A Sierpinski/Riesel-like problem - Page 76

LaurV · 2020-06-20, 02:13

Quote:

Originally Posted by Uncwilly

what is your reason for quoting huge blocks of text that you posted on the same day?

Hanlon's Razor

sweety439 · 2020-06-21, 13:40

update newest files

sweety439 · 2020-06-22, 23:00

Update zip files for 1st, 2nd, and 3rd conjectures for bases <= 32 (except 2, 3, 6, 15, 22, 24, 28, 30) and bases 64, 128, 256

sweety439 · 2020-06-24, 23:56

Update files.

sweety439 · 2020-06-25, 00:02

Extended to base 539

Note: I only searched the k <= 5000000, if there are <16 Sierpinski/Riesel k's <= 5000000, then this text file only show the Sierpinski/Riesel k's <= 5000000 for this Sierpinski/Riesel base (if there are no Sierpinski/Riesel k's <= 5000000, then this text file do not show any Sierpinski/Riesel k's <= 5000000 for this Sierpinski/Riesel base), also, I only searched the exponent n <= 2000 (for (k*b^n+-1)/gcd(k+-1,b-1), + for Sierpinski, - for Riesel) and only searched the primes <= 100000 (for the prime factor of (k*b^n+-1)/gcd(k+-1,b-1), + for Sierpinski, - for Riesel), thus this text file wrongly shows 1 as Sierpinski number base 125, although (1*125^n+1)/gcd(1+1,125-1) has no covering set, but since (1*125^n+1)/gcd(1+1,125-1) has a prime factor <= 100000 for all n <= 2000

sweety439 · 2020-06-25, 07:19

This project is from CRUS, extended to the k such that gcd(k+-1,b-1) (+ for Sierpinski, - for Riesel) is not 1. Since k*b^n+-1 is always divisible by gcd(k+-1,b-1), it is to simply take out this factor and find and prove the smallest value of k for (k*b^n+-1)/gcd(k+-1,b-1) which is 'Sierpinski value' (+1 form) or 'Riesel value' (-1 form) that is not prime for all values of n >= 1.

sweety439 · 2020-06-25, 07:19

Sierpinski problem base b: Finding and proving the smallest k such that gcd(k+1,b-1)=1 and k*b^n+1 is not prime for all integers n>=1.

Riesel problem base b: Finding and proving the smallest k such that gcd(k-1,b-1)=1 and k*b^n-1 is not prime for all integers n>=1.

Extended Sierpinski problem base b: Finding and proving the smallest k such that (k*b^n+1)/gcd(k+1,b-1) is not prime for all integers n>=1.

Extended Riesel problem base b: Finding and proving the smallest k such that (k*b^n-1)/gcd(k-1,b-1) is not prime for all integers n>=1.

With this effort, we aim to prove many of the Riesel and Sierpinski conjectures for bases <= 128 and bases 256, 512, 1024.

Project definition:

For every base (b) for the forms (k*b^n+1)/gcd(k+1,b-1) and (k*b^n-1)/gcd(k-1,b-1), there exists a unique value of k for each form that has been conjectured to be the lowest 'Sierpinski value' (+1 form) or 'Riesel value' (-1 form) that is composite for all values of n >= 1.

Goal:

Prove the conjectures by finding at least one (probable) prime (if only PRP, prove its primality) for all lower values of k. Many of the conjectures have already been proven but much more work is needed to prove additional bases. Proving them all is not possible but we aim to prove many of them.

There are many conjectures where only ONE k needs a (probable) prime (and many more that need only two). If you find it, you could be the one to prove a conjecture! This is a big deal to us here.
Algebraic factors have been found for many k's, which prove them composite for all n, allowing them to be removed from searches.
Notes:

All n must be >= 1.

k-values that 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.

See "table of Riesel problems.txt" and "table of Sierpinski problems.txt" for:

The covering set of the conjectured k for each Sierpinski/Riesel base.
The k's that make a full covering set with all or partial algebraic factors for each Sierpinski/Riesel base.
The remaining k's to find prime for each Sierpinski/Riesel base.
The top 10 k's with largest first primes for each Sierpinski/Riesel base.
This project is to solve the Sierpinski/Riesel conjectures for bases b <= 128 and bases b = 256, 512, 1024. (this project will be extended to bases b <= 2048 in future)

sweety439 · 2020-06-25, 21:25

Original CRUS project definition:

For every base (b) for the forms k*b^n+1 and k*b^n-1, there exists a unique value of k for each form that has been conjectured to be the lowest 'Sierpinski value' (+1 form) or 'Riesel value' (-1 form) that is not prime for all values of n >= 1. k's that have a trivial factor (one factor the same) for all n-values (this trivial factor is gcd(k+-1,b-1) (+ for Sierpinski, - for Riesel)) are not considered. (Thus, only the k's such that gcd(k+-1,b-1) = 1 are considered) The project is finding and proving this value of k.

This project extends the original CRUS project to the k's such that gcd(k+-1,b-1) (+ for Sierpinski, - for Riesel) is not 1. For these k's, we can deal with the fact that k*b^n+-1 is always divisible by gcd(k+-1,b-1), it is to simply take out this factor and find and prove the value of k for each form that has been conjectured to be the lowest 'Sierpinski value' (+1 form) or 'Riesel value' (-1 form) that is not prime for all values of n >= 1.

Goal:

Prove the conjectures for bases b<=2048 by finding at least one (probable) prime (if only PRP, prove its primality) for all lower values of k. Many of the conjectures have already been proven but much more work is needed to prove additional bases. Proving them all is not possible but we aim to prove many of them.

