A new definition of primality!
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From: Tim Little on 24 Jul 2010 03:07 On 2010-07-24, Sebastian Garth <sebastiangarth(a)gmail.com> wrote: > If you're asserting that, generally speaking, ((Q mod s(N, 1)) mod N) > = (Q mod N), always, for all Q, then that's just plain wrong. I'm not. It holds for all odd N (as s(N,1) is divisible by N), and the even case is irrelevant as the equality never holds. The simplified equation is true iff the original one is. - Tim
From: Sebastian Garth on 24 Jul 2010 04:03 On Jul 24, 2:07 am, Tim Little <t...(a)little-possums.net> wrote: > On 2010-07-24, Sebastian Garth <sebastianga...(a)gmail.com> wrote: > > > If you're asserting that, generally speaking, ((Q mod s(N, 1)) mod N) > > = (Q mod N), always, for all Q, then that's just plain wrong. > > I'm not. It holds for all odd N (as s(N,1) is divisible by N), and > the even case is irrelevant as the equality never holds. The > simplified equation is true iff the original one is. > > - Tim Of course! Yes, originally my equation accounted for all N > 2 (for the sake of "completeness"). Thanks for the clarification. Cheers, - Sebastian
From: Frederick Williams on 25 Jul 2010 08:17 Sebastian Garth wrote: > > In an earlier thread, I put forward a conjecture that generalized > Fermat's Little Theorem. Specifically: > > For all N > 2, IFF gcd(s(N, N - 1) mod s(N, 1), N) = 1 then N is > either a prime or a Carmichael number, where s(N, E) is the sum of > powers (eg: 1^E + 2^E ... + N^E). What does Iff ... then ... mean? I am familiar with If ... then ... and ... iff ... -- I can't go on, I'll go on.
From: spudnik on 25 Jul 2010 20:25 if & only if, that is to say, Liebniz's neccesity & sufficiency, to be used in any literate manner! > Iff ... then ... --les ducs d'oil! http://tarpley.net
From: Gerry Myerson on 28 Jul 2010 02:26
In article <050907ee-6b56-4100-8423-08c2f3d419d9(a)d37g2000yqm.googlegroups.com>, Sebastian Garth <sebastiangarth(a)gmail.com> wrote: > In an earlier thread, I put forward a conjecture that generalized > Fermat's Little Theorem. Specifically: > > For all N > 2, IFF gcd(s(N, N - 1) mod s(N, 1), N) = 1 then N is > either a prime or a Carmichael number, where s(N, E) is the sum of > powers (eg: 1^E + 2^E ... + N^E). > > Using a related concept, I can now make a statement that generalizes > *all* prime numbers: > > For all N > 2, IFF ((s(N, N - 1) mod s(N, 1)) + 1) mod N = 0 then N is > definitely prime, where s(N, E) is the sum of powers (eg: 1^E + > 2^E ... + N^E). > > AFAIK, the only other theorem that achieves a similar level of > concision is Wilson's Theorem, so the implications of this equation > may be quite significant (eg: may lead to much better primality > tests). You may be interested in looking up Giuga's conjecture. -- Gerry Myerson (gerry(a)maths.mq.edi.ai) (i -> u for email) |