Integer as Difference between Two Squares

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Formulation 1

Let $n$ be a positive integer.

Then $n$ can be expressed as:

$n = a^2 - b^2$

if and only if $n$ has at least two distinct divisors of the same parity that multiply to $n$.

Formulation 2

Any integer can be expressed as the difference of two squares if and only if that integer is NOT $n \equiv 2 \pmod 4$