My instinct was that you can still call it proof by exhaustion if you divide the values into classes and do the proof for each class.

> Similar way you'd prove four color theorem.

And this seems to support that idea? It's the same thing as proof by cases.

So the proof that n^2 + n is even for all integers is exhaustive: you do one proof for even integers, and another one for odd ones. But we wouldn't generally use the term "exhaustion" there because the vibes are wrong.