Prove that the sum of the squares of two odd integers cannot be a perfect square.
The author wrote the follow proof:
Assume, to the contrary, that there exist odd integers and such that , where . Then and , where . Thus
where is odd integer. If is even, then for some integer and so , where is an even integers; while is odd, than is odd. Produce a contradiction in each case.
Remark: I just don't see how z can be anything else be even. To me it's obvious explicitly implied that is even, which by some theorem, is also even.
Question: Which part of the proof gave the author the reason to believe that it's odd? Further how in the case of it being even be a contradiction to being a perfect square?
I just can't see it. Could someone please show me the author's intent?