20080601, 12:38  #1 
Just call me Henry
"David"
Sep 2007
Cambridge (GMT/BST)
2^{2}·3·479 Posts 
k=1 thru k=12
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. Last fiddled with by gd_barnes on 20201125 at 03:51 Reason: update status 
20080601, 15:50  #2 
Just call me Henry
"David"
Sep 2007
Cambridge (GMT/BST)
2^{2}×3×479 Posts 
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 
20080602, 07:16  #3 
May 2007
Kansas; USA
7^{2}×11×19 Posts 
Riesel bases always have a trivial factor of b1 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 kvalues in the 'come join us' thread. Gary 
20080602, 13:16  #4 
Jun 2003
Oxford, UK
3560_{8} Posts 
For the infinity of bases, the smallest Sierpinski k may take any integer value except 2^x1, 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 
20081102, 22:43  #5  
May 2007
Kansas; USA
7^{2}·11·19 Posts 
Quote:
Is this the lowest base where k=2 is the Sierpinski number? 

20081103, 03:06  #6 
Jun 2003
Oxford, UK
770_{16} Posts 
This k was generated from looking at (x^2)^n1 factorisations covering set is 3,5,17,257,641,65537,6700417 which I think is 32cover. I do not think anyone has claimed it is the smallest k, it just comes from the smallestcover.

20081103, 04:49  #7  
May 2007
Kansas; USA
24001_{8} Posts 
Quote:
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^n1 at n=10K: Code:
b 107 170 278 303 383 515 522 578 581 590 647 662 698 704 845 938 969 989 992 1019 Code:
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 Last fiddled with by gd_barnes on 20081111 at 07:06 Reason: add additional primes 

20081103, 07:55  #8 
Just call me Henry
"David"
Sep 2007
Cambridge (GMT/BST)
2^{2}·3·479 Posts 
wouldnt a pfgw script work

20081103, 11:07  #9 
May 2007
Kansas; USA
7^{2}×11×19 Posts 

20081103, 14:46  #10 
Just call me Henry
"David"
Sep 2007
Cambridge (GMT/BST)
1674_{16} Posts 

20081104, 04:31  #11 
Jun 2003
Oxford, UK
2^{4}×7×17 Posts 
Somebody should also be looking at the theory  by checking higher (x^2)^21 factorisations, to see whether a smaller k is feasible, by running through bigcover.exe