Notes:

All n must be >= 1.

k-values that 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 · 2020-06-25, 21:27

Ranges:

k >= 1

b >= 2

n >= 1

sweety439 · 2020-06-25, 21:29

Since these conjectures extend to the k such that gcd(k+-1,b-1) (+ for Sierpinski, - for Riesel) is not 1, so these conjectures are called extended Sierpinski/Riesel conjectures base b

sweety439 · 2020-06-26, 01:31

The 1st, 2nd, 3rd, and 4th conjectures for Sierpinski/Riesel bases <=64 and 128 and 256 with smaller conjectures

2020-06-25, 07:19	#831
sweety439 "99(4^34019)99 palind" Nov 2016 (P^81993)SZ base 36 2×13×113 Posts	This project is from CRUS, extended to the k such that gcd(k+-1,b-1) (+ for Sierpinski, - for Riesel) is not 1. Since kb^n+-1 is always divisible by gcd(k+-1,b-1), it is to simply take out this factor and find and prove the smallest value of k for (kb^n+-1)/gcd(k+-1,b-1) which is 'Sierpinski value' (+1 form) or 'Riesel value' (-1 form) that is not prime for all values of n >= 1. Last fiddled with by sweety439 on 2020-06-25 at 07:22

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2020-06-25, 07:19	#832
sweety439 "99(4^34019)99 palind" Nov 2016 (P^81993)SZ base 36 2×13×113 Posts	Sierpinski problem base b: Finding and proving the smallest k such that gcd(k+1,b-1)=1 and kb^n+1 is not prime for all integers n>=1. Riesel problem base b: Finding and proving the smallest k such that gcd(k-1,b-1)=1 and kb^n-1 is not prime for all integers n>=1. Extended Sierpinski problem base b: Finding and proving the smallest k such that (kb^n+1)/gcd(k+1,b-1) is not prime for all integers n>=1. Extended Riesel problem base b: Finding and proving the smallest k such that (kb^n-1)/gcd(k-1,b-1) is not prime for all integers n>=1. With this effort, we aim to prove many of the Riesel and Sierpinski conjectures for bases <= 128 and bases 256, 512, 1024. Project definition: For every base (b) for the forms (kb^n+1)/gcd(k+1,b-1) and (kb^n-1)/gcd(k-1,b-1), there exists a unique value of k for each form that has been conjectured to be the lowest 'Sierpinski value' (+1 form) or 'Riesel value' (-1 form) that is composite for all values of n >= 1. Goal: Prove the conjectures by finding at least one (probable) prime (if only PRP, prove its primality) for all lower values of k. Many of the conjectures have already been proven but much more work is needed to prove additional bases. Proving them all is not possible but we aim to prove many of them. There are many conjectures where only ONE k needs a (probable) prime (and many more that need only two). If you find it, you could be the one to prove a conjecture! This is a big deal to us here. Algebraic factors have been found for many k's, which prove them composite for all n, allowing them to be removed from searches. Notes: All n must be >= 1. k-values that 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. See "table of Riesel problems.txt" and "table of Sierpinski problems.txt" for: The covering set of the conjectured k for each Sierpinski/Riesel base. The k's that make a full covering set with all or partial algebraic factors for each Sierpinski/Riesel base. The remaining k's to find prime for each Sierpinski/Riesel base. The top 10 k's with largest first primes for each Sierpinski/Riesel base. This project is to solve the Sierpinski/Riesel conjectures for bases b <= 128 and bases b = 256, 512, 1024. (this project will be extended to bases b <= 2048 in future)

2020-06-25, 21:25	#833
sweety439 "99(4^34019)99 palind" Nov 2016 (P^81993)SZ base 36 2·13·113 Posts	Original CRUS project definition: For every base (b) for the forms kb^n+1 and kb^n-1, there exists a unique value of k for each form that has been conjectured to be the lowest 'Sierpinski value' (+1 form) or 'Riesel value' (-1 form) that is not prime for all values of n >= 1. k's that have a trivial factor (one factor the same) for all n-values (this trivial factor is gcd(k+-1,b-1) (+ for Sierpinski, - for Riesel)) are not considered. (Thus, only the k's such that gcd(k+-1,b-1) = 1 are considered) The project is finding and proving this value of k. This project extends the original CRUS project to the k's such that gcd(k+-1,b-1) (+ for Sierpinski, - for Riesel) is not 1. For these k's, we can deal with the fact that k*b^n+-1 is always divisible by gcd(k+-1,b-1), it is to simply take out this factor and find and prove the value of k for each form that has been conjectured to be the lowest 'Sierpinski value' (+1 form) or 'Riesel value' (-1 form) that is not prime for all values of n >= 1. Goal: Prove the conjectures for bases b<=2048 by finding at least one (probable) prime (if only PRP, prove its primality) for all lower values of k. Many of the conjectures have already been proven but much more work is needed to prove additional bases. Proving them all is not possible but we aim to prove many of them. Notes: All n must be >= 1. k-values that 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.

2020-06-25, 21:27	#834
sweety439 "99(4^34019)99 palind" Nov 2016 (P^81993)SZ base 36 2×13×113 Posts	Ranges: k >= 1 b >= 2 n >= 1

2020-06-25, 21:29	#835
sweety439 "99(4^34019)99 palind" Nov 2016 (P^81993)SZ base 36 2·13·113 Posts	Since these conjectures extend to the k such that gcd(k+-1,b-1) (+ for Sierpinski, - for Riesel) is not 1, so these conjectures are called extended Sierpinski/Riesel conjectures base b