On the Nonexistence of Odd Perfect Numbers

In this article, we show how to prove that an odd perfect number with eight distinct prime factors is divisible by 5. A perfect number N is equal to twice the sum of its divisors: σ(N) = 2N . The theory of perfect numbers when N is even is well known: Euclid proved that if 2 − 1 is prime, then 2p−1(2p − 1) is perfect, and Euler proved that every one is of this type. These numbers have seen a great deal of attention, ranging from very ancient numerology (Saint Augustine considered 6 to be a truly perfect number, since God fashioned the Earth in precisely this many days). They were also very important to the Greeks and to Fermat, whose investigations led him to his little theorem. Today, we have found 38 Mersenne primes (those of the form 2− 1); the latest, found on June 1, 1999 by Nayan Hajratwala, was part of the Great Internet Mersenne Prime Search (GIMPS) (see http://www.mersenne.org/); it is