An extension of the Euclid-Euler theorem to certain α-perfect numbers

Abstract

In a posthumously published work, Euler proved that all even perfect numbers are of the form 2^(p-1)(2^p−1), where 2^p−1 is a prime number. In this article, we extend Euler’s method for certain α-perfect numbers for which Euler’s result can be generalized. In particular, we use Euler’s method to prove that if N is a 3-perfect number divisible by 6; then either 2 || N or 3 || N. As well, we prove that if N is a 5/2-perfect number divisible by 5, then 2^4 || N, 5^2 || N, and 31^2| N. Finally, for p ∈ {17, 257, 65537}, we prove that there are no 2p/(p−1)-perfect numbers divisible by p.