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From: bs@faron.UUCP (Robert D. Silverman)
Newsgroups: net.math
Subject: Mersenne Primes (in general)
Message-ID: <359@faron.UUCP>
Date: Tue, 1-Oct-85 07:55:22 EDT
Article-I.D.: faron.359
Posted: Tue Oct  1 07:55:22 1985
Date-Received: Wed, 2-Oct-85 20:33:23 EDT
Distribution: net
Organization: The MITRE Coporation, 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)