mersenneforum.org A Sierpinski/Riesel-like problem
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2020-07-03, 04:33   #848
sweety439

Nov 2016

22×691 Posts

Quote:
 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

2020-07-03, 04:49   #849
sweety439

Nov 2016

22×691 Posts

Quote:
 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

 2020-07-03, 04:53 #850 sweety439   Nov 2016 1010110011002 Posts 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).
2020-07-03, 09:44   #851
sweety439

Nov 2016

53148 Posts

Quote:
 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

Last fiddled with by sweety439 on 2020-07-03 at 10:07

 2020-07-03, 12:53 #852 sweety439   Nov 2016 22×691 Posts (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) Last fiddled with by sweety439 on 2020-07-03 at 13:48
 2020-07-04, 04:53 #853 sweety439   Nov 2016 22·691 Posts k is Sierpinski number base b if.... Code: 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 Last fiddled with by sweety439 on 2020-07-04 at 15:56
2020-07-04, 12:03   #854
sweety439

Nov 2016

22·691 Posts

Quote:
 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.

2020-07-04, 13:53   #855
sweety439

Nov 2016

22·691 Posts

Quote:
 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

Last fiddled with by sweety439 on 2020-07-04 at 13:54

 2020-07-04, 15:17 #856 sweety439   Nov 2016 22·691 Posts 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) Last fiddled with by sweety439 on 2020-07-04 at 15:18
 2020-07-04, 15:36 #857 sweety439   Nov 2016 ACC16 Posts 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)
 2020-07-04, 16:09 #858 sweety439   Nov 2016 1010110011002 Posts k is Riesel number base b if.... Code: 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 Last fiddled with by sweety439 on 2020-07-06 at 02:34

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