Minimal set of the strings for primes with at least two digits - Page 35

sweety439 · 2022-08-13, 20:22

Integers b>=2 sorted by A062955(b):

2 (1), 3 (4), 4 (6), 6 (10), 5 (16), 8 (28), 7&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&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), ...

Integers b>=2 sorted by number of minimal primes (starting with b+1) base b: (not sure if 26 and 28 are before 17 and 21)

2 (1), 3 (3), 4 (5), 6 (11), 5 (22), 7 (71), 8 (75), 10 (77), 12 (106), 9 (151), 18 (549), 14 (650), 11 (1068), 15 (1284), 16 (2347), 30 (2619), 13 (3196~3197), 20 (3314), 24 (3409), 22 (8003), 17 (10405~10428), 21 (13373~13395), ...

Integers b>=2 sorted by length of largest minimal prime (starting with b+1) base b:

2 (2), 3&4 (3), 6 (5), 7 (17), 10 (31), 12 (42), 5 (96), 15 (157), 8 (221), 9 (1161), 18&20 (6271), 24 (8134), 14 (19699), 22 (22003), 30 (34206), 11 (62669), 16 (116139), ...

Integers b>=2 sorted by value of largest minimal prime (starting with b+1) base b:

2 (3), 3 (13), 4 (41), 6 (5209), 7 ((7^17-5)/2, 15 decimal digits), 10 (5*10^30+27, 31 decimal digits), 12 (4*12^41+91, 45 decimal digits), 5 (5^95+8, 67 decimal digits), 15 ((15^157+59)/2, 185 decimal digits), 8 ((4*8^221+17)/7, 200 decimal digits), 9 (3*9^1160+10, 1108 decimal digits), 18 (12*18^6270+221, 7872 decimal digits), 20 (16*20^6270+13, 8159 decimal digits), 24 (13249*24^8131-49, 11227 decimal digits), 14 (5*14^19698-1, 22578 decimal digits), 22 ((251*22^22002-335)/21, 29538 decimal digits), 30 (25*30^34205-1, 50527 decimal digits), 11 ((57*11^62668-7)/10, 65263 decimal digits), 16 ((16^116139+619)/5, 139845 decimal digits), ...

These three sequences are conjectured to be similar, the integers b = 7 and b = 15 for the third sequence is relatively small since they (as well as b = 3) are high-weight bases (like CRUS bases R7, R15, S7, S15, they are high-weight bases), i.e. they are very "primeful", while b = 5 and b = 8 and b = 11 and b = 14 are relatively large, since they are low-weight bases (like CRUS, bases == 2 mod 3 are low-weight bases), although this does not hold for b = 20, which is also == 2 mod 3

bases b such that the number of minimal primes (starting with b+1) base b is in given range:

1: 2
2: none
3~4: 3
5~8: 4
9~16: 6
17~32: 5
33~64: none
65~128: 7, 8, 10, 12
129~256: 9
2^8+1~2^9: none
2^9+1~2^10: 18, 14
2^10+1~2^11: 11, 15
2^11+1~2^12: 16, 30, 13, 20, 24
2^12+1~2^13: 22
2^13+1~2^14: 17, 21, ...

bases b such that the length of largest minimal prime (starting with b+1) base b is in given range:

1: none
2: 2
3~4: 3, 4
5~8: 6
9~16: none
17~32: 7, 10
33~64: 12
65~128: 5
129~256: 15, 8
2^8+1~2^9: none
2^9+1~2^10: none
2^10+1~2^11: 9
2^11+1~2^12: none
2^12+1~2^13: 18, 20, 24
2^13+1~2^14: none
2^14+1~2^15: 14, 22
2^15+1~2^16: 30, 11, ...

An interesting thing between this new minimal prime problem and the original minimal prime (i.e. prime > base is not required) problem: (for bases b=14 and b=15)

For this new minimal prime problem:

base 14 "number of minimal primes" (650) < base 15 "number of minimal primes" (1284)
base 14 "length of largest minimal prime" (19699) > base 15 "length of largest minimal prime" (157)

But for the original minimal prime (i.e. prime > base is not required) problem:

base 14 "number of minimal primes" (240) > base 15 "number of minimal primes" (100)
base 14 "length of largest minimal prime" (86) < base 15 "length of largest minimal prime" (107)

sweety439 · 2022-08-13, 20:35

Quote:

Originally Posted by sweety439

Number of totally digits of minimal primes (start with b+1) in base b

Sum of all minimal primes (start with b+1) in base b

Product of all minimal primes (start with b+1) in base b

Base 2:

1 primes, totally 2 digits, sum, product

Base 3:

3 primes, totally 7 digits, sum, product

Base 4:

5 primes, totally 11 digits, sum, product

Base 5:

22 primes, totally 169 digits, sum, product

Base 6:

11 primes, totally 29 digits, sum, product

Base 7:

71 primes, totally 288 digits, sum, product

Base 8:

75 primes, totally 523 digits, sum, product

Base 9:

