Let $m$ be a positive integer of the form $m = 2s$, where $s$ is an odd integer. Prove that there do not exist positive integers $x$ and $y$ such that $x^2 - y^2 = m.$
Proclaimed Solution via Proof by Contradiction: Assume, to the contrary, that there exist positive integers $x$ and $y$ such that $(x - y)(x + y)= m = 2s \quad \text{ where $s$ is an odd integer.} \tag{*}$ We consider two cases, according to whether $x$ and $y$ are of the same parity or of opposite parity. $\boxed{\text{Case 1:}}$ If $x$ and $y$ are of the same parity, then both $x - y$ and $x + y$ are even.
$\boxed{\text{Case 2:}}$ If $x$ and $y$ are of opposite parity, then both $x - y$ and $x + y$ are odd. Produce a contradiction in each case.
$\Large{1.}$ The product of two even numbers is always even. Thus, $(x - y)(x + y)$ is even. Then $(*)$ becomes: $\text{even = 2(odd)}$. Where is the contradiction?
The analogous solution works for Case 2? The product of two odds is always odd. Then $(*)$ becomes: $\text{odd = ($2s$ = even)}$. Contradiction.
$\Large{2.}$ What is the motivation behind considering "whether $x$ and $y$ are of the same parity or of opposite parity"?
Source: Exercise 5.22, P124 of Mathematical Proofs, 2nd ed. by Chartrand et al