A new definition of primality!
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From: Sebastian Garth on 23 Jul 2010 11:47 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).
From: Santiago Egido on 23 Jul 2010 10:12 This is a really cute result. I have written a little program for everybody to check online: http://www.wiris.com/portal/es/content/primality-test Santiago Egido
From: spudnik on 23 Jul 2010 15:50 yes, the double mod is interesting ... and, of course, it dreives from my observation that one can put the modulo on either side of an equation. just kidding. thus: the climate is changing very rapidly in the Anthropocene, and not according to any computerized simulacra & selective reporting; see Fred Singer's retrospective metastudy on glaciers e.g., iff the God-am palimpsest hasn't crumbled to dust in desertification. thus: anyway, with regard to the Big Bang "theory," the key term is "observable," and we've already got a couple of scopes in orbit, and even one that does infrared, constituting the sort-of-known known, the never-ending manifest of Olber's paradox, which now include's "Doppler's." > Approach could trace as much as 50% of the observable universe. thus: NB, teleportation is a conversion from matter to energy & back; have fun with that! > However if humans could also be teleported great distances thus: it isn't a tax; it's "free-er trade," and that is shown by what the PRC is apparently doing. this was also mooted by the head of the California EPA, who gave a catered "briefing" at the local library to invited guests -- I just happened to see it, noticed in the library lobby -- when he said that the PRC was really going gangbusters on greenstuff, but was merely wating for the USA to legislate, to admit it, which obviously they have been doing of late. now, whether or not there is a more-considered element of taxation in the Chinese model, I don't know. I do know, Ssanta Monica heavily promotes the cargo-cult of PVs and CFLs and every thing from Southwest Asia! the voluntary market in the USA is tens of billions of dolllars per annum in hedging, apparently began by the Presdient in 2003 as CCX, when the Senator was pooling foundation bucks to get it started. there is also an ICE, which I think was began in 2005 (from a little article in the WSUrinal); Waxman's biill, I think that is now before the Senate, mandatorizes the scheme. I would prefer to call it, the next-and-maybe-last bailout of Wall St. and "the City," because this is an international market, and both CCE and ICE are legally bound in British law, not US. thus quoth: China can learn from America's voluntary trading mechanism and the European quota trading mechanism. China's relative carbon emission goal can be transferred into the absolute emission-reduction target. Carbon emission trading can start with the energy-intensive industries like power, steel, construction materials, chemical industries. "In a market-based mechanism, an emissions trading scheme provides economic incentives for reducing pollution," said Zhang. "Companies unable to limit their emissions have to pay a much higher price for extra credits on the exchange." --les ducs d'oil! http://tarpley.net
From: Tim Little on 23 Jul 2010 21:28 On 2010-07-23, Sebastian Garth <sebastiangarth(a)gmail.com> wrote: > ((s(N, N - 1) mod s(N, 1)) + 1) mod N = 0 You could simplify that to Sum_(k<N) k^(N-1) = -1 mod N as the intermediate mod s(N,1) step does nothing useful. > 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). A good primality test is one that is fast for very large numbers. Wilson's Theorem is useless for primality testing as it requires more than O(n) time, which is slower than trial division. So is yours. - Tim
From: Sebastian Garth on 24 Jul 2010 02:55
On Jul 23, 8:28 pm, Tim Little <t...(a)little-possums.net> wrote: > On 2010-07-23, Sebastian Garth <sebastianga...(a)gmail.com> wrote: > > > ((s(N, N - 1) mod s(N, 1)) + 1) mod N = 0 > > You could simplify that to Sum_(k<N) k^(N-1) = -1 mod N as the > intermediate mod s(N,1) step does nothing useful. 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. Eg, ((29 mod s(6, 1)) mod 6) != (29 mod 6). Or did I miss your point? On Jul 23, 8:28 pm, Tim Little <t...(a)little-possums.net> wrote: > A good primality test is one that is fast for very large numbers. > Wilson's Theorem is useless for primality testing as it requires more > than O(n) time, which is slower than trial division. So is yours. I never claimed to have produced an efficient primality test - I was simply implying that the insight thus gained *may* lead to better ones. |