151 primes, totally 3004 digits, sum, product

Base 10:

77 primes, totally 310 digits, sum, product

Base 11:

1068 primes, totally 75414 digits, sum, product

Base 12:

106 primes, totally 433 digits, sum, product

Base 14:

650 primes, totally 25404 digits, sum, product

Base 15:

1284 primes, totally 8286 digits, sum, product

Base 18:

Conjecture: the sum of all minimal primes (start with b+1) base b is always in https://oeis.org/A063538, i.e. it must have a prime factor >= its square root, this has been verified for bases 2, 3, 4, 5, 6, 7, 8, 10, 12, 15, but this is very hard to prove or disprove, since proving or disproving this requires factoring large numbers.

sweety439 · 2022-08-13, 20:40

For the interest of "greatest prime factor ^2-1" sequences:

* N-1 primality proving
* N+1 primality proving
* P-1 integer factorization method
* P+1 integer factorization method

https://oeis.org/A087713 (greatest prime factor of p^2-1)
https://oeis.org/A024710 (the same sequence (start with p=11) of the A087713, which is the greatest prime factor of A024702)
https://oeis.org/A024702 ((p^2-1)/24)
https://oeis.org/A084920 (p^2-1)
https://oeis.org/A001248 (p^2)
https://oeis.org/A001318 (generalized pentagonal numbers, (n^2-1)/24)

sweety439 · 2022-10-14, 11:02

The data files are in https://github.com/xayahrainie4793/quasi-mepn-data

Now, bases 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 18, 20, 24 are solved, and bases 11, 16, 22, 30 are also solved if strong probable primes are allowed.

sweety439 · 2022-10-28, 03:34

Fully proven bases (i.e. include the primality proven of the PRPs) and almost proven bases (i.e. proven except the primality proven of the PRPs) sorted by the size of the largest primes in the set for the base:

Fully proven bases (sorted by largest to smallest): (in fact, b=14 has the largest prime: 5×14^19698−1 (22578 decimal digits), but since this prime can be easily proven prime using the N+1 primality proving, thus not counted)

1. b=24, 13249×24^8131−49, 11227 decimal digits, primality certificate

2. b=20, 16×20^6270+13, 8159 decimal digits, primality certificate

3. b=18, 12×18^6270+221, 7872 decimal digits, primality certificate

4. b=9, 3×9^1160+10, 1108 decimal digits, primality certificate

Almost proven bases (sorted by smallest to largest, i.e. base 22 would be the next possible fully proven bases, then base 30):

1. b=22, (251×22^22002−335)/21, 29538 decimal digits

2. b=30, 18×30^24609+13, 36352 decimal digits

3. b=11, (57×11^62668−7)/10, 65263 decimal digits

4. b=16, (16^116139+619)/5, 139845 decimal digits

sweety439 · 2022-11-07, 23:21

Nice news!!!

9(5^197420) (base 13) is probable prime!!! (its formula is (113*13^197420-5)/12, and at the time of its discovery it is the 1051st largest PRP in the PRP top)

Now base 13 has only one unsolved family (A{3}A)!!! Also bases 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 18, 20, 24 are already completely solved, and bases 11, 16, 22, 30 are also already solved if PRP are allowed, thus the base 13 family A{3}A is the only one unsolved family for bases up to 16 (if PRP are allowed)!!!

sweety439 · 2022-11-14, 00:19

New link for GitHub: https://github.com/xayahrainie4793/m...-prime-numbers (I renamed it, I think that this name is better)

sweety439 · 2022-11-16, 10:56

For the numbers < 10^25000, I require them to be definitely primes (i.e. not merely probable primes), N-1 proving or N+1 proving or ECPP proving or CHG proving can be used

For the numbers between 10^25000 and 10^500000, I require them to be strong PRP to all prime bases p <= 61 (see https://oeis.org/A014233), Lucas PRP with parameters (P, Q) defined by Selfridge's Method A (see https://oeis.org/A217255 and http://ntheory.org/pseudoprimes.html), trial factored to 2^64

For the numbers > 10^500000, I only require them to be strong PRP to base 2 and 3

(I require less conditions for larger numbers, because when passed the primality tests, the property that the number is in fact composite is less when the number is larger, see https://primes.utm.edu/notes/prp_prob.html and https://www.ams.org/journals/mcom/19...-0982368-4.pdf)

Only use Fermat tests (base b) is dangerous, especially these three types of numbers:

1. Numbers of the form Phi(n,b)/gcd(Phi(n,b),n) (where Phi is the cyclotomic polynomial) and their divisors (although this case is less dangerous since we can do Fermat tests to two bases which are not rational power of the other, such as 2 and 3, to avoid them to be PRP)

2. Numbers of the form (k+1)*(2*k+1) with k+1 and 2*k+1 both primes, and numbers of the form (k+1)*(3*k+1) with k+1 and 3*k+1 both primes (this case is even dangerous when the strong tests are used, however, this case is very easy to check: just check if 8*N+1 or 3*N+1 is square, since such numbers are hexagonal numbers and octagonal numbers, respectively)

