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From: doug@arizona.UUCP
Newsgroups: net.math
Subject: More On Mersenne Numbers
Message-ID: <5604@arizona.UUCP>
Date: Sat, 22-Oct-83 03:26:11 EDT
Article-I.D.: arizona.5604
Posted: Sat Oct 22 03:26:11 1983
Date-Received: Sun, 23-Oct-83 08:34:23 EDT
Organization: CS Dept, U of Arizona, Tucson
Lines: 21


When first set upon to write about Mersenne primes, I didn't stop at
correcting the existing errors.  I boldly forged ahead adding my own
blunders to the list of those already written.  In particular, (2^17)-1
is not 8191, (2^13)-1 is.  Similarly, (2^17)-1 = 131 071, (2^19)-1 = 524 287,
and (2^67)-1 = 147 573 952 589 676 412 927.  Note that 13 is not in the
form (2^(2*n)) +1 or +3 and (2^13)-1 is prime.

Also, I asked for a counter example to Mersenne's conjecture, and I have
one.  67 is prime, (2^6)+3=67, and (2^67)-1 is not prime.  Hence, Mersene's
conjecture is false.  (2^67)-1 was proved composite nearly 100 years ago,
I believe by a man named Lucas.

Descartes proved in 1638 that every even perfect number is of Euclid's type,
and every odd perfect number is of the form p(s^2), where p is prime.  J.J.
Sylvester concluded that there is no odd perfect number (opn - I'm getting
tired of writing it) with fewer than six distinct prime factors, (eight if
not divisible by three), no opn is of the form (a^n)(b^m), and no opn is
divisible by 105.  Since this was nearly 100 years ago, I'm sure that the
list of restrictions by now has grown longer.
                                                           Pase