2111, 3031, 3055, 3334, 3503, 3505, 3545, 4504, 4555, 5011, 5455, 5545, 5554, 6034, 6634, 11111, 11201, 30011, 30101, 31001, 31111, 33001, 33311, 35555, 40054, 100121, 150001, 300053, 351101, 531101, 1100021, 33333301, 5100000001, 33333333333333331} as subsequences, prove that n is composite.

* If n is an integer which is > 8, and the base 8 representation of n contains none of {13, 15, 21, 23, 27, 35, 37, 45, 51, 53, 57, 65, 73, 75, 107, 111, 117, 141, 147, 161, 177, 225, 255, 301, 343, 361, 401, 407, 417, 431, 433, 463, 467, 471, 631, 643, 661, 667, 701, 711, 717, 747, 767, 3331, 3411, 4043, 4443, 4611, 5205, 6007, 6101, 6441, 6477, 6707, 6777, 7461, 7641, 47777, 60171, 60411, 60741, 444641, 500025, 505525, 3344441, 4444477, 5500525, 5550525, 55555025, 444444441, 744444441, 77774444441, 7777777777771, 555555555555525, 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447} as subsequences, prove that n is composite.

* If n is an integer which is > 12, and the base 12 representation of n contains none of {11, 15, 17, 1B, 25, 27, 31, 35, 37, 3B, 45, 4B, 51, 57, 5B, 61, 67, 6B, 75, 81, 85, 87, 8B, 91, 95, A7, AB, B5, B7, 221, 241, 2A1, 2B1, 2BB, 401, 421, 447, 471, 497, 565, 655, 665, 701, 70B, 721, 747, 771, 77B, 797, 7A1, 7BB, 907, 90B, 9BB, A41, B21, B2B, 2001, 200B, 202B, 222B, 229B, 292B, 299B, 4441, 4707, 4777, 6A05, 6AA5, 729B, 7441, 7B41, 929B, 9777, 992B, 9947, 997B, 9997, A0A1, A201, A605, A6A5, AA65, B001, B0B1, BB01, BB41, 600A5, 7999B, 9999B, AAAA1, B04A1, B0B9B, BAA01, BAAA1, BB09B, BBBB1, 44AAA1, A00065, BBBAA1, AAA0001, B00099B, AA000001, BBBBBB99B, B0000000000000000000000000009B, 400000000000000000000000000000000000000077} as subsequences, prove that n is composite.

etc.

sweety439 2022-01-24 13:06

Thus, the total proof of base 10 includes these proofs:

* Prove that all of 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027 are primes. (of course, they are > 10, thus this part (i.e. all these numbers are > 10) needs no proof) (we can use ECPP (such as [URL="http://www.ellipsa.eu/public/primo/primo.html"]Primo[/URL]) to prove that the largest two numbers are defined primes (i.e. not merely PRPs), in this case of base 10, the largest number has only 31 digits and can be proved primality in <1 second, but in other case, such as base 13, 14, and 16, there are numbers > 10^10000 in the set, thus ECPP (or N-1, N+1, if this prime -1 or +1 can be trivially factored, such as the case of base 14, the largest prime 5*14^19698-1 in this set) is need to prove their primality)

* Prove that all proper subsequence of all elements in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} which are > 10 are composite. (this is the easiest part of all these parts, as we can use either [URL="https://en.wikipedia.org/wiki/Trial_division"]trial division[/URL] or [URL="https://en.wikipedia.org/wiki/Fermat_primality_test"]Fermat test[/URL] to prove their compositeness (if these numbers have small prime divisors, or these numbers fails the Fermat primality tests, then they are defined composite), unless the numbers are [URL="https://en.wikipedia.org/wiki/Fermat_pseudoprime"]Fermat pseudoprimes[/URL] to many bases (say bases 2, 3, 5, 7, 11) with no small divisors (say < 2^32), in this case, we need to run either [URL="https://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test"]Miller–Rabin primality test[/URL] or [URL="https://en.wikipedia.org/wiki/Lucas_primality_test"]Lucas primality test[/URL] to prove their compositeness)

