Integer as Difference between Two Squares/Formulation 2

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

Proof

 * Each integer will be in one of the 4 sets of residue classes modulo $4$:
 * $n \equiv 0 \pmod 4$
 * $n \equiv 1 \pmod 4$
 * $n \equiv 2 \pmod 4$
 * $n \equiv 3 \pmod 4$


 * For $n \equiv 2 \pmod 4$, it is impossible to represent such an integer as the difference of two squares.

Taking the squares of each of the residue classes, we have:
 * $0^2 \equiv 0 \pmod 4$
 * $1^2 \equiv 1 \pmod 4$
 * $2^2 \equiv 0 \pmod 4$
 * $3^2 \equiv 1 \pmod 4$


 * Therefore, when taking the difference of two squares, $n \equiv 2 \pmod 4$ is never a result.
 * $0 - 0 \equiv 0 \pmod 4$
 * $1 - 1 \equiv 0 \pmod 4$
 * $0 - 1 \equiv 3 \pmod 4$
 * $1 - 0 \equiv 1 \pmod 4$