Sierpinski / Riesel - Base 22

[geoff]

Originally Posted by gd_barnes

Thanks for responding and clarifying Geoff. I'm wondering what properties those were for base 5 and would they apply to at least 'some' other bases such as base 22?

Roughly, if the definition of `base B Sierpinski number' is `X is a base B Sierpinski number if X*B^n+1 is never prime for n >= L' then for a given X, once L is reduced below a certain level reducing it further doesn't add any further cases that need to be checked, because X*B^L+1 will not be an integer and therefore not a prime. As more primes are found the level at which L is set can be raised without altering the list of open candidates.

[robert44444uk]

Some notes from unpublished work by Prof Caldwell

Definition of a Sierpinski number

An integer k > 1 is a Sierpinski number base b if gcd(k+1,b-1) = 1 and
k.bn+1 is composite for all n > 0.
• gcd(k+1,b-1) = 1 avoids trivial covers (1-covers).
• k > 1 avoids leading Generalized Fermat divisors. (May use Strong Sierpinski for GFN’s included, and may toss out further GFN’s in the weak case above.)
• n > 0 avoids removing k = p-1 as a multiplier for all primes p and all bases b. (Shouldn’t b be involved in the choice of k?)

[gd_barnes]

Originally Posted by robert44444uk

Some notes from unpublished work by Prof Caldwell

Definition of a Sierpinski number

An integer k > 1 is a Sierpinski number base b if gcd(k+1,b-1) = 1 and
k.bn+1 is composite for all n > 0.
• gcd(k+1,b-1) = 1 avoids trivial covers (1-covers).
• k > 1 avoids leading Generalized Fermat divisors. (May use Strong Sierpinski for GFN’s included, and may toss out further GFN’s in the weak case above.)
• n > 0 avoids removing k = p-1 as a multiplier for all primes p and all bases b. (Shouldn’t b be involved in the choice of k?)

Thanks for the info. I understand everything but the last two here. On the 2nd and 3rd ones, can you give specific examples on what you are referring to. Pardon the ignorance. I can understand best by examples. I suspect the 2nd one is related to the issue of 22*22^n+1 and 484*22^n+1 but am not sure.

So by extension on the definition of a Sieprpinski number base b, we can draw the same conclusion on Riesel numbers that have 1-covers by changing k+1 to k-1. That is:

An integer k > 1 is a Riesel number base b if gcd(k-1,b-1) = 1 and
k.bn-1 is composite for all n > 0.
• gcd(k-1,b-1) = 1 avoids trivial covers (1-covers).

In my research and creation of web pages for all conjectures base 2-32, I had already observed this and will put it in 'sentence format' for ease of everyone else's reference (mainly my own) :

A k cannot be a Riesel nor a Sierpinski k if all values of n have a single trival factor. And the elimination of these k's is directly related to the factorization of the base minus 1. Siting specific examples for both types, we have:

Riesel:
Base 2 (none, i.e. there is no factorization of 1)
Base 3 (k==1 mod 2 eliminated with a factor of 2, i.e. the factorization of 2 is simply 2).
Base 4 (k==1 mod 3 eliminated with a factor of 3, i.e. the factorization of 3 is simply 3).
Base 5 (k==1 mod 2 eliminated with a factor of 2, i.e. the factorization of 4 is 2*2)
Base 6 (k==1 mod 5 eliminated with a factor of 5).
Base 7 (k==1 mod 2 eliminated with a factor of 2 AND k==1 mod 3 eliminated with a factor of 3, i.e. the factorization of 6 is 2*3).

Sierpinski:
Bases 2 & 3; same as above
Base 4 (k==2 mod 3 eliminated with a factor of 3).
Base 5; same as above
Base 6; (k==4 mod 5 eliminated with a factor of 5).
Base 7 (k==1 mod 2 eliminated with a factor of 2 AND k==2 mod 3 eliminated with a factor of 3, i.e. the factorization of 6 is 2*3).

And continuing to larger Sierpinski bases we have base 30 where only k==28 mod 29 are gone with a factor of 29 and base 31 where k==1 mod 2 are gone with a factor of 2, k==2 mod 3 are gone with a factor of 3, and k==4 mod 5 are gone with a factor of 5.

Robert, I'm going to send you links to web pages in a PM that I have already created with a lot of information for bases 2-32 for this. I have come up with a most unusual proof for the base 12 Riesel conjecture and I think part of it may be related to the last item that you are referring to here. I will post the proof in the next post here for everyone to review.

You did a lot of coordination on these conjectures and if you could, I'd like to get your feedback on the pages before opening them up to everyone. I also have a reservation page to get the ball rolling again on this.


Thanks,
Gary

[gd_barnes]

I have placed a proof of the Riesel Base 12 conjecture in the 'Sierpinski/Riesel bases 6 to 18' thread.

[gd_barnes]

Originally Posted by kar_bon

something like this:

call:
pfgw -tc -q"1468*11^26258+1"

output:
PFGW Version 20031222.Win_Dev (Beta 'caveat utilitor') [FFT v22.13 w/P4]

Primality testing 1468*11^26258+1 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 2
Running N-1 test using base 3
Running N+1 test using discriminant 23, base 1+sqrt(23)
Calling N-1 BLS with factored part 100.00% and helper 0.02% (300.02% proof)
1468*11^26258+1 is prime! (813.9536s+0.1133s)

karsten

Hi Karsten,

Thanks for the feedback. I tried Axn1/Rogue's suggestion of using the -tm switch along with the -f0 switch and it speeded up the search on your prime by nearly 10 times!! Apparently the -f0 switch causes it to not do any trial factoring, which would certainly be unneccesary for a probable prime. -tc apparently does both a +1 and -1 test so it also does more testing than is needed. -tm does just what you need here...it only does a -1 test, which is what is needed for a +1 probable prime. Here is the output:

Primality testing 1468*11^26258+1 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 2
Calling Brillhart-Lehmer-Selfridge with factored part 99.99%
1468*11^26258+1 is prime! (83.0808s+0.0065s)

The input to PFGW was: "pfgw -tm -f0 -q1468*11^26258+1"

Particulars of the test:

It was run on a 1.66 Ghz Dell Core duo laptop. (It seems to run about as fast as a 3 Ghz P4.)

My version of PFGW does not accept the quotes around the equation.

Per the README file, I am using PFGW v1.2 Release (January 30, 2005). I downloaded it 2-3 months ago.


I hope this saves you a little time too! It did me...I needed to test a much larger prime for base 17 that took almost 6 times as long as the one you tested, i.e. 92*17^51311+1, which took 476.1328s+0.0087s.

Thanks Rogue, Axn1, and Karsten for helping!


Gary

[gd_barnes]

All base 22 searchers,

All conjectures for bases > 2 except those being worked by other major projects are now being coordinated in the new "Conjectures 'R Us" effort in this Open Projects forum. Please report all future reservations and statuses for base 22 in the reservations/statuses thread for that effort. Web pages have been created that show all current relavent info.

After a couple of days, I'll request that this thread be locked to avoid any duplication of future effort.


Thanks,
Gary