* Prove that all numbers > 10 not containing any element in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} as subsequence are composite. (for this part, we use either [URL="http://irvinemclean.com/maths/siercvr.htm"]covering congruences[/URL] or [URL="https://en.wikipedia.org/wiki/Factorization_of_polynomials"]algebraic factorization[/URL] (or combine of them, such as the base 12 family {B}9B and the base 14 family 8{D}) to prove that all numbers in a given family (may be non-simple family, such as many families in base 29 and 41) are composite)

sweety439 2022-01-25 00:24

A conjecture between this minimal prime problem (i.e. start with b+1) and [URL="https://cs.uwaterloo.ca/~cbright/reports/mepn.pdf"]the original minimal prime problem[/URL] (i.e. p > b is not required): In all bases other than 2, 3, 6, the largest minimal prime (start with b+1) is not minimal prime when p > b is not required, equivalently, the largest minimal prime (start with b+1) always contain either at least one prime digit or (contain the string "10" as subsequence and the base (b) is prime), also equivalently, the largest minimal prime (start with b+1) is not equal (must be larger than) the largest minimal prime when p > b is not required.

For the bases such that the original minimal prime problem (i.e. p > b is not required) is solved, this conjecture is verified in these bases b:

* b=5: this new minimal prime problem is also solved, and 1(0^93)13 contain the prime digit 3 and contain the string "10" as subsequence and the base (5) is prime
* b=7: this new minimal prime problem is also solved, and (3^16)1 contain the prime digit 3
* b=8: this new minimal prime problem is also solved, and (4^220)7 contain the prime digit 7
* b=9: there is minimal prime (start with b+1) > the largest original minimal prime, such as 3(0^1158)11
* b=10: this new minimal prime problem is also solved, and 5(0^28)27 contain the prime digits 2, 5, 7
* b=11: there is minimal prime (start with b+1) > the largest original minimal prime, such as 55(7^1011)
* b=12: this new minimal prime problem is also solved, and 4(0^39)77 contain the prime digit 7
* b=13: not verified (may be false), since there is no known minimal prime (start with b+1) which is > the largest original minimal prime
* b=14: there is minimal prime (start with b+1) > the largest original minimal prime, such as 4(D^19698)
* b=15: there is minimal prime (start with b+1) > the largest original minimal prime, such as (7^155)97
* b=16: there is minimal prime (start with b+1) > the largest original minimal prime, such as 5B(C^3700)D (D(B^32234) is only probable prime, not proven prime)
* b=17: original minimal prime problem not solved (there is an unsolved family F1{9}), but there are two known minimal primes (start with b+1) > the largest known original minimal prime (i.e. 4(9^111333)): 97(0^166047)1 and F7(0^186767)1
* b=18: there is minimal prime (start with b+1) > the largest original minimal prime, such as 8(0^298)B
* b=20: there is minimal prime (start with b+1) > the largest original minimal prime, such as C(D^2449)
* b=22: (I do not know whether there is minimal prime (start with b+1) > the largest original minimal prime or not, but it is very likely)
* b=24: there is minimal prime (start with b+1) > the largest original minimal prime, such as 2(0^313)7
* b=30: there is minimal prime (start with b+1) > the largest original minimal prime, such as O(T^34205)
* b=42: there is minimal prime (start with b+1) > the largest original minimal prime, such as 2(f^2523)
* b=60: (original minimal prime problem seems to be solved, and there is an unsolved family Z{x} in the minimal prime (start with b+1) problem)

sweety439 2022-01-28 23:07

[QUOTE=sweety439;569239]These are families I am interested: (of the form (a*b^n+c)/gcd(a+c,b-1) for fixed a>=1, b>=2, c != 0, gcd(a,c) = 1, gcd(b,c) = 1 and variable n) (although some of these families do not always product minimal primes (start with b+1))
[/QUOTE]
These (a*b^n+c)/gcd(a+c,b-1) (a>=1, b>=2, c != 0, gcd(a,c) = 1, gcd(b,c) = 1) families are [I]dual[/I] families (for the definition, see [URL="http://www.kurims.kyoto-u.ac.jp/EMIS/journals/INTEGERS/papers/i61/i61.pdf"]http://www.kurims.kyoto-u.ac.jp/EMIS/journals/INTEGERS/papers/i61/i61.pdf[/URL] and [URL="https://oeis.org/A076336/a076336c.html"]https://oeis.org/A076336/a076336c.html[/URL]):

