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Old 2020-07-03, 04:33   #848
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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
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Old 2020-07-03, 04:49   #849
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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
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Old 2020-07-03, 04:53   #850
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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).
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Old 2020-07-03, 09:44   #851
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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
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Old 2020-07-03, 12:53   #852
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(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
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Old 2020-07-04, 04:53   #853
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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
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Old 2020-07-04, 12:03   #854
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Quote:
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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.
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Old 2020-07-04, 13:53   #855
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Quote:
Originally Posted by sweety439 View Post
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
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Old 2020-07-04, 15:17   #856
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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
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Old 2020-07-04, 15:36   #857
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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)
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Old 2020-07-04, 16:09   #858
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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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