20080519, 21:27  #1 
Jan 2006
Hungary
2^{2}×67 Posts 
Riesel/Sierp #'s for bases 3, 7, and 15
Hidiho,
I've done some programming this week and this is what I found: Base 7 cover set = 5, 13, 19, 43, 73, 181, 193, 1201 Smallest Riesel = 408034255082 Base 15 cover set = 13, 17, 113, 211, 241, 1489, 3877 Smallest Riesel = 36370321851498 I'll be tinkering a bit more with my code and then I'll show it here on the forum. Laters, Willem. 
20080520, 03:57  #2  
May 2007
Kansas; USA
23676_{8} Posts 
Quote:
Can you do the same thing for the Sierp side on both bases? Gary 

20080520, 05:59  #3 
May 2007
Kansas; USA
23676_{8} Posts 
I've now confirmed these to be correct, although cannot guarantee that they are the lowest Riesel values. Proofs:
408034255082*7^n1: Code:
Factor noccurrences nremaining 19 n==(1 mod 3) n==(0,2 mod 3) 5 n==(3 mod 4) n==(0,2,5,6,8,9 mod 12) 43 n==(2 mod 6) n==(0,5,6,9 mod 12) 1201 n==(1 mod 8) n==(0,5,6,12,18,21 mod 24) 13 n==(6 mod 12) n==(0,5,12,21 mod 24) 181 n==(0 mod 12) n==(5,21 mod 24) 73 n==(5 mod 24) n==(21 mod 24) 193 n==(21 mod 24) (none) Code:
Factor noccurrences nremaining 241 n==(1 mod 3) n==(0,2 mod 3) 113 n==(2 mod 4) n==(0,3,5,8,9,11 mod 12) 211 n==(3 mod 6) n==(0,5,8,11 mod 12) 17 n==(4 mod 8) n==(0,5,8,11,17,23 mod 24) 1489 n==(0 mod 8) n==(5,11 mod 12) 13 n==(5 mod 12) n==(11 mod 12) 3877 n==(11 mod 12) (none) The Riesel conjecture web pages have now been updated. Gary Last fiddled with by gd_barnes on 20080520 at 06:01 
20080520, 11:13  #4 
Jan 2006
Hungary
2^{2}×67 Posts 
They are the lowest for this cover set.
There may be different cover sets that repeat every 24n. But those also do not give a lower riesel. I will check a bit deeper (36n or 48n) but my program isn't ready for that yet. I need to improve on the efficiency before I can tackle base 3. The proposed cover set for that one repeats every 144n. Laters, Willem. 
20080520, 12:16  #5 
Jun 2003
Oxford, UK
3×5^{4} Posts 
Good work Siemelink. The real challenge is in base 3, where I would like to think there is a really much lower Sierpinski and Riesel.

20080520, 15:08  #6 
Jul 2003
wear a mask
10101101011_{2} Posts 
Wow! That is really great work. Having studied these ideas in the past, I always appreciate seeing someone find lower Riesel and Sierpinski k's and the corresponding covering sets.
Congrats. P.S. You may be able to "brute force" the base 7 result now. This can be done with looping in NewPGen and/or pfgw, I believe. Last fiddled with by masser on 20080520 at 15:12 
20080520, 17:26  #7 
"Robert Gerbicz"
Oct 2005
Hungary
10101011110_{2} Posts 
1*7^n1 is also composite for every positive n, so k=1 would be the smallest Riesel value, or what are you searching? If you accept only even k values, then 4*7^n1 is also composite for every positive n, because it's >3 and divisible by 3.
Last fiddled with by R. Gerbicz on 20080520 at 17:26 
20080520, 17:38  #8  
May 2007
Kansas; USA
2·5,087 Posts 
Quote:
Therefore for Riesel base 7, we only consider k==(0 mod 6) and k==(2 mod 6). Taking it further, for Sierp base 7, we would only consider k==(0 mod 6) and k==(4 mod 6). Gary Last fiddled with by gd_barnes on 20080520 at 17:40 

20080521, 22:01  #9 
Jan 2006
Hungary
2^{2}·67 Posts 
Tada!
Smallest Riesel for base 3 = 1910197852104712
Cover set = {5, 7, 13, 17, 41, 73, 97, 193, 577, 6481} With factor in sequence length 48: 5 6481 13 7 5 13 41 73 13 7 17 13 5 97 13 7 5 13 193 73 13 7 41 13 5 6481 13 7 5 13 41 73 13 7 193 13 5 577 13 7 5 13 17 73 13 7 41 13 
20080521, 22:13  #10 
Jan 2006
Hungary
414_{8} Posts 
Spoke too soon!
Smallest Riesel for base 3 = 1200424637252
Cover set = {5, 7, 13, 19, 37, 41, 73, 757, 6481} With factor in sequence length 72: 13 19 6481 13 5 7 13 37 5 13 73 7 13 757 41 13 5 7 13 19 5 13 41 7 13 37 6481 13 5 7 13 757 5 13 73 7 13 19 41 13 5 7 13 37 5 13 41 7 13 757 6481 13 5 7 13 19 5 13 757 7 13 37 41 13 5 7 13 757 5 13 41 7 
20080522, 00:25  #11 
Jul 2003
wear a mask
1387_{10} Posts 
Riesel base 3
The notation is slightly different on the linked webpage, but the point is that
2*31532322469*3^n1 is always composite. http://tech.groups.yahoo.com/group/p...m/message/4698 Last fiddled with by masser on 20080522 at 00:28 
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