Newsgroups: rec.puzzles
From: hoey@zogwarg.etl.army.mil (Dan Hoey)
Date: 28 Jan 91 17:14:07 GMT
Subject: Re: Perfect numbers

iain@minster.york.ac.uk writes:

>Can any one supply me with large perfect numbers ?
>(By large I mean greater that 8589869056)

All known perfect numbers are of the form (2^p - 1) 2^(p-1), where
2^p-1 is prime.  Such primes are called Mersenne primes and the list
as of mid-1990 is p=2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521,
607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937,
21701, 23209, 44497, 86243, 110503, 132049, and 216091.  Any other
Mersenne primes must have 150000<p<216089 or p>350000.  Any other
perfect numbers must be odd, and as of 1989 it is known they must be
greater than 10^300.

Thanks to Chip Kerchner and Bob Silverman for posting these status
reports last year.

Dan