More on the Nonexistence of Odd Perfect Numbers of a Certain Form

Abstract

Euler showed that if an odd perfect number exists, it must be of the form N = p α q 2β 1 1 . . . q 2β k k , where p, q 1 , . . . , q k are distinct odd primes, α, β 1 , . . . , β k ∈ N, with p ≡ α ≡ 1 (mod 4). In 2005, Evans and Pearlman showed that N is not perfect, if 3|N or 7|N and each β i ≡ 2 (mod 5). We improve on this result by removing the hypothesis that 3|N or 7|N and show that N is not perfect, simply, if each β i ≡ 2 (mod 5).