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  • Is there any interest in a collaboration to build a truly massive database? No. It's faster to generate primes than to download them. A good generator is http://primesieve.org/ for example. Best regards, Andrey
    Andrey Kulsha Aug 2, 2014
  •  But this sequence has 3 consecutive primes {23,29,31}. What is the probability of such a occurrence? 02:03:05 03:05:07 05:07:11 07:11:13 11:13:17 13:17:19 17:19:23 19:23:29 23:29:31 pi(24)/86400 = 1/9600 Best regards, Andrey
    Andrey Kulsha Dec 28, 2013
  •  Here is the extra step. Let a = 1 + r*p. Then S := (a^p-1)/(a-1) = sum(k=0,p-1,a^k) = sum(k=0,p-1,1+k*r*p) mod p^2 = p mod p^2, since sum(k=0,p-1,k)=p*(p-1)/2 and p is odd. Hence S/p = 1 mod p. Happy now? That was so simple. Thanks. Andrey :-)
    Andrey Kulsha Dec 23, 2013
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  •  > Conjecture: > if p divides a-1, then (a^p-1)/(a-1)/p is p-tight. This follows from Theorem 48 of Daniel Shanks' delightful book: https://archive.org/details/SolvedAndUnsolvedProblemsInNumberTheory Thanks for the book. Maybe I misunderstand something, but it's quite obvious that p divides (a^p-1)/(a-1) when a = 1 (mod p), because (a^p-1)/(a-1) = sum(a^k,{k,0,p-1}). The...
    Andrey Kulsha Dec 21, 2013
  • Given an odd prime p, let's call the positive integer "p-tight" if all its prime factors are 1 (mod 2p). Theorem: if p does not divide a-1, then (a^p-1)/(a-1) is p-tight. (trivial proof follows from FLT) Conjecture: if p divides a-1, then (a^p-1)/(a-1)/p is p-tight. Any suggestions for a proof? Thanks a lot, Andrey
    Andrey Kulsha Dec 20, 2013
  • PrimeSieve by Kim Walisch finds all quadruplets up to 1e12 just in one minute: http://code.google.com/p/primesieve/ Best regards, Andrey
    Andrey Kulsha Dec 6, 2013
  • No one interested? :( --- In primenumbers@^$1, Andrey Kulsha wrote: > > The gap of 1368 is known between primes > 4105079953458040849 and 4105079953458042217: > http://www.trnicely.net/gaps/gaplist.html > > To prove the first occurence, we need to check > a relatively small range from 4e18 to 4.10508e18 > because 4e18 is already searched: > http://sweet.ua.pt/tos/goldbach.html...
    andrey_601 Jul 30, 2013
  • The gap of 1368 is known between primes 4105079953458040849 and 4105079953458042217: http://www.trnicely.net/gaps/gaplist.html To prove the first occurence, we need to check a relatively small range from 4e18 to 4.10508e18 because 4e18 is already searched: http://sweet.ua.pt/tos/goldbach.html It seems the search could be optimized if we focus exactly on 1368. I'd provide some...
    Andrey Kulsha Jul 27, 2013
  • > Summing the geometric series in Andrey Kulsha's conjecture > http://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind0107&L=nmbrthry&P=3477 > we arrive at a rather attractive conjectural heuristic. That conjecture seems to be wrong. BTW, the new correct link to it is https://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind0107&L=NMBRTHRY&P=21148 [there's a typo, f(g,x) should be instead of f(g,p...
    Andrey Kulsha Aug 29, 2012
  • If there are k composites preceding Nth prime, then the proposed merit is k / (log N)^2 There are 17 known gaps with merit > 1: merit k N 1.15817 3 5 1.13821 33 218 1.13709 1131 49749629143527 1.10242 13 31 1.07487 71 3386 1.05228 209 1319946 1.05033 455 1094330260 1.03633 905 6822667965941 1.03522 147 149690 1.03436 111 31546 1.03257 765 662221289044 1.02282 1441 20004097201301080...
    Andrey Kulsha May 9, 2012