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From: johnf@apollo.uucp (John Francis)
Newsgroups: net.math
Subject: Re: Perfect Numbers
Message-ID: <2a7d48cd.917@apollo.uucp>
Date: Mon, 2-Dec-85 15:26:32 EST
Article-I.D.: apollo.2a7d48cd.917
Posted: Mon Dec  2 15:26:32 1985
Date-Received: Thu, 5-Dec-85 05:39:36 EST
Organization: Apollo Computer, Chelmsford, Mass.
Lines: 14

>    ... numbers of the form  x = 2^(n-1) * (2^n - 1)
>   are perfect if n is a prime number.  ...

This is false - the smallest counter-example occurs for n=11.  The true statement
is that 2^(n-1) * (2^n - 1) is a perfect number if (2^n - 1) is prime (a Mersenne
prime). A necessary condition is that n is prime, but this is not sufficent.

These perfect numbers are known as the Euclid numbers, and it is known that all
even perfect numbers are of this form. To the best of my knowledge nobody has yet
proved that there are no odd perfect numbers, but there are stringent conditions
that any such number will have to meet - it must have at least six different prime
factors, and can not be less than 1.4 x 10^14.

(References: Hardy & Wright - The Theory of Numbers)