A Sierpinski/Riesel-like problem

[sweety439]

Originally Posted by sweety439

Update the newest text file for some Sierpinski bases.

Note: I don't have the prime for S46 k=283, I only know this prime is between n=15K and n=25K.

[sweety439]

Update newest text files for SR2 (some large n's are given by https://www.rieselprime.de/).

[sweety439]

Update the newest zip file.

[sweety439]

(4*115^4223-1)/3 is (probable) prime!!!

We solved k=4 for the smallest Riesel base with k=4 remaining!!! (115 was the smallest Riesel base without known (probable) prime for k=4, excluding the bases b = 14 mod 15 (for such bases, k=4 has a covering set {3, 5}) and the bases b = m^2 (for such bases, k=4 has full algebra factors: 4*(m^2)^n-1 = (2*m^n-1) * (2*m^n+1)) and the bases b = 4 mod 5 (for such bases, k=4 has partial algebra factors: even n factors to (2*b^(n/2)-1) * (2*b^(n/2)+1), odd n has factor of 5). Now, the smallest Riesel base with k=4 remaining is 178.

Note: R72 does not have k=4 remaining, 4*72^1119849-1 is prime, see CRUS.

This is the text file for Riesel k=4 for all bases 2<=b<=256, tested to at least n=2000, there are 3 remain Riesel bases 2<=b<=256 for k=4: R178, R223 and R232 (the n's for R72, R212 and R218 are given by CRUS). In fact, I know exactly which Sierpinski/Riesel bases 2<=b<=1024 have k=1, k=2, k=3, and k=4 remaining at n=1000, even including the non-tested Sierpinski/Riesel bases, since I have tested these k's for these Sierpinski/Riesel bases to at least n=1000 (without comparing with CRUS). Besides, k=1, k=2, k=3, and k=4 for all Sierpinski/Riesel bases 2<=b<=256 have been tested to at least n=2000 by me (also without comparing with CRUS). (all of the CK's for all Sierpinski/Riesel bases 2<=b<=1024 are >= 4, i.e. no Sierpinski/Riesel bases 2<=b<=1024 have CK = 1, 2, or 3. Besides, a Sierpinski/Riesel base 2<=b<=1024 have CK = 4 if and only if b = 14 mod 15)

Since for Riesel k=2, all GCD are 1, thus for Riesel k=2, a prime for base b is the same as that for CRUS (i.e. the original problem) for base b.

R107 is an interesting base, it is not only the smallest Riesel base with k=2 remaining at n=2000, but also the second smallest Riesel base with k=3 remaining at n=2000. (the smallest Riesel base with k=3 remaining at n=2000 is 42, but 3*42^2523-1 is prime).

Another interesting base is S899, this base is the only Sierpinski/Riesel base 2<=b<=1024 with all k=1, k=2, and k=3 remaining at n=1000. Besides, the CK for S899 is only 4, thus, all k < CK for this base are remaining at n=1000 (S899 is the only such Sierpinski/Riesel base 2<=b<=1024).

The bases which are excluded for the k's are:

Sierpinski k=1:

b = m^r with odd r > 1 proven composite by full algebra factors.

Sierpinski k=2:

none.

Sierpinski k=3:

none.

Sierpinski k=4:

b = 14 mod 15: covering set {3, 5}.
b = m^4 proven composite by full algebra factors.

Riesel k=1:

b = m^r with r > 1 proven composite by full algebra factors.

Riesel k=2:

none.

Riesel k=3:

none.

Riesel k=4:

b = 14 mod 15: covering set {3, 5}.
b = m^2 proven composite by full algebra factors.
b = 4 mod 5: odd n, factor of 5; even n, algebraic factors.

[sweety439]

(11*256^5702+1)/3 is (probable) prime!!!

S256 is proven!!!

S256 may be the last power of 2 Sierpinski base which is proven by this project, since all the other power of 2 Sierpinski bases b<=1024 (S2, S32, S128, S512 and S1024) cannot be proven in our lifetime, i.e. these bases cannot be proven with current technology, since these bases have GFN's remain. (such k's for these bases are: S2, k=65536; S32, k=4; S128, k=16; S512, k=2, 4 and 16; S1024, k=4 and 16)

[sweety439]

(79*121^4545-1)/6 is (probable) prime!!!

R121 is proven!!!

[sweety439]

Reserve SR108 to n=2000, only test the k's not in CRUS.

Also reserve S67 and S105.

[sweety439]

I will not update any word file in the future, for the information for the remaining k with no known (probable) prime, k's with full or partial algebra factors, top 10 primes, etc. see http://www.mersennewiki.org/index.ph..._definition%29 (Sierpinski bases) and http://www.mersennewiki.org/index.ph..._definition%29 (Riesel bases), the format of the tables in these two websites are the same as the format of the tables in the word files.

This is the word files for the newest status for bases 2<=b<=64 (except base 15).

[sweety439]

Found the prime 36*105^675+1.

Thus, S105 is now 1k base.

[sweety439]

Update newest zip file to include the newest status for SR42, SR48, SR60, S105, S113, R115, R121 and S256.

[sweety439]

Originally Posted by sweety439

Reserve SR108 to n=2000, only test the k's not in CRUS.

Also reserve S67 and S105.

Found these (probable) primes:

(8987*108^1737+1)/107
(14444*108^1938+1)/107
(18831*108^1596+1)/107
(21613*108^1076+1)/107
(3532*108^1779-1)/107

Thus, S108 has now only k=20543 remain for k = 106 mod 107, R108 has 3 k's remain for k = 1 mod 107: 5351, 6528 and 13162.