* b^n+1 (self dual) (b == 0 mod 2)
* (b^n+1)/2 (self dual) (b == 1 mod 2)
* (b^n-1)/(b-1) (self dual)
* 2*b^n+1 and b^n+2 (b == 3, 5 mod 6)
* 2*b^n+1 and (b/2)*b^n+1 (b == 0, 2 mod 6)
* (2*b^n+1)/3 and (b^n+2)/3 (b == 1 mod 6)
* (2*b^n+1)/3 and ((b/2)*b^n+1)/3 (b == 4 mod 6)
* 2*b^n-1 and b^n-2 (b == 1 mod 2)
* 2*b^n-1 and (b/2)*b^n-1 (b == 0 mod 2)
* 3*b^n+1 and b^n+3 (b == 2, 4 mod 6)
* 3*b^n+1 and (b/3)*b^n+1 (b == 0 mod 6)
* (3*b^n+1)/2 and (b^n+3)/2 (b == 7, 11 mod 12)
* (3*b^n+1)/2 and ((b/3)*b^n+1)/2 (b == 3 mod 12)
* (3*b^n+1)/4 and (b^n+3)/4 (b == 1, 5 mod 12)
* (3*b^n+1)/4 and ((b/3)*b^n+1)/4 (b == 9 mod 12)
* 3*b^n-1 and b^n-3 (b == 2, 4 mod 6)
* 3*b^n-1 and (b/3)*b^n-1 (b == 0 mod 6)
* (3*b^n-1)/2 and (b^n-3)/2 (b == 1, 5 mod 6)
* (3*b^n-1)/2 and ((b/3)*b^n-1)/2 (b == 3 mod 6)
* (b-1)*b^n-1 and b^n-(b-1)
* (b-1)*b^n+1 and b^n+(b-1)
* (b+1)*b^n-1 and b^n-(b+1)
* (b+1)*b^n+1 and b^n+(b+1) (b == 0, 2 mod 3)
* ((b+1)*b^n+1)/3 and (b^n+(b+1))/3 (b == 1 mod 3)
* ((b-2)*b^n+1)/(b-1) and (b^n+(b-2))/(b-1) (b == 1 mod 2)
* ((b-2)*b^n+1)/(b-1) and ((1/[URL="https://oeis.org/A006519"]A006519[/URL](b-2))*b^n+[URL="https://oeis.org/A000265"]A000265[/URL](b-2))/(b-1) (b == 0 mod 2)
* ((2*b-1)*b^n-1)/(b-1) and (b^n-(2*b-1))/(b-1)

Note: 1/[URL="https://oeis.org/A006519"]A006519[/URL](b-2) is not integer (for b == 0 mod 2), we should start with the smallest n such that (1/[URL="https://oeis.org/A006519"]A006519[/URL](b-2))*b^n is integer

sweety439 2022-02-02 13:54

y{z} and {z}1 are dual families (their algebraic forms are (b-1)*b^n-1 and b^n-(b-1))
families {z}1 and z{0}1 with length 2 are the same number z1 (its value is b^n-b+1)
z{0}1 and 1{0}z are dual families (their algebraic forms are (b-1)*b^n+1 and b^n+(b-1))
families 1{0}z and 1{z} with length 2 are the same number 1z (its value is 2*b-1)
1{z} and {z}y are dual families when the base (b) is odd (their algebraic forms are 2*b^n-1 and b^n-2)

{1} is self dual (its algebraic form is (b^n-1)/(b-1))
families {1} and 1{0}1 with length 2 are the same number 11 (its value is b+1)
1{0}1 is self dual (its algebraic form is (b^n+1)/2)
families {1} and 1{0}11 and 11{0}1 with length 3 are the same number 111 (its value is b^n+b+1)
1{0}11 and 11{0}1 are dual families (their algebraic forms are b^n+(b+1) and (b+1)*b^n+1)

2{0}1 and 1{0}2 are dual families (their algebraic forms are 2*b^n+1 and b^n+2)

sweety439 2022-02-03 01:25

(Note: families 1{0}11 and 11{0}1 only works for length >= 3, since both they have 3 digits not in {})

