Euclid-Euler Heuristics for Perfect Numbers

Abstract

An odd perfect number $N$ is said to be given in Eulerian form if $N=q^k{n}^2$ where $q$ is prime with $q\equiv k\equiv 1(\mathrm{mod}4)$ and $\gcd (q,n)=1$. Similarly, an even perfect number $M$ is said to be given in Euclidean form if $M=(2^p-1)\cdot 2^{p-1}$ 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.

9 pages

References in corpus (1)

• On a Curious Biconditional Involving Divisors of Odd Perfect Numbers

Cited by in corpus (3)

• The Non-Euler Part of a Spoof Odd Perfect Number is Not Almost Perfect
• A Sufficient Condition for Disproving Descartes's Conjecture on Odd Perfect Numbers
• A short proof for the Descartes-Frenicle-Sorli conjecture on odd perfect numbers