Sum of Squares of Two Odd Integers is not Square

Theorem
Let $m$ and $n$ be odd integers.

Then $m^2 + n^2$ is not a square number.

Proof
$m^2 + n^2$ is a square number.

Because $m$ and $n$ are both odd, we have:

From Parity of Integer equals Parity of its Square, $m^2$ and $n^2$ are both odd integers.

Hence $m^2 + n^2$ is an even integer.

But from Square Modulo 4, $m^2 + n^2$ is even $x^2 \equiv 0 \pmod 4$.

This contradicts the deduction that $m^2 + n^2 \equiv 2 \pmod 4$.

Hence by Proof by Contradiction it follows that $m^2 + n^2$ cannot be square.