OEIS sequences of the smallest primes in these families:

y{z}: [URL="https://oeis.org/A122396"]A122396[/URL] (prime bases, numbers are added by 1)
{z}1: [URL="https://oeis.org/A113516"]A113516[/URL], [URL="https://oeis.org/A343589"]A343589[/URL] (corresponding primes)
z{0}1: [URL="https://oeis.org/A305531"]A305531[/URL], [URL="https://oeis.org/A087139"]A087139[/URL] (prime bases, numbers are added by 1)
1{0}z: [URL="https://oeis.org/A076845"]A076845[/URL], [URL="https://oeis.org/A076846"]A076846[/URL] (corresponding primes), [URL="https://oeis.org/A078178"]A078178[/URL] (length >= 3), [URL="https://oeis.org/A078179"]A078179[/URL] (length >= 3, corresponding primes)
1{z}: [URL="https://oeis.org/A119591"]A119591[/URL], [URL="https://oeis.org/A098873"]A098873[/URL] (bases == 0 mod 6)
{z}y: [URL="https://oeis.org/A250200"]A250200[/URL], [URL="https://oeis.org/A255707"]A255707[/URL] (length 1 (i.e. the single-digit prime "y") is allowed), [URL="https://oeis.org/A292201"]A292201[/URL] (length 1 (i.e. the single-digit prime "y") is allowed, prime bases)

{1}: [URL="https://oeis.org/A084740"]A084740[/URL], [URL="https://oeis.org/A084738"]A084738[/URL] (corresponding primes), [URL="https://oeis.org/A128164"]A128164[/URL] (length >= 3), [URL="https://oeis.org/A285642"]A285642[/URL] (length >= 3, corresponding primes), [URL="https://oeis.org/A065854"]A065854[/URL] (prime bases), [URL="https://oeis.org/A279068"]A279068[/URL] (prime bases, corresponding primes)
1{0}1: [URL="https://oeis.org/A079706"]A079706[/URL], [URL="https://oeis.org/A084712"]A084712[/URL] (corresponding primes) [URL="https://oeis.org/A228101"]A228101[/URL] (exponent of 2 for "length-1") [URL="https://oeis.org/A123669"]A123669[/URL] (length >= 3, corresponding primes)
1{0}11: [URL="https://oeis.org/A346149"]A346149[/URL], [URL="https://oeis.org/A346154"]A346154[/URL] (corresponding primes)

2{0}1: [URL="https://oeis.org/A119624"]A119624[/URL], [URL="https://oeis.org/A253178"]A253178[/URL] (bases not == 1 mod 3), [URL="https://oeis.org/A098872"]A098872[/URL] (bases == 0 mod 6)
1{0}2: [URL="https://oeis.org/A138066"]A138066[/URL], [URL="https://oeis.org/A084713"]A084713[/URL] (corresponding primes), [URL="https://oeis.org/A138067"]A138067[/URL] (length >= 3)

Also, the bases that the numbers listed in previous post are primes:

z1: [URL="https://oeis.org/A055494"]A055494[/URL]
1z: [URL="https://oeis.org/A006254"]A006254[/URL]
11: [URL="https://oeis.org/A006093"]A006093[/URL]
111: [URL="https://oeis.org/A002384"]A002384[/URL]

sweety439 2022-02-03 01:35

[QUOTE=sweety439;593116]we can consider the set of "strings of base b digits" which have no "prime strings > base" as subsequences.

In base 2, such strings are {0} (i.e. 000...000 with any number of 0's) and {0}1{0} (i.e. 000...0001000...000 with any number of 0's before the 1 and any number of 0's after the 1), i.e. all strings with at most one 1

In base 3, such strings are {0}1{0} and {0}1{0}1{0} and {0,2} (i.e. any combinations of any number of 0's and any number of 2's)

In base 10, the set of such strings are not simply to write, however, if "primes > base" is not needed, then such strings are any strings n such that [URL="https://oeis.org/A039995"]A039995[/URL](n) = 0 (not [URL="https://oeis.org/A062115"]A062115[/URL], since [URL="https://oeis.org/A062115"]A062115[/URL] is for [URL="https://en.wikipedia.org/wiki/Substring"]substring[/URL] instead of [URL="https://en.wikipedia.org/wiki/Subsequence"]subsequence[/URL], i.e. [URL="https://oeis.org/A062115"]A062115[/URL] is the numbers n such that [URL="https://oeis.org/A039997"]A039997[/URL](n) = 0 instead of the numbers n such that [URL="https://oeis.org/A039995"]A039995[/URL](n) = 0) with any number (including 0) of leading zeros.