3. Carmichael numbers (in this case, we can use strong tests, since there are no "strong Carmichael numbers", all composites are strong pseudoprimes to at most 1/4 of the bases coprime to them)

sweety439 · 2022-11-16, 11:17

https://mersenneforum.org/showpost.p...2&postcount=66

The numbers in x{y}z (where x and z are strings (may be empty) of digits in base b, y is a digit in base b) families are of the form (a×bⁿ+c)/gcd(a+c,b−1) for some fixed a, b, c such that a ≥ 1, b ≥ 2 (b is the base), c ≠ 0, gcd(a,c) = 1, gcd(b,c) = 1. Except in the special case c = ±1 and gcd(a+c,b−1) = 1, when n is large the known primality tests for such a number are too inefficient to run. In this case one must resort to a probable primality test such as a Miller–Rabin primality test or a Baillie–PSW primality test, unless a divisor of the number can be found. Since we are testing many numbers in an exponential sequence, it is possible to use a sieving process to find divisors rather than using trial division.

And when we sieve the sequence (a*b^n+c)/gcd(a+c,b-1) (which is equivalent to sieve the family x{y}z), then .... (below, r is linear functions of n, m is constant like a, b, c)

General:

1. If (a*b^n+c)/gcd(a+c,b-1) can be written as (m^r-1)/(m-1); display a warning message on the screen that this form is a generalized repunit number and could better be factored or sieved with another program (remove all composite r, and only sieve with the primes p == 1 mod r).

2. If (a*b^n+c)/gcd(a+c,b-1) can be written as (m^r+1)/(m+1); display a warning message on the screen that this form is a generalized Wagstaff number and could better be factored or sieved with another program (remove all composite r, and only sieve with the primes p == 1 mod 2*r).

3. If (a*b^n+c)/gcd(a+c,b-1) can be written as m^r+1 or (m^r+1)/2; display a warning message on the screen that this form is a generalized Fermat number and could better be factored or sieved with another program (remove all non-power-of-2 r, and no need to sieve, and use trial division with the primes == 1 mod 2*r).

Remove all n cases:

1. If a, b, -c are all squares; remove all n.

2. If a, b, c are all r-th powers for an odd r > 1; remove all n.

3. If b and 4*a*c are both 4-th powers; remove all n. These are Aurifeuillian factors.

The above should all be checked first before preceding.

Remove partial n cases:

1. If a and -c are both squares; remove all n == (0 mod 2).

2. If a and c are both r-th powers for an odd r > 1; for each such r, remove all n == (0 mod r).

3. If 4*a*c is 4-th power; remove all n == (0 mod 4).

4. If a*c and 4*b are both 4-th powers; remove all n == (1 mod 2).

5. If a*c is 4-th power and 2*b is square; remove all n == (2 mod 4).

#3, 4, and 5 are more Aurifeuillian factors.

Coordination with existing code:

1. If all n's are removed by algebraic factors for all sequences (a*b^n+c)/gcd(a+c,b-1), program should stop immediately.

2. If some n's are removed by algebraic factors, program continues sieving for removing the numbers with small prime factors.

3. Program should be able to handle input of one or multiple k's for a new base at the screen or in a file. Some (a*b^n+c)/gcd(a+c,b-1) could have algebraic factors while others do not.

4. Program should be able to handle an already sieved file as input, check the file for algebraic factors, remove them, and then continue sieving more deeply. Once again some (a*b^n+c)/gcd(a+c,b-1) could have algebraic factors while others do not.

sweety439 · 2022-11-21, 12:31

Quote:

Originally Posted by sweety439

Integers b>=2 sorted by A062955(b):

2 (1), 3 (4), 4 (6), 6 (10), 5 (16), 8 (28), 7&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&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), ...

Integers b>=2 sorted by number of minimal primes (starting with b+1) base b: (not sure if 26 and 28 are before 17 and 21)

2 (1), 3 (3), 4 (5), 6 (11), 5 (22), 7 (71), 8 (75), 10 (77), 12 (106), 9 (151), 18 (549), 14 (650), 11 (1068), 15 (1284), 16 (2347), 30 (2619), 13 (3196~3197), 20 (3314), 24 (3409), 22 (8003), 17 (10405~10428), 21 (13373~13395), ...

Integers b>=2 sorted by length of largest minimal prime (starting with b+1) base b:

2 (2), 3&4 (3), 6 (5), 7 (17), 10 (31), 12 (42), 5 (96), 15 (157), 8 (221), 9 (1161), 18&20 (6271), 24 (8134), 14 (19699), 22 (22003), 30 (34206), 11 (62669), 16 (116139), ...

