On the multiplicative structure of odd perfect numbers

Abstract

Starting with Euler's theorem that any odd perfect number n has the form n = p e p i 2 e i … p k 2 e k , where p, p 1,…, p k are distinct odd primes and p ≡ e ≡ 1 (mod 4), we show that extensive subsets of these numbers (so described) can be eliminated from consideration. A typical result says: if p e , p i 2 e i ,…, p r 2 e r are all of the prime-power divisors of such an n with p ≡ p i ≡ 1 (mod 4), then the ordered set { e 1,…, e r } contains an even number or odd number of odd numbers according as e ≡ p or e ≡ p (mod 8).