Such strings are called [B]primefree strings[/B] in this post.[/QUOTE]

The largest "non-obvious" primefree strings in base b:

b=2: none
b=3: 11 (4 in decimal)
b=4: 21 (9 in decimal)
b=5: 1(0^92)13 (the largest minimal prime (start with b+1) is 1(0^93)13)
b=6: 4041 (889 in decimal) (if 4041 is prime, then the largest "non-obvious" primefree strings would be 441 (169 in decimal))
b=7: (3^15)1 (the largest minimal prime (start with b+1) is (3^16)1)
b=8: (4^219)7 (the largest minimal prime (start with b+1) is (4^220)7)

sweety439 2022-02-04 07:29

OEIS sequence and top prime/PRP for the families:

[CODE]
base 2:

{1} [URL="https://oeis.org/A000043"]OEIS[/URL] [URL="http://primes.utm.edu/primes/search.php?Description=%5E2%5E%-1&Style=HTML"]top[/URL]
1{0}1 [URL="https://oeis.org/A%3F%3F%3F%3F%3F%3F"]OEIS[/URL] [URL="http://primes.utm.edu/primes/search.php?Description=%5E2%5E%%2B1&Style=HTML"]top[/URL] [URL="http://www.prothsearch.com/fermat.html"]status[/URL]

base 3:

{1} [URL="https://oeis.org/A028491"]OEIS[/URL] [URL="http://www.primenumbers.net/prptop/searchform.php?form=%283%5En-1%29%2F2&action=Search"]top[/URL]
1{2} [URL="https://oeis.org/A003307"]OEIS[/URL] [URL="http://primes.utm.edu/primes/search.php?Description=%5E2*3%5E%-1&Style=HTML"]top[/URL]
{1}2 [URL="https://oeis.org/A171381"]OEIS[/URL] [URL="http://www.primenumbers.net/prptop/searchform.php?form=%283%5En%2B1%29%2F2&action=Search"]top[/URL] [URL="http://www.prothsearch.com/GFN03.html"]status[/URL]
2{1} [URL="https://oeis.org/A%3F%3F%3F%3F%3F%3F"]OEIS[/URL] [URL="http://www.primenumbers.net/prptop/searchform.php?form=%285*3%5En-1%29%2F2&action=Search"]top[/URL]
{2}1 [URL="https://oeis.org/A014224"]OEIS[/URL] [URL="http://www.primenumbers.net/prptop/searchform.php?form=3%5En-2&action=Search"]top[/URL]
1{0}2 [URL="https://oeis.org/A051783"]OEIS[/URL] [URL="http://www.primenumbers.net/prptop/searchform.php?form=3%5En%2B2&action=Search"]top[/URL]
2{0}1 [URL="https://oeis.org/A003306"]OEIS[/URL] [URL="http://primes.utm.edu/primes/search.php?Description=%5E2*3%5E%%2B1&Style=HTML"]top[/URL]

base 4 (perfect power bases can be converted to their "ground bases" (references: [URL="https://www.rose-hulman.edu/~rickert/Compositeseq/#b9d4"]the case base 3 families *{1} converted to base 9=3^2 families *{4}[/URL] [URL="https://en.wikipedia.org/wiki/Truncatable_prime#Other_bases"]the case n base 10 digits is equivalent to one base 10^n digit, in research of truncatable primes (instead of minimal primes) to other bases[/URL] [URL="https://en.wikipedia.org/wiki/Talk:P-adic_number/Archive_1#%22The_reason_for_this_property_turns_out_to_be_that_10_is_a_composite_number_which_is_not_a_power_of_a_prime.%22"]the case of p-adic numbers, one base p^n digit is equivalent to n base p digits[/URL], also see the [URL="https://en.wikipedia.org/wiki/Automorphic_number"]automorphic numbers[/URL] to base b and base b^n, they are equivalent)):

