A Sierpinski/Riesel-like problem

[sweety439]

Originally Posted by sweety439

Reserve some 1k bases (S17, S27, S51, S56, R38, R54)

and found the prime 38*51^4881+1

the first 4 conjectures of S51 are all proven!!!

Reserve R56

[sweety439]

Originally Posted by sweety439

and found the prime 38*51^4881+1

the first 4 conjectures of S51 are all proven!!!

Reserve R56

I skipped these bases since they are already reserved by other projects:

S16: see post #463, already at n=15K

S38: reserved by Prime Grid's GFN primes search, already at n=2^24-1

S50: reserved by Prime Grid's GFN primes search, already at n=2^24-1

R10: reserved by http://www.worldofnumbers.com/em197.htm (case d=3, k=817) and https://www.rose-hulman.edu/~rickert/Compositeseq/ (case b=10, d=3, k=817), already at n=554789

R12: see post #664, already at n=21760

R32: reserved by CRUS (case R1024, k=29), already at n=500K

R49: reserved by https://github.com/RaymondDevillers/primes (see the "left49" file) (case b=49 family R{G}), already at n=10K

[sweety439]

Conjecture 1 (the strong Sierpinski conjecture): For b>=2, k>=1, if there is an n such that:

(1) k*b^n is neither a perfect odd power (i.e. k*b^n is not of the form m^r with odd r>1) nor of the form 4*m^4.
(2) gcd((k*b^n+1)/gcd(k+1,b-1),(b^(945*2^s)-1)/(b-1)) = 1 for all s, i.e. for all s, every prime factor of (k*b^n+1)/gcd(k+1,b-1) does not divide (b^(945*2^s)-1)/(b-1). (i.e. ord_p(b) is not of the form 2^r (r>=0 if p = 2 or p = 3, r>=1 if p>=5) or m*2^r (m divides 945, r>=0) for every prime factor p of (k*b^n+1)/gcd(k+1,b-1)).
(3) this (k,b) pair is not the case: b = q^m, k = q^r, where q is not of the form t^s with odd s>1, and m and r have no common odd prime factor, and the exponent of highest power of 2 dividing r >= the exponent of highest power of 2 dividing m, and the equation 2^x = r (mod m) has no solution. (the first 6 Sierpinski bases with k's which are this case are 128, 2187, 16384, 32768, 78125 and 131072)

Then there are infinitely many primes of the form (k*b^n+1)/gcd(k+1,b-1).

Conjecture 2 (the strong Riesel conjecture): For b>=2, k>=1, if there is an n such that:

(1) k*b^n is not a perfect power (i.e. k*b^n is not of the form m^r with r>1).
(2) gcd((k*b^n-1)/gcd(k-1,b-1),(b^(945*2^s)-1)/(b-1)) = 1 for all s, i.e. for all s, every prime factor of (k*b^n-1)/gcd(k-1,b-1) does not divide (b^(945*2^s)-1)/(b-1). (i.e. ord_p(b) is not of the form 2^r (r>=0 if p = 2 or p = 3, r>=1 if p>=5) or m*2^r (m divides 945, r>=0) for every prime factor p of (k*b^n-1)/gcd(k-1,b-1)).

Then there are infinitely many primes of the form (k*b^n-1)/gcd(k-1,b-1).

[sweety439]

Originally Posted by sweety439

Reserve some 1k bases (S17, S27, S51, S56, R38, R54)

(33*27^7876+1)/2 is (probable) prime

the first 4 conjectures of S27 are all proven!!!

reserve R57

[sweety439]

(281*57^5610-1)/56 is (probable) prime

the first 4 conjectures of R57 are all proven!!!

reserve R49 (the corresponding page only searched it to 10K, I double check it and reserve it for n>10K)

[sweety439]

k is Sierpinski number base b if....