Integers b>=2 sorted by value of largest minimal prime (starting with b+1) base b:

2 (3), 3 (13), 4 (41), 6 (5209), 7 ((7^17-5)/2, 15 decimal digits), 10 (5*10^30+27, 31 decimal digits), 12 (4*12^41+91, 45 decimal digits), 5 (5^95+8, 67 decimal digits), 15 ((15^157+59)/2, 185 decimal digits), 8 ((4*8^221+17)/7, 200 decimal digits), 9 (3*9^1160+10, 1108 decimal digits), 18 (12*18^6270+221, 7872 decimal digits), 20 (16*20^6270+13, 8159 decimal digits), 24 (13249*24^8131-49, 11227 decimal digits), 14 (5*14^19698-1, 22578 decimal digits), 22 ((251*22^22002-335)/21, 29538 decimal digits), 30 (25*30^34205-1, 50527 decimal digits), 11 ((57*11^62668-7)/10, 65263 decimal digits), 16 ((16^116139+619)/5, 139845 decimal digits), ...

Records for number of minimal primes (starting with b+1) base b:

2 (1), 3 (3), 4 (5), 5 (22), 7 (71), 8 (75), 9 (151), 11 (1068), 13 (3196 or 3197, 3197 assuming the heuristic argument that all unsolved families have a prime), 17 (10408~10428), 19 (31410~31435), 23 (65144~65276), 25 (100000+), …

Conjecture: All primes are in this sequence

Conjecture: All terms are primes or powers of primes (the converse is not true, the only counterexample among them is 16, which has only 2347 minimal primes (starting with b+1), less than base 13’s 3197 minimal primes (starting with b+1))

Records for length of largest minimal prime (starting with b+1) base b:

2 (2), 3 (3), 5 (96), 8 (221), 9 (1161), 11 (62669, PRP), 13 (>=197421, PRP), …

We should not conjecture that all primes except 7 are in this sequence, there is a possibility that the length of largest minimal prime (starting with b+1) base b = 17 is larger than length of largest minimal prime (starting with b+1) base b = 19, since the base 17 family F1{9} was searched to length 1000000 with no prime or PRP found, since you can see the case for base b = 5 and b = 7, the length of largest minimal prime (starting with b+1) base b = 5 is as long as 96, which is as large as about 6 times the length of largest minimal prime (starting with b+1) base b = 7 (17)

sweety439 · 2022-11-22, 04:40

See https://github.com/xayahrainie4793/m...main/README.md for more information of this project, including the condensed table (currently the condensed table is only available for bases 2~24, 26, 28, 30, 36)

Known large minimal primes (or PRPs) not in the data in https://github.com/xayahrainie4793/m...-prime-numbers: (also large minimal primes (or PRPs) for base 25 and base 27 in https://github.com/curtisbright/mepn-data)

Base 17:

4(9^111333), the algebraic form is (73*17^111333-9)/16, which is only PRP
97(0^166047)1, the algebraic form is 160*17^166048+1, which is definitely prime (reference: http://www.noprimeleftbehind.net/cru...ctures.htm#S17)
F7(0^186767)1, the algebraic form is 262*17^186768+1, which is definitely prime (reference: http://www.noprimeleftbehind.net/cru...ctures.htm#S17)

thus base 17 in fact has only 17 unsolved families

Base 19:

FG(6^110984), the algebraic form is (904*19^110984-1)/3, which is only PRP
1E7(0^122896)1, the algebraic form is 634*19^122897+1, which is definitely prime (reference: http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm#S19)

Base 21:

4(0^47333)9G, the algebraic form is 4*21^47335+205, which is only PRP
C(F^479147)0K, the algebraic form is (51*21^479149-1243)/4, which is only PRP

Base 23:

AI(F^21143), the algebraic form is (5471*23^21143-15)/22, which is only PRP
K9A(E^23275), the algebraic form is (118774*23^23275-7)/11, which is only PRP
96(E^25511), the algebraic form is (2350*23^25511-7)/11, which is only PRP
8(0^119214)1, the algebraic form is 8*23^119215+1, which is definitely prime
9(E^800873), the algebraic form is (106*23^800873-7)/11, which is only PRP

Base 25:

1J71(0^96272)1, the algebraic form is 27676*25^96273+1, which is definitely prime

71JD(0^458549)1, the algebraic form is 110488*25^458550+1, which is definitely prime (reference: http://www.primegrid.com/forum_thread.php?id=5087, 110488*5^917100+1 found by Ronny Willig on 25 March 2013)

DKJ(0^246808)1, the algebraic form is 8644*25^246809+1, which is definitely prime

KJD(0^63399)1, the algebraic form is 12988*25^63400+1, which is definitely prime

Base 32:

NU(0^661863)1, the algebraic form is 766*32^661864+1, which is definitely prime (reference: http://www.prothsearch.com/riesel1a.html, k=383)

Base 33:

13(0^23614)1, the algebraic form is 36*33^23615+1, which is definitely prime (reference: http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm#S33)
N7(0^610411)1, the algebraic form is 766*33^610412+1, which is definitely prime (reference: http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm#S33)

Base 35:

1B(0^56061)1, the algebraic form is 46*35^56062+1, which is definitely prime (reference: http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm#S35)