1{3} = {1} (number of 1's in {} is even) in base 2 [URL="https://oeis.org/A146768"]OEIS[/URL] [URL="http://primes.utm.edu/primes/search.php?Description=%5E2%5E%-1&Style=HTML"]top[/URL]
2{3} = 10{1} (number of 1's in {} is even) in base 2 [URL="https://oeis.org/A272057"]OEIS[/URL] [URL="http://primes.utm.edu/primes/search.php?Description=%5E3*2%5E%-1&Style=HTML"]top[/URL]
{2}3 = {10}11 in base 2 [URL="https://oeis.org/A127936"]OEIS[/URL] [URL="http://www.primenumbers.net/prptop/searchform.php?form=%282%5En%2B1%29%2F3&action=Search"]top[/URL]
{3}1 = {1}01 (number of 1's in {} is even) in base 2 [URL="https://oeis.org/A059266"]OEIS[/URL] [URL="http://www.primenumbers.net/prptop/searchform.php?form=2%5En-3&action=Search"]top[/URL]
1{0}1 = 1{0}1 (number of 0's in {} is odd) in base 2 [URL="https://oeis.org/A%3F%3F%3F%3F%3F%3F"]OEIS[/URL] [URL="http://primes.utm.edu/primes/search.php?Description=%5E2%5E%%2B1&Style=HTML"]top[/URL] [URL="http://www.prothsearch.com/fermat.html"]status[/URL]
1{0}3 = 1{0}11 (number of 0's in {} is even) in base 2 [URL="https://oeis.org/A089437"]OEIS[/URL] [URL="http://www.primenumbers.net/prptop/searchform.php?form=2%5En%2B3&action=Search"]top[/URL]
3{0}1 = 11{0}1 (number of 0's in {} is odd) in base 2 [URL="https://oeis.org/A326655"]OEIS[/URL] [URL="http://primes.utm.edu/primes/search.php?Description=%5E3*2%5E%%2B1&Style=HTML"]top[/URL]

base 5:

{1} [URL="https://oeis.org/A004061"]OEIS[/URL] [URL="http://www.primenumbers.net/prptop/searchform.php?form=%285%5En-1%29%2F4&action=Search"]top[/URL]
1{4} [URL="https://oeis.org/A120375"]OEIS[/URL] [URL="http://primes.utm.edu/primes/search.php?Description=%5E2*5%5E%-1&Style=HTML"]top[/URL]
{2}3 [URL="https://oeis.org/A%3F%3F%3F%3F%3F%3F"]OEIS[/URL] [URL="http://www.primenumbers.net/prptop/searchform.php?form=%285%5En%2B1%29%2F2&action=Search"]top[/URL] [URL="http://www.prothsearch.com/GFN05.html"]status[/URL]
3{4} [URL="https://oeis.org/A046865"]OEIS[/URL] [URL="http://primes.utm.edu/primes/search.php?Description=%5E4*5%5E%-1&Style=HTML"]top[/URL]
{4}1 [URL="https://oeis.org/A059613"]OEIS[/URL] [URL="http://www.primenumbers.net/prptop/searchform.php?form=5%5En-4&action=Search"]top[/URL]
{4}3 [URL="https://oeis.org/A109080"]OEIS[/URL] [URL="http://www.primenumbers.net/prptop/searchform.php?form=5%5En-2&action=Search"]top[/URL]
1{0}2 [URL="https://oeis.org/A087885"]OEIS[/URL] [URL="http://www.primenumbers.net/prptop/searchform.php?form=5%5En%2B2&action=Search"]top[/URL]
1{0}4 [URL="https://oeis.org/A124621"]OEIS[/URL] [URL="http://www.primenumbers.net/prptop/searchform.php?form=5%5En%2B4&action=Search"]top[/URL]
2{0}1 [URL="https://oeis.org/A058934"]OEIS[/URL] [URL="http://primes.utm.edu/primes/search.php?Description=%5E2*5%5E%%2B1&Style=HTML"]top[/URL]
4{0}1 [URL="https://oeis.org/A204322"]OEIS[/URL] [URL="http://primes.utm.edu/primes/search.php?Description=%5E4*5%5E%%2B1&Style=HTML"]top[/URL]

base 6:

