Euclid-Euler Heuristics for Perfect Numbers

2013/10/14 by Dris, Jose Arnaldo B.
#11A05 (Primary) #11J25 #11J99 (Secondary) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1310.5616

Abstract

An odd perfect number N is said to be given in Eulerian form if N = q^kn² where q is prime with q ≡ k ≡ 1 \pmod 4 and gcd(q,n) = 1. Similarly, an even perfect number M is said to be given in Euclidean form if M = (2^p - 1)⋅2^{p - 1} where p and 2^p - 1 are primes. In this article, we show how simple considerations surrounding the differences between the underlying properties of the Eulerian and Euclidean forms of perfect numbers give rise to what we will call the Euclid-Euler heuristics for perfect numbers.

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