Base 36:

(P^81993)SZ, the algebraic form is (5*36^81995+821)/7, which is only PRP

Known test limits for some families larger than the test limits for the corresponding bases listed in https://github.com/xayahrainie4793/m...-prime-numbers:

Base 17:

F1{9}: tested to length 1000048 (reference: https://github.com/curtisbright/mepn...a/sieve.17.txt)

Base 19:

EE1{6}: tested to length 707350 (reference: https://github.com/curtisbright/mepn...a/sieve.19.txt)

Base 21:

G{0}FK: tested to length 506722 (reference: https://github.com/curtisbright/mepn...a/sieve.21.txt)

Base 23:

H3{0}1: tested to length 700003 (reference: http://www.noprimeleftbehind.net/cru...tures.htm#S529)

JH{0}1: tested to length 700003 (reference: http://www.noprimeleftbehind.net/cru...tures.htm#S529)

Base 25:

D71J{0}1: tested to length 350004 (reference: http://www.noprimeleftbehind.net/cru...25-reserve.htm)

EF{O}: tested to length 300002 (reference: http://www.noprimeleftbehind.net/cru...25-reserve.htm)

Base 31:

{F}G: tested to length 16777215 (reference: http://factordb.com/index.php?query=...%29%2B1%29%2F2 and http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt)

Base 32:

4{0}1: tested to length 1717986918 (reference: http://www.prothsearch.com/fermat.html)

G{0}1: tested to length 3435973836 (reference: http://www.prothsearch.com/fermat.html)

NG{0}1: tested to length 1800001 (reference: http://www.prothsearch.com/riesel1.html, k=47)

UG{0}1: tested to length 720001 (reference: http://www.prothsearch.com/riesel1.html, k=61)

2022-08-13, 20:22	#375
sweety439 "99(4^34019)99 palind" Nov 2016 (P^81993)SZ base 36 2·7·263 Posts	Integers b>=2 sorted by A062955(b): 2 (1), 3 (4), 4 (6), 6 (10), 5 (16), 8 (28), 7&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&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), ... Integers b>=2 sorted by number of minimal primes (starting with b+1) base b: (not sure if 26 and 28 are before 17 and 21) 2 (1), 3 (3), 4 (5), 6 (11), 5 (22), 7 (71), 8 (75), 10 (77), 12 (106), 9 (151), 18 (549), 14 (650), 11 (1068), 15 (1284), 16 (2347), 30 (2619), 13 (3196~3197), 20 (3314), 24 (3409), 22 (8003), 17 (10405~10428), 21 (13373~13395), ... Integers b>=2 sorted by length of largest minimal prime (starting with b+1) base b: 2 (2), 3&4 (3), 6 (5), 7 (17), 10 (31), 12 (42), 5 (96), 15 (157), 8 (221), 9 (1161), 18&20 (6271), 24 (8134), 14 (19699), 22 (22003), 30 (34206), 11 (62669), 16 (116139), ... Integers b>=2 sorted by value of largest minimal prime (starting with b+1) base b: 2 (3), 3 (13), 4 (41), 6 (5209), 7 ((7^17-5)/2, 15 decimal digits), 10 (510^30+27, 31 decimal digits), 12 (412^41+91, 45 decimal digits), 5 (5^95+8, 67 decimal digits), 15 ((15^157+59)/2, 185 decimal digits), 8 ((48^221+17)/7, 200 decimal digits), 9 (39^1160+10, 1108 decimal digits), 18 (1218^6270+221, 7872 decimal digits), 20 (1620^6270+13, 8159 decimal digits), 24 (1324924^8131-49, 11227 decimal digits), 14 (514^19698-1, 22578 decimal digits), 22 ((25122^22002-335)/21, 29538 decimal digits), 30 (2530^34205-1, 50527 decimal digits), 11 ((5711^62668-7)/10, 65263 decimal digits), 16 ((16^116139+619)/5, 139845 decimal digits), ... These three sequences are conjectured to be similar, the integers b = 7 and b = 15 for the third sequence is relatively small since they (as well as b = 3) are high-weight bases (like CRUS bases R7, R15, S7, S15, they are high-weight bases), i.e. they are very "primeful", while b = 5 and b = 8 and b = 11 and b = 14 are relatively large, since they are low-weight bases (like CRUS, bases == 2 mod 3 are low-weight bases), although this does not hold for b = 20, which is also == 2 mod 3 bases b such that the number of minimal primes (starting with b+1) base b is in given range: 1: 2 2: none 3~4: 3 5~8: 4 9~16: 6 17~32: 5 33~64: none 65~128: 7, 8, 10, 12 129~256: 9 2^8+1~2^9: none 2^9+1~2^10: 18, 14 2^10+1~2^11: 11, 15 2^11+1~2^12: 16, 30, 13, 20, 24 2^12+1~2^13: 22 2^13+1~2^14: 17, 21, ... bases b such that the length of largest minimal prime (starting with b+1) base b is in given range: 1: none 2: 2 3~4: 3, 4 5~8: 6 9~16: none 17~32: 7, 10 33~64: 12 65~128: 5 129~256: 15, 8 2^8+1~2^9: none 2^9+1~2^10: none 2^10+1~2^11: 9 2^11+1~2^12: none 2^12+1~2^13: 18, 20, 24 2^13+1~2^14: none 2^14+1~2^15: 14, 22 2^15+1~2^16: 30, 11, ... An interesting thing between this new minimal prime problem and the original minimal prime (i.e. prime > base is not required) problem: (for bases b=14 and b=15) For this new minimal prime problem: base 14 "number of minimal primes" (650) < base 15 "number of minimal primes" (1284) base 14 "length of largest minimal prime" (19699) > base 15 "length of largest minimal prime" (157) But for the original minimal prime (i.e. prime > base is not required) problem: base 14 "number of minimal primes" (240) > base 15 "number of minimal primes" (100) base 14 "length of largest minimal prime" (86) < base 15 "length of largest minimal prime" (107) Last fiddled with by sweety439 on 2022-11-20 at 15:14*

