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From: dale@linc.cis.upenn.edu (Dale Miller)
Newsgroups: comp.lang.prolog
Subject: Re: perfect numbers
Message-ID: <7093@netnews.upenn.edu>
Date: 11 Jan 89 18:17:36 GMT
References: <1564@kulcs.kulcs.uucp>
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Reply-To: dale@linc.cis.upenn.edu.UUCP (Dale Miller)
Organization: University of Pennsylvania
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The following results are known about perfect numbers.

Theorem: An even number is perfect if and only if it is of the form
2^n*(2^(n+1) - 1) where (2^(n+1) - 1) is a prime (such primes are
called Mersenne primes).

No odd perfect numbers are known.  There is the following result,
however.

Theorem: If an odd number is perfect, it must have at least four
distinct prime factors and be larger than 2,000,000.

This result has been strengthen greatly in recent years (more prime
factors would need to be present and the lower bound has been raised
considerably) but I don't remember the exact results.

If your Prolog program happens to find an odd perfect number, be sure
to publish it!