k=1 thru k=12

[henryzz]

I don't really know where to post this but does anyone know if anyone has tested riesel or sierp numbers with k=1? It seems that something overlooked to me
of because gimps is doing this for base2 but what testing has been done on k=1 other bases?

Admin edit: All primes and remaining k's/bases/search depths for k=1 thru 12 and bases<=1030 are attached to this post.

[henryzz]

forget reisel it always has a factor b+1
there also seems to to algebraic factors of sierp except if n is a power of 2 but i cant quite place my finger on it

[gd_barnes]

Riesel bases always have a trivial factor of b-1 rather than b+1. Technically k=1 IS considered in the Riesel base 2 conjecture because you cannot have a trivial factor of 1 since it is not considered prime. But k=1 has a prime at n=2 and hence is quickly eliminated.

For Sierp, k=1 always make Generalized Fermat #'s (GFNs). GFNs are forms that can reduce to b^n+1, hence k's where k=b^q and q>=0 are also not considered.

We do not consider GFNs in testing because n must be 2^q to make a prime, resulting in few possibilities of primes. Most mathematicians agree that the number of primes of such forms is finite.

See the project definition for more details about exclusions and inclusions of k-values in the 'come join us' thread.


Gary

[robert44444uk]

For the infinity of bases, the smallest Sierpinski k may take any integer value except 2^x-1, x=integer.

There are generating functions to discover instances of certain values such as k=2,5,65 which do not appear for small bases.

This is down to the work of Chris Caldwell and his last year students.

for example k=2 for b=19590496078830101320305728

[gd_barnes]

Originally Posted by robert44444uk

For the infinity of bases, the smallest Sierpinski k may take any integer value except 2^x-1, x=integer.

There are generating functions to discover instances of certain values such as k=2,5,65 which do not appear for small bases.

This is down to the work of Chris Caldwell and his last year students.

for example k=2 for b=19590496078830101320305728

Is this the lowest base where k=2 is the Sierpinski number?

[robert44444uk]

Originally Posted by gd_barnes

Is this the lowest base where k=2 is the Sierpinski number?

This k was generated from looking at (x^2)^n-1 factorisations -covering set is 3,5,17,257,641,65537,6700417 which I think is 32-cover. I do not think anyone has claimed it is the smallest k, it just comes from the smallest-cover.

[gd_barnes]

Originally Posted by robert44444uk

This k was generated from looking at (x^2)^n-1 factorisations -covering set is 3,5,17,257,641,65537,6700417 which I think is 32-cover. I do not think anyone has claimed it is the smallest k, it just comes from the smallest-cover.

OK, very good. I asked because I'm undertaking an effort on 2 slow cores to see which small bases do not yield an easy prime for k=2. I started with the Riesel side and am testing bases 2 to 1024.

Here are the 20 Riesel bases <= 1024 remaining that do NOT have a prime of the form 2*b^n-1 at n=10K:

 b
107
170
278
303
383
515
522
578
581
590
647
662
698
704
845
938
969
989
992
1019

Here are the primes for n>=1000 found for the effort:

  b   (n)
 785 (9670)
 233 (8620)
 618 (8610)
 627 (7176)
 872 (6036)
 716 (4870)
 298 (4202)
 572 (3804)
 380 (3786)
 254 (2866)
 669 (2787)
 551 (2718)
 276 (2484)
 382 (2324)
 968 (1750)
 550 (1380)
 434 (1166)
1013 (1116)
 734 (1082)
 215 (1072)

I'm going to take it up to n=10K and then work on the Sierp side to the same depth. The hard part about the effort is that each base has to be sieved individually. AFAIK sr(x)sieve will not sieve more than one base at a time.

Obviously PROVING that the lowest base that has a Sierp k=2 would not be possible using the brute force approach such as this but it would be quite possible for higher values of k.

If anyone else has any input or info. for searches done like this with a fixed k and variable base, please post it here.

I will edit this post with additional primes found and update the search limit as I progress.

Admin edit: Effort has now been completed to n=10K. 20 bases remain.


Gary

[henryzz]

wouldnt a pfgw script work

[gd_barnes]

Originally Posted by henryzz

wouldnt a pfgw script work

How might one sieve using PFGW? I'm not referring to factoring like would be done with the -f100 or -f10000 option.

Sieving is the issue when attempting to search this way.

[henryzz]

Originally Posted by gd_barnes

How might one sieve using PFGW? I'm not referring to factoring like would be done with the -f100 or -f10000 option.

Sieving is the issue when attempting to search this way.

yes u would have to skip sieving and do trial factoring instead

[robert44444uk]

Somebody should also be looking at the theory - by checking higher (x^2)^2-1 factorisations, to see whether a smaller k is feasible, by running through bigcover.exe