Squares mod 100

Two neat little identities for squares mod 100.

And

These can be algebraically shown to be true

What about the generic case of a modulus $k$? What is the specific $m$ to be added to $n$ to make the squares congruent?

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The trivial case is for $m^2 \equiv 0 \mod k$ and $2mn \equiv 0 \mod k$. Let’s just ignore the possibility that both of those aren’t true and yet still $(m^2 + 2mn) \equiv 0 \mod k$

The smallest natural number $m$ such that $2mn \equiv 0 \mod k$ for any integer $n$ is clearly $k/2$ if $k$ is even, and $k$ if $k$ is odd.

The smallest natural number $m$ such that $m^2\equiv 0 \mod k$ is a bit trickier.

$m^2 = k^2$ Obviously works, but the case of $m^2 = (k/2)^2$ only works if $k$ is divisible by $4$

So for

If $k$ is divisible by $4$

Otherwise: