Well, of course if

is odd, then

is odd, and if

is even, then

is even. But without wanting to sound pedantic, I'd say that's a contrapositive proof in disguise. I'd like to

*start* reasoning about

and

*end* at

.

I just discussed it with a colleague, and we came up with this.

By the fundamental theorem of arithmetic,

has a unique prime factorization, so

, for certain primes

, and positive whole numbers

. (Assume these primes are listed in ascending order.) Because

is even,

and because it is a square, we have that for all

,

. (This shows why, if

is even,

.) So

, which means

is even. QED.