k         b
1         (none)
2         (no such b < 201446503145165177)
3         (no such b < 158503)
4         == 14 mod 15
5         == 11 mod 12
6         == 34 mod 35
7         == 5 mod 24 or == 11 mod 12
8         == 20 mod 21 or == 47, 83 mod 195 or == 467 mod 73815 or == 722 mod 1551615
9         == 19 mod 20
10        == 32 mod 33
11        == 5 mod 24 or == 14 mod 15 or == 19 mod 20
12        == 142 mod 143 or == 296, 901 mod 19019 or 562, 828, 900, 1166 mod 1729 or == 563 mod 250705 or == 597, 1143 mod 1885
13        == 20 mod 21 or == 27 mod 28 or == 132, 293 mod 595
14        == 38 mod 39 or == 64 mod 65
15        == 13 mod 14 but not == 1 mod 16
16        == 38, 47, 98, 242 mod 255 or == 50 mod 51 or == 84 mod 85
17        == 11 mod 12 or == 278, 302 mod 435 or == 283, 355, 367, 607, 907 mod 1638 or == 373, 445, 646, 718 mod 819
18        == 322 mod 323 or == 398, 512 mod 1235
19        == 11 mod 12 or == 14 mod 15 or == 29 mod 40
20        == 56 mod 57 or == 132 mod 133
21        == 43 mod 44 or == 54 mod 55
22        == 68 mod 69 or == 160 mod 161
23        (== 21 mod 22 but not == 1 mod 8) or == 32 mod 33 or == 41 mod 48 or == 83 mod 530 or == 182 mod 795
24        == 114 mod 115
25        == 38 mod 39 or == 51 mod 52

[sweety439]

Originally Posted by sweety439

Reserve some 1k bases (S17, S27, S51, S56, R38, R54)

No (probable) found for these bases except S27, S51, R57, these bases are likely tested to at least n=10K, bases released.

[sweety439]

Originally Posted by sweety439

All tested to n=1024.

k's that proven composite by algebra factors:

R243:

k = m^5
k = m^2 with m = 11 or 50 mod 61

R729:

k = m^2
k = m^3

S243:

k = m^5

S729:

k = m^3

In fact, R243 k=81 is already tested to n=443060 with no (probable) prime found, since (81*243^n-1)/gcd(81-1,243-1) = (3^(5*n+4)-1)/2, but no known terms in A028491 is = 4 mod 5

[sweety439]

Special cases of (k*b^n+-1)/gcd(k+-1,b-1) (+ for Sierpinski, - for Riesel):

* gcd(k+-1,b-1) (+ for Sierpinski, - for Riesel) = 1: the same as the original Sierpinski/Riesel problem in CRUS

* Riesel case k=1: the smallest generalized repunit prime base b (see A084740 and http://www.fermatquotient.com/PrimSerien/GenRepu.txt)

* Riesel case k=b-1: the smallest Williams prime base (b-1)

* Sierpinski case k=1 and b even: the smallest generalized Fermat prime base b (see http://www.noprimeleftbehind.net/crus/GFN-primes.htm and http://jeppesn.dk/generalized-fermat.html)

* Sierpinski case k=1 and b odd: the smallest generalized half Fermat prime base b (see http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt)

[sweety439]

we allow n=1 or n=2 or n=3 or n=4 or ..., but not allow n=0 or n=-1 or n=-2 or n=-3 or ... for (k*b^n+-1)/gcd(k+-1,b-1)

[sweety439]

k is Riesel number base b if....

k         b
1         (none)
2         (none)
3         (none)
4         == 14 mod 15
5         == 11 mod 12
6         == 34 mod 35
7         == 11 mod 12
8         == 20 mod 21 or == 83, 307 mod 455
9         == 19 mod 20 or == 29 mod 40
10        == 32 mod 33
11        == 14 mod 15 or == 19 mod 20
12        == 142 mod 143 or == 307 mod 1595 or == 901 mod 19019
13        == 5 mod 24 or == 20 mod 21 or == 27 mod 28 or == 38, 47 mod 255
14        == 8, 47, 83, 122 mod 195 or == 38 mod 39 or == 64 mod 65
15        == 27 mod 28
16        == 50 mod 51 or == 84 mod 85