Outline of a proof that every odd perfect number has at least eight prime factors

Abstract

An argument is outlined which demonstrates that every odd perfect number is divisible by at least eight distinct primes.

Odd perfect numbers have a prime factor exceeding $10^8$

“…Chein [2] and Hagis [7] independently showed that n must have at least 8 distinct prime factors, and this bound was recently improved to 9 by Nielsen [18]. Hare [9] showed that n must have totally at least 47 prime factors, and he recently improved this bound to 75 in [10].…”

Improved Techniques for Lower Bounds for Odd Perfect Numbers

“…= 0 (mod 2). It is easy to show that /' > 2, and we make implicit use of this below; in fact, it is known that j >1 (Hagis [6]). Each py , and q , are called components of N.…”

Sketch of a proof that an odd perfect number relatively prime to 3 has at least eleven prime factors

“…Throughout this paper N will represent an odd perfect number, and o>(N) will denote the number of distinct prime factors of TV. It was shown in [1] that w(7V) > 8, while if 3 J TV it was proved by Kishore [6] that u(N)> 10. The purpose of the present paper is to sketch a proof of the following improvement of Kishore's result.…”