mersenneforum.org Sierpinski/ Riesel bases 6 to 18
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 2007-01-06, 07:55 #23 ltd     Apr 2003 22·193 Posts Change my reservation i will sieve all noted k for base 17 (riesel and sierpinski) Lars
 2007-01-06, 08:40 #24 robert44444uk     Jun 2003 Oxford, UK 35628 Posts Base 6 Did a little work this afternoon of base 6. Using the covering set [7,43,37,31,13] repeating every 24n provides a Sierpinski number 243417. I will try to do the Riesel later. Note that the alternative set [7,43,37,31,97] repeating every 24n could also provide a lower Sierpinski value. However 243417 is at 0.73% of the product of this set's cover primes, and there are only 24 values to check, if I was any good at statistics I could tell you what the probability is, but I am not!! Regards Robert Smith
 2007-01-06, 08:53 #25 Citrix     Jun 2003 32×52×7 Posts Candidates for 10 804*10^n+1 1024*10^n+1 2157*10^n+1 2311*10^n+1 2607*10^n+1 2661*10^n+1 2683*10^n+1 3301*10^n+1 3312*10^n+1 3345*10^n+1 3981*10^n+1 4069*10^n+1 4863*10^n+1 5028*10^n+1 5125*10^n+1 5512*10^n+1 5556*10^n+1 5565*10^n+1 6172*10^n+1 6687*10^n+1 6841*10^n+1 7404*10^n+1 7459*10^n+1 7534*10^n+1 7666*10^n+1 7809*10^n+1 7866*10^n+1 8194*10^n+1 8425*10^n+1 8454*10^n+1 8667*10^n+1 8724*10^n+1 8889*10^n+1 8922*10^n+1 8953*10^n+1 9021*10^n+1 9043*10^n+1 9175*10^n+1 9351*10^n+1 1343*10^n-1 1506*10^n-1 1803*10^n-1 1935*10^n-1 2111*10^n-1 2276*10^n-1 2333*10^n-1 3015*10^n-1 3332*10^n-1 3356*10^n-1 4016*10^n-1 4421*10^n-1 4478*10^n-1 4577*10^n-1 5499*10^n-1 5897*10^n-1 6588*10^n-1 6633*10^n-1 6665*10^n-1 7019*10^n-1 7602*10^n-1 8174*10^n-1 8579*10^n-1 9461*10^n-1 9701*10^n-1 9824*10^n-1 10176*10^n-1 candidates for 16 riesel (upto 10,000) 450*16^n-1 1343*16^n-1 1803*16^n-1 1935*16^n-1 2333*16^n-1 3015*16^n-1 3332*16^n-1 4478*16^n-1 4500*16^n-1 4577*16^n-1 5499*16^n-1 5897*16^n-1 6588*16^n-1 6633*16^n-1 6665*16^n-1 7019*16^n-1 7602*16^n-1 8174*16^n-1 8579*16^n-1
 2007-01-06, 11:34 #26 robert44444uk     Jun 2003 Oxford, UK 77216 Posts Riesel candidate for base 6 After a bit of fiddling about with [7,43,37,31,13] came up with the riesel candidate 213410 for base 6. 133946 is trivial. Regards Robert Smith
 2007-01-06, 12:07 #27 robert44444uk     Jun 2003 Oxford, UK 35628 Posts Simple to prove Sierpinskis and Riesels for the following bases are simple to find and should be relatively simple to prove as they have 2 prime factors in b^2-1 which are not in b-1, and therefore have cover from these new prime factors, every 2n. 14 was the first such case, proven with S-4, R-4 20 is next with S-8, R-8, both proven The others exhibiting this small covering set, less than base =100, and which therefore should be relatively simple to prove are 29, 32, 34, 38, 41, 44, 50, 54, 56, 59, 62, 64, 65, 68, 69, 74, 76, 77, 83, 84, 86, 89, 90, 92, 94, 98 Someone might want to just run these quickly to prove them. Then we might bash on to find Sierpinskis and Riesels for all other bases up to 100. Tomorrow I start work again so time I can spend on this will be limited. Regards Robert Smith
2007-01-06, 12:49   #28
rogue

"Mark"
Apr 2003
Between here and the

2·3·17·59 Posts

Quote:
 Originally Posted by robert44444uk The problem is that, for some values, there is no mooted covering set less than 10^16, therefore the suggested method is inefficient for these bases. Tacking 3 and 15 needs another approach, but I am at a loss to think how to do efficiently.
I agree with that. My method works well if the Riesel/Sierpinski number is relatively small. Once you hit numbers that large it could take weeks, months, or even years to find one.

2007-01-06, 14:14   #29
robert44444uk

Jun 2003
Oxford, UK

2·953 Posts
Alternative approach

Quote:
 Originally Posted by rogue I agree with that. My method works well if the Riesel/Sierpinski number is relatively small. Once you hit numbers that large it could take weeks, months, or even years to find one.
I have been thinking that covering sets should comprise some prime factor(s) which is/are small enough to appear often enough to allow the creation of the cover. I also noticed that there are relationships of modF, for prime factors F of b^n-1 which are predictable. If n=2, then the sierpinski/ riesel are all either 1modF or(F-1)modF. In fact for n=N, then 2/N of the sierpinskis/ riesels are 1ModF or (F-1)modF. This might allow some type of sieving to take place which would allow some concentration of numbers to be tested.

So an approach would be to define, in principle, what properties covering sets must have. For example it is highly unlikely that a covering set would have no P which has a mulitplicative order in base b of less than 5. But can we prove this is the case? If we can, then we can say for certain that the covering set must have at least one of the prime factors of b^2-1, b^3-1 or b^4-1. Then we should be able to sieve out some k from providing a covering set, and only test those which meet the mod criteria.

Just a few musings on an approach.

Regards

Robert Smith

 2007-01-06, 17:03 #30 ltd     Apr 2003 22×193 Posts First one down: 88*17^4868+1 is prime. Lars Edit: Next one down: 44*17^6488-1 is prime. Riesel side done. Last fiddled with by ltd on 2007-01-06 at 17:32
2007-01-06, 18:16   #31
Xentar

Sep 2006

101110102 Posts

Think, I will try base 18

Quote:
 Sierpinski value is 398. 4 k candidates seek primes 18,324,122,381

 2007-01-06, 23:31 #32 Citrix     Jun 2003 32×52×7 Posts For base 16, I found a covering set [17,13,7,241] So S/R must be less than 372827 For base 32 [3,7,13,17,241] Last fiddled with by Citrix on 2007-01-06 at 23:32
 2007-01-06, 23:54 #33 rogue     "Mark" Apr 2003 Between here and the 2·3·17·59 Posts I think that it would make sense to either put up a website or make a sticky thread with the current status for each base. It is beginning to be difficult to follow this thread.

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