2022-11-07, 23:21	#380
sweety439 "99(4^34019)99 palind" Nov 2016 (P^81993)SZ base 36 7142₈ Posts	Nice news!!! 9(5^197420) (base 13) is probable prime!!! (its formula is (11313^197420-5)/12, and at the time of its discovery it is the 1051st largest PRP in the PRP top) Now base 13 has only one unsolved family (A{3}A)!!! Also bases 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 18, 20, 24 are already completely solved, and bases 11, 16, 22, 30 are also already solved if PRP are allowed, thus the base 13 family A{3}A is the only one unsolved family for bases up to 16 (if PRP are allowed)!!! Last fiddled with by sweety439 on 2022-11-07 at 23:24*

2022-11-16, 11:17	#383
sweety439 "99(4^34019)99 palind" Nov 2016 (P^81993)SZ base 36 7142₈ Posts	https://mersenneforum.org/showpost.p...2&postcount=66 The numbers in x{y}z (where x and z are strings (may be empty) of digits in base b, y is a digit in base b) families are of the form (a×bⁿ+c)/gcd(a+c,b−1) for some fixed a, b, c such that a ≥ 1, b ≥ 2 (b is the base), c ≠ 0, gcd(a,c) = 1, gcd(b,c) = 1. Except in the special case c = ±1 and gcd(a+c,b−1) = 1, when n is large the known primality tests for such a number are too inefficient to run. In this case one must resort to a probable primality test such as a Miller–Rabin primality test or a Baillie–PSW primality test, unless a divisor of the number can be found. Since we are testing many numbers in an exponential sequence, it is possible to use a sieving process to find divisors rather than using trial division. And when we sieve the sequence (ab^n+c)/gcd(a+c,b-1) (which is equivalent to sieve the family x{y}z), then .... (below, r is linear functions of n, m is constant like a, b, c) General: 1. If (ab^n+c)/gcd(a+c,b-1) can be written as (m^r-1)/(m-1); display a warning message on the screen that this form is a generalized repunit number and could better be factored or sieved with another program (remove all composite r, and only sieve with the primes p == 1 mod r). 2. If (ab^n+c)/gcd(a+c,b-1) can be written as (m^r+1)/(m+1); display a warning message on the screen that this form is a generalized Wagstaff number and could better be factored or sieved with another program (remove all composite r, and only sieve with the primes p == 1 mod 2r). 3. If (ab^n+c)/gcd(a+c,b-1) can be written as m^r+1 or (m^r+1)/2; display a warning message on the screen that this form is a generalized Fermat number and could better be factored or sieved with another program (remove all non-power-of-2 r, and no need to sieve, and use trial division with the primes == 1 mod 2r). Remove all n cases: 1. If a, b, -c are all squares; remove all n. 2. If a, b, c are all r-th powers for an odd r > 1; remove all n. 3. If b and 4ac are both 4-th powers; remove all n. These are Aurifeuillian factors. The above should all be checked first before preceding. Remove partial n cases: 1. If a and -c are both squares; remove all n == (0 mod 2). 2. If a and c are both r-th powers for an odd r > 1; for each such r, remove all n == (0 mod r). 3. If 4ac is 4-th power; remove all n == (0 mod 4). 4. If ac and 4b are both 4-th powers; remove all n == (1 mod 2). 5. If ac is 4-th power and 2b is square; remove all n == (2 mod 4). #3, 4, and 5 are more Aurifeuillian factors. Coordination with existing code: 1. If all n's are removed by algebraic factors for all sequences (ab^n+c)/gcd(a+c,b-1), program should stop immediately. 2. If some n's are removed by algebraic factors, program continues sieving for removing the numbers with small prime factors. 3. Program should be able to handle input of one or multiple k's for a new base at the screen or in a file. Some (ab^n+c)/gcd(a+c,b-1) could have algebraic factors while others do not. 4. Program should be able to handle an already sieved file as input, check the file for algebraic factors, remove them, and then continue sieving more deeply. Once again some (ab^n+c)/gcd(a+c,b-1) could have algebraic factors while others do not. Last fiddled with by sweety439 on 2022-11-16 at 11:42*

