Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP Posting-Version: version B 2.10.2 9/18/84; site linus.UUCP Path: utzoo!linus!bs From: bs@linus.UUCP (Robert D. Silverman) Newsgroups: net.math Subject: Mersenne Primes Message-ID: <576@linus.UUCP> Date: Mon, 30-Sep-85 08:09:37 EDT Article-I.D.: linus.576 Posted: Mon Sep 30 08:09:37 1985 Date-Received: Wed, 2-Oct-85 10:08:09 EDT Distribution: net Organization: The MITRE Corporation, Bedford, MA Lines: 29 The exponents for the thirty known Mersenne primes are: 1,2,3,5,7,13,17,19,31,61,89,107,127,521,607,1279,2203,2281,3217,4253,9941,11213, 19937,21701,23209,44497,86243,132049,216091 A number of the form 2^p - 1 is prime iff its rank of apparition in the sequence: 2 2 x = x - 2 n+1 n Is p-2 where x = 4. 0 That is to say 2^p -1 is prime iff the p-2 term in the above sequence is divisible by 2^p-1. Thus, to do the prime test one need only compute the sequence mod 2^p - 1 and see if the p-2 term is zero. One can compute the sequence with just multiplication and addition by noting that: n n n 2 A + B = (2 - 1)A + B + A = B + A mod (2 - 1) One 'merely' needs a very fast routine which will multiply 65,000 digit numbers. I believe that Mr. Slowinski, who found the prime, uses some sort of fast fourier transform multiplication routine. Bob Silverman (they call me Mr.9)