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Path: utzoo!linus!decvax!harpo!utah-cs!arizona!doug
From: doug@arizona.UUCP
Newsgroups: net.math
Subject: More On Mersenne Numbers
Message-ID: <5582@arizona.UUCP>
Date: Fri, 21-Oct-83 14:37:45 EDT
Article-I.D.: arizona.5582
Posted: Fri Oct 21 14:37:45 1983
Date-Received: Tue, 25-Oct-83 00:47:12 EDT
Organization: CS Dept, U of Arizona, Tucson
Lines: 22


N being prime does not guarantee 2^n -1 will be prime.  Take a look a n=11.  
2^11 -1 = 2047 = 23*89.  Euclid's theorem, ((2^n)-1)*(2^(n-1)) is perfect 
iff (2^n)-1 is prime, has been proven on a number of occasions 
(see History of the Theory of Numbers, vol. 1, pp 3, 5, 10).
If someone has a counter example, I would of course be very interested in
seeing it...

Mersenne stated that (2^n)-1 is a prime if n is a prime which exceeds by 3 
or less a power of 2 with an even exponent (which alas, does not qualify 11).  
The first 6 of these are: 
           (2^3)-1=7,           (2^5)-1=31,            (2^7)-1=127, 
          (2^17)-1=8191,       (2^19)-1=131071,       (2^67)-1=524287.  

Notice that: 
             3=(2^0)+3,           5=(2^2)+1,            7=(2^2)+3, 
            17=(2^4)+1,          19=(2^4)+3,           67=(2^6)+3.

If someone has a counter example for that, I would also be very interested in
seeing it....
                                                         Pase