2022-11-22, 04:40	#385
sweety439 "99(4^34019)99 palind" Nov 2016 (P^81993)SZ base 36 2×7×263 Posts	See https://github.com/xayahrainie4793/m...main/README.md for more information of this project, including the condensed table (currently the condensed table is only available for bases 2~24, 26, 28, 30, 36) Known large minimal primes (or PRPs) not in the data in https://github.com/xayahrainie4793/m...-prime-numbers: (also large minimal primes (or PRPs) for base 25 and base 27 in https://github.com/curtisbright/mepn-data) Base 17: 4(9^111333), the algebraic form is (7317^111333-9)/16, which is only PRP 97(0^166047)1, the algebraic form is 16017^166048+1, which is definitely prime (reference: http://www.noprimeleftbehind.net/cru...ctures.htm#S17) F7(0^186767)1, the algebraic form is 26217^186768+1, which is definitely prime (reference: http://www.noprimeleftbehind.net/cru...ctures.htm#S17) thus base 17 in fact has only 17 unsolved families Base 19: FG(6^110984), the algebraic form is (90419^110984-1)/3, which is only PRP 1E7(0^122896)1, the algebraic form is 63419^122897+1, which is definitely prime (reference: http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm#S19) Base 21: 4(0^47333)9G, the algebraic form is 421^47335+205, which is only PRP C(F^479147)0K, the algebraic form is (5121^479149-1243)/4, which is only PRP Base 23: AI(F^21143), the algebraic form is (547123^21143-15)/22, which is only PRP K9A(E^23275), the algebraic form is (11877423^23275-7)/11, which is only PRP 96(E^25511), the algebraic form is (235023^25511-7)/11, which is only PRP 8(0^119214)1, the algebraic form is 823^119215+1, which is definitely prime 9(E^800873), the algebraic form is (10623^800873-7)/11, which is only PRP Base 25: 1J71(0^96272)1, the algebraic form is 2767625^96273+1, which is definitely prime 71JD(0^458549)1, the algebraic form is 11048825^458550+1, which is definitely prime (reference: http://www.primegrid.com/forum_thread.php?id=5087, 1104885^917100+1 found by Ronny Willig on 25 March 2013) DKJ(0^246808)1, the algebraic form is 864425^246809+1, which is definitely prime KJD(0^63399)1, the algebraic form is 1298825^63400+1, which is definitely prime Base 32: NU(0^661863)1, the algebraic form is 76632^661864+1, which is definitely prime (reference: http://www.prothsearch.com/riesel1a.html, k=383) Base 33: 13(0^23614)1, the algebraic form is 3633^23615+1, which is definitely prime (reference: http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm#S33) N7(0^610411)1, the algebraic form is 76633^610412+1, which is definitely prime (reference: http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm#S33) Base 35: 1B(0^56061)1, the algebraic form is 4635^56062+1, which is definitely prime (reference: http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm#S35) Base 36: (P^81993)SZ, the algebraic form is (536^81995+821)/7, which is only PRP Known test limits for some families larger than the test limits for the corresponding bases listed in https://github.com/xayahrainie4793/m...-prime-numbers: Base 17: F1{9}: tested to length 1000048 (reference: https://github.com/curtisbright/mepn...a/sieve.17.txt) Base 19: EE1{6}: tested to length 707350 (reference: https://github.com/curtisbright/mepn...a/sieve.19.txt) Base 21: G{0}FK: tested to length 506722 (reference: https://github.com/curtisbright/mepn...a/sieve.21.txt) Base 23: H3{0}1: tested to length 700003 (reference: http://www.noprimeleftbehind.net/cru...tures.htm#S529) JH{0}1: tested to length 700003 (reference: http://www.noprimeleftbehind.net/cru...tures.htm#S529) Base 25: D71J{0}1: tested to length 350004 (reference: http://www.noprimeleftbehind.net/cru...25-reserve.htm) EF{O}: tested to length 300002 (reference: http://www.noprimeleftbehind.net/cru...25-reserve.htm) Base 31: {F}G: tested to length 16777215 (reference: http://factordb.com/index.php?query=...%29%2B1%29%2F2 and http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt) Base 32: 4{0}1: tested to length 1717986918 (reference: http://www.prothsearch.com/fermat.html) G{0}1: tested to length 3435973836 (reference: http://www.prothsearch.com/fermat.html) NG{0}1: tested to length 1800001 (reference: http://www.prothsearch.com/riesel1.html, k=47) UG{0}1: tested to length 720001 (reference: http://www.prothsearch.com/riesel1.html, k=61) Last fiddled with by sweety439 on 2022-11-26 at 06:17

