On the multiplicative structure of odd perfect numbers

AbstractStarting with Euler's theorem that any odd perfect number n has the form n = pepi2ei … pk2ek, where p, p1,…,pk 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 pe, pi2ei,…,pr2er are all of the prime-power divisors of such an n with p ≡ pi ≡ 1 (mod 4), then the ordered set {e1,…,er} contains an even number or odd number of odd numbers according as e ≡ p or e ≡ p (mod 8)