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From: doug@arizona.UUCP (Doug Pase)
Newsgroups: net.math
Subject: More On Mersenne Numbers
Message-ID: <5594@arizona.UUCP>
Date: Fri, 21-Oct-83 19:33:00 EDT
Article-I.D.: arizona.5594
Posted: Fri Oct 21 19:33:00 1983
Date-Received: Sat, 22-Oct-83 17:00:45 EDT
Organization: CS Dept, U of Arizona, Tucson
Lines: 20


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, which says ((2^n)-1)*(2^(n-1)) 
is perfect if (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