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Old 2021-11-20, 10:39   #210
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"99(4^34019)99 palind"
Nov 2016
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Quote:
Originally Posted by sweety439 View Post
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 http://www.primenumbers.net/prptop/s...&action=Search and http://www.fermatquotient.com/PrimSerien/GenRepuP.txt, 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
(note: base 9409 (i.e. b=97) has been searched to length 249998 by Fan Ming without any prime or PRP found, see this post)

Last fiddled with by sweety439 on 2021-11-24 at 09:43
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Old 2021-11-20, 10:57   #211
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This problem is better than the original minimal prime problem since this problem is regardless whether 1 is considered as prime or not, i.e. no matter 1 is considered as prime or not prime (in the beginning of the 20th century, 1 is regarded as prime), 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 (a*b^n+c)/gcd(a+c,b-1) (a>=1, b>=2, c != 0, gcd(a,c) = 1, gcd(b,c) = 1)?" for any given (a,b,c) triple satisfying that a>=1, b>=2, c != 0, gcd(a,c) = 1, gcd(b,c) = 1 (see this post), we can extend this problem to a large base which is power 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)
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Old 2021-11-21, 15:27   #212
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For the forms in https://docs.google.com/spreadsheets...RwmKME/pubhtml, 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 A096059 (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 proven 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 proven to be 29, while the largest prime is 1000004 (with 7 digits) in base 23, which is equal to 23^6+4
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Old 2021-11-23, 15:45   #213
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Every base b family is the dual (for the definition, see http://www.kurims.kyoto-u.ac.jp/EMIS...rs/i61/i61.pdf and https://oeis.org/A076336/a076336c.html and https://mersenneforum.org/showthread.php?t=10761 and https://mersenneforum.org/showthread.php?t=6545 and https://mersenneforum.org/showthread.php?t=26328 and https://mersenneforum.org/showthread.php?t=21954, also see Five or Bust forum, the dual form has same Nash weight (or difficulty), the difficulty page 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))
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 any 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)

Last fiddled with by sweety439 on 2021-11-24 at 17:58
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Old 2021-11-23, 18:13   #214
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See https://stdkmd.net/nrr/prime/primesize.txt, 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:

http://www.prothsearch.net/cullen.html (broken link: from wayback machine cached copy) (b=2)
http://guenter.loeh.name/gc/status.html (3<=b<=100)
https://harvey563.tripod.com/GClist.txt (101<=b<=10000)
http://www.primzahlenarchiv.de/index.html (broken link: from wayback machine cached copy) (101<=b<=200)
http://www.primzahlenarchiv.de/news.html (broken link: from wayback machine cached copy) (101<=b<=200)
http://www.primzahlenarchiv.de/status.html (broken link: from wayback machine cached copy) (101<=b<=200)
https://www.rieselprime.de/ziki/Gen_Cullen_prime_table (all available bases b)
http://www.primegrid.com/forum_thread.php?id=7073 (primegrid search page)
http://www.primegrid.com/stats_cullen_llr.php (primegrid status page, b=2)
http://www.primegrid.com/stats_gcw_llr.php (primegrid status page, b>=3)
https://primes.utm.edu/top20/page.php?id=6 (top 20 page, b=2)
https://primes.utm.edu/top20/page.php?id=42 (top 20 page, b>=3)
https://oeis.org/A240234 (smallest n for given base b)
https://oeis.org/A327660 (smallest n>b-2 for given base b)

Generalized Woodall primes:

http://www.prothsearch.net/woodall.html (broken link: from wayback machine cached copy) (b=2)
https://harvey563.tripod.com/GWlist.txt (3<=b<=10000)
https://harvey563.tripod.com/General...dallPrimes.txt (3<=b<=10000)
http://science.kennesaw.edu/~jdemaio/generali.htm (broken link: from wayback machine cached copy) (3<=b<=100)
https://www.rieselprime.de/ziki/Gen_Woodall_prime_table (all available bases b)
http://www.primegrid.com/forum_thread.php?id=7073 (primegrid search page)
http://www.primegrid.com/stats_woodall_llr.php (primegrid status page, b=2)
http://www.primegrid.com/stats_gcw_llr.php (primegrid status page, b>=3)
https://primes.utm.edu/top20/page.php?id=7 (top 20 page, b=2)
https://primes.utm.edu/top20/page.php?id=45 (top 20 page, b>=3)
https://oeis.org/A240235 (smallest n for given base b)
https://oeis.org/A327661 (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:

