mersenneforum.org Riesel/Sierp #'s for bases 3, 7, and 15
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 2008-05-19, 21:27 #1 Siemelink     Jan 2006 Hungary 1000011002 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.
2008-05-20, 03:57   #2
gd_barnes

May 2007
Kansas; USA

1022610 Posts

Quote:
 Originally Posted by Siemelink 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.
If you are correct, this is HUGE, especially for base 7 where the conjecture dropped substantially! I'll do some verification myself and if they are correct, I will change the web pages.

Can you do the same thing for the Sierp side on both bases?

Gary

 2008-05-20, 05:59 #3 gd_barnes     May 2007 Kansas; USA 27F216 Posts I've now confirmed these to be correct, although cannot guarantee that they are the lowest Riesel values. Proofs: 408034255082*7^n-1: Code: Factor n-occurrences n-remaining 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) 36370321851498*15^n-1: Code: Factor n-occurrences n-remaining 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) A nice piece of programming Willem! The Riesel conjecture web pages have now been updated. Gary Last fiddled with by gd_barnes on 2008-05-20 at 06:01
 2008-05-20, 11:13 #4 Siemelink     Jan 2006 Hungary 22·67 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.
 2008-05-20, 12:16 #5 robert44444uk     Jun 2003 Oxford, UK 2·13·73 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.
 2008-05-20, 15:08 #6 masser     Jul 2003 wear a mask 22·192 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 2008-05-20 at 15:12
2008-05-20, 17:26   #7
R. Gerbicz

"Robert Gerbicz"
Oct 2005
Hungary

17·83 Posts

Quote:
 Originally Posted by gd_barnes I've now confirmed these to be correct, although cannot guarantee that they are the lowest Riesel values.
1*7^n-1 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^n-1 is also composite for every positive n, because it's >3 and divisible by 3.

Last fiddled with by R. Gerbicz on 2008-05-20 at 17:26

2008-05-20, 17:38   #8
gd_barnes

May 2007
Kansas; USA

237628 Posts

Quote:
 Originally Posted by R. Gerbicz 1*7^n-1 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^n-1 is also composite for every positive n, because it's >3 and divisible by 3.
k=1 and k=4 have trivial factors of 3 for all n-values and hence are not considered. For Riesel base 7, we do not consider k==(1 mod 3) where all n-values have a trivial factor of 3 nor k==(1 mod 2) where all n-values have a trivial factor of 2.

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 2008-05-20 at 17:40

 2008-05-21, 22:01 #9 Siemelink     Jan 2006 Hungary 26810 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
 2008-05-21, 22:13 #10 Siemelink     Jan 2006 Hungary 22·67 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
 2008-05-22, 00:25 #11 masser     Jul 2003 wear a mask 22×192 Posts Riesel base 3 The notation is slightly different on the linked webpage, but the point is that 2*31532322469*3^n-1 is always composite. http://tech.groups.yahoo.com/group/p...m/message/4698 Last fiddled with by masser on 2008-05-22 at 00:28

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