  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 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.
