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Path: utzoo!linus!bs
From: bs@linus.UUCP (Robert D. Silverman)
Newsgroups: net.math
Subject: Re: Mersenne Primes (in general)
Message-ID: <579@linus.UUCP>
Date: Wed, 2-Oct-85 09:22:43 EDT
Article-I.D.: linus.579
Posted: Wed Oct  2 09:22:43 1985
Date-Received: Fri, 4-Oct-85 03:38:19 EDT
References: <359@faron.UUCP>
Distribution: net
Organization: The MITRE Corporation, Bedford, MA
Lines: 42

> 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)

Sorry about the typo in the last posting: The proper prime testing sequence
should be:


	       2
	x   = x  - 2
         n+1   n
 
My previous posting mistakenly had an exponent 2 on the left hand side.

Bob Silverman  (they call me Mr. 9)