Similar Threads
Thread	Thread Starter	Forum	Replies	Last Post
generalized minimal (probable) primes	sweety439	sweety439	140	2022-12-20 07:08
Minimal set of the strings for primes with at least two digits	sweety439	sweety439	6	2019-11-25 07:37
Digit strings containing primes	davar55	Puzzles	13	2018-03-15 14:46
Primes from powers of 2 strings.	Flatlander	Puzzles	40	2011-02-10 09:42
Strings of Digits	davar55	Puzzles	5	2008-11-02 00:08

2022-08-13, 20:40	#377
sweety439 "99(4^34019)99 palind" Nov 2016 (P^81993)SZ base 36 7142₈ Posts	For the interest of "greatest prime factor ^2-1" sequences: * N-1 primality proving * N+1 primality proving * P-1 integer factorization method * P+1 integer factorization method https://oeis.org/A087713 (greatest prime factor of p^2-1) https://oeis.org/A024710 (the same sequence (start with p=11) of the A087713, which is the greatest prime factor of A024702) https://oeis.org/A024702 ((p^2-1)/24) https://oeis.org/A084920 (p^2-1) https://oeis.org/A001248 (p^2) https://oeis.org/A001318 (generalized pentagonal numbers, (n^2-1)/24)

2022-10-14, 11:02	#378
sweety439 "99(4^34019)99 palind" Nov 2016 (P^81993)SZ base 36 2×7×263 Posts	The data files are in https://github.com/xayahrainie4793/quasi-mepn-data Now, bases 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 18, 20, 24 are solved, and bases 11, 16, 22, 30 are also solved if strong probable primes are allowed.

2022-10-28, 03:34	#379
sweety439 "99(4^34019)99 palind" Nov 2016 (P^81993)SZ base 36 2·7·263 Posts	Fully proven bases (i.e. include the primality proven of the PRPs) and almost proven bases (i.e. proven except the primality proven of the PRPs) sorted by the size of the largest primes in the set for the base: Fully proven bases (sorted by largest to smallest): (in fact, b=14 has the largest prime: 5×14^19698−1 (22578 decimal digits), but since this prime can be easily proven prime using the N+1 primality proving, thus not counted) 1. b=24, 13249×24^8131−49, 11227 decimal digits, primality certificate 2. b=20, 16×20^6270+13, 8159 decimal digits, primality certificate 3. b=18, 12×18^6270+221, 7872 decimal digits, primality certificate 4. b=9, 3×9^1160+10, 1108 decimal digits, primality certificate Almost proven bases (sorted by smallest to largest, i.e. base 22 would be the next possible fully proven bases, then base 30): 1. b=22, (251×22^22002−335)/21, 29538 decimal digits 2. b=30, 18×30^24609+13, 36352 decimal digits 3. b=11, (57×11^62668−7)/10, 65263 decimal digits 4. b=16, (16^116139+619)/5, 139845 decimal digits

2022-11-14, 00:19	#381
sweety439 "99(4^34019)99 palind" Nov 2016 (P^81993)SZ base 36 2×7×263 Posts	New link for GitHub: https://github.com/xayahrainie4793/m...-prime-numbers (I renamed it, I think that this name is better)

2022-11-16, 10:56	#382
sweety439 "99(4^34019)99 palind" Nov 2016 (P^81993)SZ base 36 E62₁₆ Posts	For the numbers < 10^25000, I require them to be definitely primes (i.e. not merely probable primes), N-1 proving or N+1 proving or ECPP proving or CHG proving can be used For the numbers between 10^25000 and 10^500000, I require them to be strong PRP to all prime bases p <= 61 (see https://oeis.org/A014233), Lucas PRP with parameters (P, Q) defined by Selfridge's Method A (see https://oeis.org/A217255 and http://ntheory.org/pseudoprimes.html), trial factored to 2^64 For the numbers > 10^500000, I only require them to be strong PRP to base 2 and 3 (I require less conditions for larger numbers, because when passed the primality tests, the property that the number is in fact composite is less when the number is larger, see https://primes.utm.edu/notes/prp_prob.html and https://www.ams.org/journals/mcom/19...-0982368-4.pdf) Only use Fermat tests (base b) is dangerous, especially these three types of numbers: 1. Numbers of the form Phi(n,b)/gcd(Phi(n,b),n) (where Phi is the cyclotomic polynomial) and their divisors (although this case is less dangerous since we can do Fermat tests to two bases which are not rational power of the other, such as 2 and 3, to avoid them to be PRP) 2. Numbers of the form (k+1)(2k+1) with k+1 and 2k+1 both primes, and numbers of the form (k+1)(3k+1) with k+1 and 3k+1 both primes (this case is even dangerous when the strong tests are used, however, this case is very easy to check: just check if 8N+1 or 3N+1 is square, since such numbers are hexagonal numbers and octagonal numbers, respectively) 3. Carmichael numbers (in this case, we can use strong tests, since there are no "strong Carmichael numbers", all composites are strong pseudoprimes to at most 1/4 of the bases coprime to them)