User group for PFGW & PrimeForm programs Yahoo Group Sierpinski / Riesel base 2^x+1 =============================================== Robert Message 1 of 4 Sep 26, 2004 ----------------------------------------------- Following on to my post on Sierpinski/ Riesel base 5, it seems worth exploring the various Sierpinski and Riesel numbers base 2^x+1, where x is an integer. These (3,5,9,17,33...) are the only bases which do not provide trivial even k solutions. Why is that? No doubt someone will tell me! It is, of course, only a conjecture at this stage. Even so, the base sets are generally simple. My (and others) results so far are (I hope my table posts sensibly): Number Candidates left Covering set Sierp Riesel Sierp Riesel Sierpinski Riesel 5 159986 346802 230 c600 3,7,13,31,601 3,7,13,31,601 9 2344 74 4 4 5,7,13,73 5,7,13,73 17 278 86 4 0 3,5,29 3,5,29 33 1854 764 14 9 5,7,17,109 5,7,17,109 65 10 10 0 0 3,11 3,11 129 14 14 1 1 5,13 5,13 257 44 44 4 0 3,43 3,43 513 ?? ?? 1025 20 20 0 2 3,19 3,19 2049 124 124 11 11 5,41 5,41 4097 94 682? 17 110 3,5,7,13,19 3,683 513 has given me gyp, because there are so many choices of potential covering sets, but the obvious candidates do not have the right congruences. Maybe someone will solve that. The 4097 Riesel is only a maybe. The others need checking, but I think they are the smallest. Those with only two members in the covering set share the same k value, but other than that there is not much to observe. The column showing "Candidates left" is after a shallow check for primes, and therefore shows those with no small primes. But these numbers get big quickly, for example, 4*9^n-1 is not prime up to n=83606 (the only number I have checked to any depth), so actually showing that 74.9^n-1 is the Riesel number base 9 may prove beyond us, who knows! Regards Robert Smith =============================================== David Broadhurst Message 2 of 4 Sep 26, 2004 ----------------------------------------------- Robert: > These (3,5,9,17,33...) are the only bases which do not provide > trivial even k solutions. Why is that? Consider any odd base b that is _not_ of the form 2^x+1. Then b-1 is divisible by at least one odd prime, say q. So q-1 is a Sierpinski number: (q-1)*b^n+1=0 mod q for all n and q+1 is a Riesel number: (q+1)*b^n-1=0 mod q for all n since b=1 mod q. David =============================================== David Broadhurst Message 3 of 4 Sep 26, 2004 ----------------------------------------------- The only reason that b=2^x+1 is nontrivial, for x>0, is the fact that you demand the multiplier k, in k*b^n -/+ 1, to be even, i.e. k != +/-1 mod 2 So why not generalize your "exclusion principle" to exclude the trivial residue classes +/-1 modulo *all* of the prime divisors of b-1? Then you have the same game to play in every odd base, not so? For example: Search for k such that k != +1 mod 2 [you do this already!] k != +1 mod 3 [so why not this also?] k*7^n - 1 is always composite Search for k such that k != -1 mod 2 [you do this already!] k != -1 mod 3 [so why not this also?] k*7^n + 1 is always composite Search for k such that k != +1 mod 2 [you do this already!] k != +1 mod 3 [so why not this also?] k != +1 mod 5 [and this also!] k*31^n - 1 is always composite Search for k such that k != -1 mod 2 [you do this already!] k != -1 mod 3 [so why not this also?] k != -1 mod 5 [and this also!] k*31^n + 1 is always composite At present you have an exponential explosion of supposedly "non-trivial" bases because you exclude the trivial residue classes modulo only one of the prime divisors of b-1, namely modulo 2. If you want to progress sub-exponentially, the appropriate "generalized exclusion principle" should be clear from the above, not so? David (per proxy Wolfgang Pauli) =============================================== Shane Message 4 of 4 Sep 27, 2004 ----------------------------------------------- David wrote, > The only reason that b=2^x+1 is nontrivial, for x>0, > is the fact that you demand the multiplier k, in > k*b^n -/+ 1, > to be even, i.e. > k != +/-1 mod 2 Sounds like you're speaking to me. Obviously not directly, but this is verbatim what the generalized RMA sequence is. ID Number: A084924 URL: http://www.research.att.com/projects/OEIS?Anum=A084924 Non-triavial in respect to base two. Since RMA, is only for base two, and the upcoming ASA, will be for multiple bases. The real non-trivial sequence is given by ASA. Anchored-Sieving-Algorithm. =============================================== Cached by Georg Fischer at Nov 14 2019 12:47 with clean_yahoo.pl V1.4