{1} [URL="https://oeis.org/A004062"]OEIS[/URL] [URL="http://www.primenumbers.net/prptop/searchform.php?form=%286%5En-1%29%2F5&action=Search"]top[/URL]
1{5} [URL="https://oeis.org/A057472"]OEIS[/URL] [URL="http://primes.utm.edu/primes/search.php?Description=%5E2*6%5E%-1&Style=HTML"]top[/URL]
2{5} [URL="https://oeis.org/A186106"]OEIS[/URL] [URL="http://primes.utm.edu/primes/search.php?Description=%5E3*6%5E%-1&Style=HTML"]top[/URL]
3{5} [URL="https://oeis.org/A%3F%3F%3F%3F%3F%3F"]OEIS[/URL] [URL="http://primes.utm.edu/primes/search.php?Description=%5E4*6%5E%-1&Style=HTML"]top[/URL]
4{5} [URL="https://oeis.org/A079906"]OEIS[/URL] [URL="http://primes.utm.edu/primes/search.php?Description=%5E5*6%5E%-1&Style=HTML"]top[/URL]
{4}5 [URL="https://oeis.org/A248613"]OEIS[/URL] [URL="http://www.primenumbers.net/prptop/searchform.php?form=%284*6%5En%2B1%29%2F5&action=Search"]top[/URL]
{5}1 [URL="https://oeis.org/A059614"]OEIS[/URL] [URL="http://www.primenumbers.net/prptop/searchform.php?form=6%5En-5&action=Search"]top[/URL]
1{0}1 [URL="https://oeis.org/A%3F%3F%3F%3F%3F%3F"]OEIS[/URL] [URL="http://primes.utm.edu/primes/search.php?Description=%5E6%5E%%2B1&Style=HTML"]top[/URL] [URL="http://www.prothsearch.com/GFN06.html"]status[/URL]
1{0}5 [URL="https://oeis.org/A145106"]OEIS[/URL] [URL="http://www.primenumbers.net/prptop/searchform.php?form=6%5En%2B5&action=Search"]top[/URL]
2{0}1 [URL="https://oeis.org/A120023"]OEIS[/URL] [URL="http://primes.utm.edu/primes/search.php?Description=%5E2*6%5E%%2B1&Style=HTML"]top[/URL]
3{0}1 [URL="https://oeis.org/A186112"]OEIS[/URL] [URL="http://primes.utm.edu/primes/search.php?Description=%5E3*6%5E%%2B1&Style=HTML"]top[/URL]
5{0}1 [URL="https://oeis.org/A247260"]OEIS[/URL] [URL="http://primes.utm.edu/primes/search.php?Description=%5E5*6%5E%%2B1&Style=HTML"]top[/URL]
[/CODE]

sweety439 2022-02-05 05:24

1 Attachment(s)
[QUOTE=sweety439;599263]y{z} and {z}1 are dual families (their algebraic forms are (b-1)*b^n-1 and b^n-(b-1))
families {z}1 and z{0}1 with length 2 are the same number z1 (its value is b^n-b+1)
z{0}1 and 1{0}z are dual families (their algebraic forms are (b-1)*b^n+1 and b^n+(b-1))
families 1{0}z and 1{z} with length 2 are the same number 1z (its value is 2*b-1)
1{z} and {z}y are dual families when the base (b) is odd (their algebraic forms are 2*b^n-1 and b^n-2)

{1} is self dual (its algebraic form is (b^n-1)/(b-1))
families {1} and 1{0}1 with length 2 are the same number 11 (its value is b+1)
1{0}1 is self dual (its algebraic form is (b^n+1)/2)
families {1} and 1{0}11 and 11{0}1 with length 3 are the same number 111 (its value is b^n+b+1)
1{0}11 and 11{0}1 are dual families (their algebraic forms are b^n+(b+1) and (b+1)*b^n+1)

2{0}1 and 1{0}2 are dual families (their algebraic forms are 2*b^n+1 and b^n+2)[/QUOTE]

z means digit value b-1, y means digit value b-2, etc. this will made the base b expression more convenient, see this example of Phi(n,b) (where Phi is the [URL="https://en.wikipedia.org/wiki/Cyclotomic_polynomial"]cyclotomic polynomial[/URL]) written in base b, which is related to the generalized [URL="https://en.wikipedia.org/wiki/Unique_prime"]unique primes[/URL] in base b

Note that Phi(10,b) = z0z1 in base b and Phi(12,b) = zz01 in base b are sometimes minimal primes (start with b+1) base b, the only conditions (assume Phi(10,b) and Phi(12,b) are primes) are z1 (= Phi(6,b)) and z01 (related to the family z{0}1 = (b-1)*b^n+1) and zz1 (related to the family {z}1 = b^n-(b-1)) are all composites.

[URL="https://oeis.org/A085398/b085398_2.txt"]this[/URL] is a list of the smallest base b (b>=2) such that Phi(n,b) is prime, for 1<=n<=2500

(it is notable that GFN and GRU are the only simple families in base b which are also cyclotomic numbers (i.e. numbers of the form Phi(n,b)/gcd(Phi(n,b),n), which is related to the generalized unique primes in base b), GFN is when n is power of 2, GRU is when n is prime (and when n is twice an odd prime, then the family is simple family in base b^2 (instead of base b)), see [URL="https://mersenneforum.org/showpost.php?p=568817&postcount=116"]this post[/URL])


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