A Sierpinski/Riesel-like problem

[sweety439]

Current test limit:

S10 k=269: at n=30184, continue to n=100K...
S36 k=1814: at n=20256, continue to n=100K...

[sweety439]

There are PRP's to solve an extended Sierpinski/Riesel problem that have not been certified to be prime:

S61:

(43*61^2788+1)/4
(62*61^3698+1)/3

S64:

(11*64^3222+1)/3

S75:

(11*75^3071+1)/2

S105:

(191*105^5045+1)/8

S256:

(11*256^5702+1)/3

R7:

(159*7^4896-1)/2
(197*7^181761-1)/2
(313*7^5907-1)/6
(367*7^15118-1)/6

R17:

(29*17^4904-1)/4

R51:

(1*51^4229-1)/50

R67:

(25*67^2829-1)/6

R91:

(1*91^4421-1)/90
(27*91^5048-1)/2

R100:

(133*100^5496-1)/33

R107:

(3*107^4900-1)/2

R121:

(79*121^4545-1)/6

[sweety439]

Originally Posted by sweety439

k-values that are a multiple of base (b) and where (k+-1)/gcd(k+-1,b-1) (+ for Sierpinski, - for Riesel) is not prime are included in the conjectures but excluded from testing, since such k-values will have the same prime as k / b.

However, k-values that are a multiple of base (b) and where (k+-1)/gcd(k+-1,b-1) (+ for Sierpinski, - for Riesel) is prime are included from testing since the exponent n must be >=1 (n can be 1, but cannot be 0 or -1 or -2 or ...), and the same prime n=1 for k / b would be n=0 for this k but n must be >=1 hence it is not allowed so this k must continue to be searched. (of course, k-values that are not a multiple of base (b) are included from testing)

Thus, for S3, k = 42, 45, 57, 60, 66 and 72 are included from testing since although 42, 45, 57, 60, 66 and 72 are multiples of 3, but 42+1, (45+1)/2, (57+1)/2, 60+1, 66+1 and 72+1 are primes. However, k = 48, 51, 54, 63, 69 and 75 are excluded from testing since 48, 51, 54, 63, 69 and 75 are multiples of 3, and 48+1, (51+1)/2, 54+1, (63+1)/2, (69+1)/2 and (75+1)/2 are not primes. Besides, for R3, k = 42, 48, 54, 60, 63, 72 and 75 are included from testing since although 42, 48, 54, 60, 63, 72 and 75 are multiples of 3, but 42-1, 48-1, 54-1, 60-1, (63-1)/2, 72-1 and (75-1)/2 are primes. However, k = 45, 51, 57, 66 and 69 are excluded from testing since 45, 51, 57, 66 and 69 are multiples of 3, and (45-1)/2, (51-1)/2, (57-1)/2, 66-1 and (69-1)/2 are not primes.

Note: Since 1 is not prime, thus for R3, k = 3 is excluded from testing. ((3-1)/2 = 1) However, since 2 is prime, thus for S3, k = 3 is included from testing. ((3+1)/2 = 2)

k is excluded from testing if and only if k is a multiple of base (b) and (k+-1)/gcd(k+-1,b-1) (+ for Sierpinski, - for Riesel) is not prime.

[sweety439]

Originally Posted by 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.

For Riesel k=2, the extended problem is completely the same as the original problem, since for Riesel k=2, all GCD are 1.

[sweety439]

Originally Posted by sweety439

k is excluded from testing if and only if k is a multiple of base (b) and (k+-1)/gcd(k+-1,b-1) (+ for Sierpinski, - for Riesel) is not prime.

Thus, for example, for R36, all square k have algebra factors, and the smallest nonsquare k which is excluded from testing is 540, since 540 is a multiple of 36 and (540-1)/gcd(540-1,36-1) = 77 is not prime. (for all smaller nonsquare k which is a multiple of 36, (k-1)/gcd(k-1,36-1) is prime, thus these k are still included from testing)

[sweety439]

Originally Posted by sweety439

Current test limit:

S10 k=269: at n=30184, continue to n=100K...
S36 k=1814: at n=20256, continue to n=100K...

S10 k=269 at n=56080
S36 k=1814 at n=48984

Both no (probable) prime found.

[sweety439]

Reserve R118.

[sweety439]

Originally Posted by sweety439

Reserve R118.

(27*118^860-1)/13 is (probable) prime!!!

Continue reserving R118 k=43 to n=8K.

[sweety439]

Will reserve S115 and R105 (both are 3k base), all bases <= 3 k's remaining are reserved after this reservation was done.

[sweety439]

Originally Posted by sweety439

Will reserve S115 and R105 (both are 3k base), all bases <= 3 k's remaining are reserved after this reservation was done.

(50*115^798+1)/3 is (probable) prime!!!

S115 is now a 2k base.

[sweety439]

Originally Posted by sweety439

Will reserve S115 and R105 (both are 3k base), all bases <= 3 k's remaining are reserved after this reservation was done.

(265*105^1666-1)/8 is (probable) prime!!!

R105 is now a 2k base.