On the total number of prime factors of an odd perfect number

Abstract

We say n ∈ N is perfect if σ(n) = 2n, where σ(n) denotes the sum of the positive divisors of n. No odd perfect numbers are known, but it is well known that if such a number exists, it must have prime factorization of the form n = pα Φj=1k qj2βj, where p,q1,..., qk are distinct primes and p ≡ α ≡ 1 (mod 4). We prove that if βj ≡ 1 (mod 3) or βj ≡ 2 (mod 5) for all j, 1 ≤ j ≤ k, then 3|n. We also prove as our main result that Ω(n) ≥ 37, where Ω(n) = α + 2 Σj=1k βj. This improves a result of Sayers (Ω(n) ≥ 29) given in 1986.