A Sierpinski/Riesel-like problem

[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)

Since 1 is not prime, thus for a Riesel base b>=2, k = b is excluded from testing since (b-1)/gcd(b-1,b-1) = (b-1)/(b-1) = 1, thus k = b would have the same prime as k = 1.

[sweety439]

The formula for Sierpinski conjectures in CRUS is k*b^n+1
The formula for Riesel conjectures in CRUS is k*b^n-1
The formula for Sierpinski conjectures in this project is (k*b^n+1)/gcd(k+1,b-1)
The formula for Riesel conjectures in this project is (k*b^n-1)/gcd(k-1,b-1)

[sweety439]

All n must be >= 1.

k-values which make a full covering set with all or partial algebraic factors are excluded from the conjectures.

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. Such k-values will have the same prime as k / b.

[sweety439]

Originally Posted by sweety439

The CK for S726 is known to be 10923176
The CK for R726 is known to be 12751579

The CK for S1020 is known to be 95696289
The CK for R1020 is known to be 94655888

[sweety439]

Originally Posted by sweety439

The CK for S1020 is known to be 95696289
The CK for R1020 is known to be 94655888

I skipped bases 876 and 966 because the covering of the upper bound of these two bases (except the Riesel sides of 966) have a prime > 50000:

S876 and R876 have 59029
S966 has 71707

Like the status for both sides for base 728, which has 105997

[sweety439]

Now, the CK for all Sierpinski/Riesel bases <= 1024 except SR156, SR280, SR876, SR910, R946, SR960, SR966 (which have too large upper bounds) are known!!!

[sweety439]

See https://github.com/xayahrainie4793/E...0to%202048.txt (Sierpinski) and https://github.com/xayahrainie4793/E...0to%202048.txt (Riesel) for the CK's for bases 2 <= b <= 2500 and b = 4096, 8192, 16384, 32768, 65536

[sweety439]

This is the sieve file for R70

[sweety439]

Originally Posted by sweety439

2 (probable) primes found for R70:

(376*70^6484-1)/3
(496*70^4934-1)/3

k=811 still remains ....

No other (probable) primes found for R70 k = 376, 496, 811 up to n=22813

[sweety439]

Originally Posted by sweety439

This including:

* k=1 for all Sierpinski base not of the form m^r with odd r>1 nor of the form 4*m^4
* k=1 for all Riesel base not of the form m^r with r>1
* k=2 for all Sierpinski base not == 2 mod 3
* k=2 for all Sierpinski base of the form 2^r-2 or 3*2^r-2
* k=2 for all Riesel base
* k=3 for all Sierpinski base not == 3 mod 4
* k=3 for all Sierpinski base of the form 3^r-3 or 2*3^r-3 or 4*3^r-3
* k=3 for all Riesel base
* k=5 for all Sierpinski base not == 2 mod 3
* k=5 for all Sierpinski base of the form 5^r-5 or 2*5^r-5 or 3*5^r-5 or 6*5^r-5
* k=5 for all Riesel base not == 3 mod 4
* k=5 for all Riesel base of the form 5^r+5 or 2*5^r+5 or 4*5^r+5
* k=6 for all Sierpinski base == 0, 1 mod 7
* k=6 for all Sierpinski base of the form 2^r*3^s-6 or 7*2^r*3^s-6
* k=6 for all Riesel base == 0, 1 mod 5
* k=6 for all Riesel base of the form 2^r*3^s+6 or 5*2^r*3^s+6
* k=7 for all Sierpinski base not == 3, 5, 7 mod 8
* k=7 for all Sierpinski base of the form 7^r-7 or 2*7^r-7 or 3*7^r-7 or 6*7^r-7
* k=7 for all Riesel base not == 2 mod 3
* k=7 for all Riesel base of the form 7^r+7 or 2*7^r+7 or 4*7^r+7 or 8*7^r+7
* k=b-2 for all Sierpinski base b where k is not of the form m^r with odd r>1 nor of the form 4*m^4
* k=b-2 for all Riesel base b
* k=b-1 for all Sierpinski base b where k is not of the form m^r with odd r>1 nor of the form 4*m^4
* k=b-1 for all Riesel base b
* k=b+1 for all Riesel base b
* k=b+2 for all Riesel base b

This also including all k-values such that k+-1 (+ for Sierpinski, - for Riesel) divides (b-1)*b^r for some r

[sweety439]

Originally Posted by sweety439

This also including all k-values such that k+-1 (+ for Sierpinski, - for Riesel) divides (b-1)*b^r for some r

Also all k-values such that numerator(k*b^n+-1) (+ for Sierpinski, - for Riesel) divides (b-1)*b^r for some (positive or negative or 0) integer n and (positive or 0) integer r