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An odd perfect number $N$ is said to be given in Eulerian form if $N ={q^k}{n^2}$ where $q$ is prime with $q \equiv k \equiv 1 \pmod 4$ and$\gcd(q,n) = 1$. Similarly, an even perfect number $M$ is said to be given inEuclidean form if $M = (2^p - 1)\cdot{2^{p - 1}}$ where $p$ and $2^p - 1$ areprimes. In this article, we show how simple considerations surrounding thedifferences between the underlying properties of the Eulerian and Euclideanforms of perfect numbers give rise to what we will call the Euclid-Eulerheuristics for perfect numbers.

Euclid-Euler Heuristics for Perfect Numbers