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

Last fiddled with by sweety439 on 2021-11-23 at 18:31
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Old 2021-11-24, 18:45   #215
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Quote:
Originally Posted by sweety439 View Post
All GFN base b and GRU base b are strong-probable-primes to base b, so don't test with this base (see https://mersenneforum.org/showthread.php?t=10476&page=2, https://oeis.org/A171381, https://oeis.org/A028491), 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.
Our result about minimal primes (start with b+1) assume that a number which has passed Miller–Rabin tests 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 proven prime rigorously (and neither N−1 nor N+1 can be ≥33.3333% factored). The smallest composite number which passed Miller–Rabin tests to the first n prime bases are listed in https://oeis.org/A014233, 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 triangular numbers, 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 octagonal numbers, and such numbers N have 3*N+1 square numbers (the fully N-values with 3*N+1 square numbers are the generalized octagonal numbers)), or of the form (2*m+1)*(3*m+1) with 2*m+1 and 3*m+1 both primes (such numbers are pentagonal numbers with odd index, and such numbers N have 24*N+1 square numbers (the fully N-values with 24*N+1 square numbers are the generalized pentagonal numbers)) (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 divisor or 24, and generalized pentagonal numbers are important for the pentagonal number theorem)

e.g. A014233(12) = A014233(13) = 318665857834031151167461 = 399165290221 * 798330580441 = (399165290220+1) * (2*399165290220+1), and 8*318665857834031151167461+1 = 1596661160883^2 is a square, and for the currently only "probable primes that failed primality proof or had a factor" with order 21 in factordb, 1396981702787004809899378463251 = 835757651112751 * 1671515302225501 = (835757651112750+1) * (2*835757651112750+1), and 8*1396981702787004809899378463251+1 = 3343030604451003^2 is a square. (both of them are triangular numbers)

In fact, the triangular number (m+1)*(2*m+1) (if both m+1 and 2*m+1 are primes) are Fermat pseudoprime to all bases which are coprime to this triangular number and quadratic residue mod 2*m+1 (reference for a special example that base = 4, which is quadratic residue mod all numbers since it is square number)

Note that no polygonal numbers can be primes (except the trivial case, i.e. every n is the second n-gonal number), see this reference, 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.

Last fiddled with by sweety439 on 2021-11-25 at 12:35
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Old 2021-11-24, 18:59   #216
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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}, ...

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}, ...

they are the families (generalized octagonal number){1}, and since base b families can be converted to base b^n families for n>1, 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. (reference)

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}, ...

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}

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Old 2021-11-25, 11:48   #217
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Now I consider to add these families to the list, 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
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Old 2021-11-26, 14:56   #218
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Quote:
Originally Posted by sweety439 View Post
* 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
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 2

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 leading zeros or trailing zeros (or both))

Last fiddled with by sweety439 on 2021-11-26 at 20:13
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Old 2021-11-26, 17:44   #219
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"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36

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Quote:
Originally Posted by sweety439 View Post
Now I consider to add these families to the list, 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
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For 11{0}1 (the dual of 1{0}11), see https://www.rieselprime.de/ziki/Williams_prime_PP_least
For 10{z} (the dual of {z}yz), see https://www.rieselprime.de/ziki/Williams_prime_PM_least
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File Type: txt zzzzyz.txt (7.2 KB, 5 views)

Last fiddled with by sweety439 on 2021-11-26 at 17:44
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Old 2021-11-26, 22:14   #220
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"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36

22·3·7·37 Posts
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For families X{Y}Zb = (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}Zb 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 numerator(abs((a*b^n+c)/gcd(a+c,b-1))) are prime factors of b (including the case that numerator(abs((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 C{1} must be only algebra factors, not by covering congruence, since

Code:
length        factors
6       19 · 101 · 233 · 263
12      69173177 · 415039063
18      71 · 91337821853<11> · 1080830891927<13>
36      277 · 383 · 4253 · 9157 · 305497 · 2649281 · 5391097 · 141581477 · 39961446977<11>
72      337 · 499 · 1051 · 4598699002567<13> · 5169269868350933<16> · 132787369068406184845865339939<30> · 38714450336182811476979541935153363<35>
84      103 · 139 · 197 · 9277 · 500921 · 13288909 · (94-digit composite with no small factor)
and it 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 E{1} 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 ....

* {1} must be only algebra factors, not by covering congruence, since (9^n-1)/8 is divisibility sequence, by Zsigmondy's theorem, (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

* 3{1} 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

* 5{1} 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, 6{1} by both covering congruence 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
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