$x^2-y^2=2s$, s cannot be an odd integer How can we prove that if $x^2-y^2=2s$ holds, s cannot be an odd integer. What theorem in number theory should we use?

You don't need any special theorem, just reason through it:
If $x^2 - y^2$ is even, that means either both $x$ and $y$ are even or they're both odd.
If both $x$ and $y$ are even, let's say $x = 2m$ and $y = 2n$. Then $x^2 - y^2 = 4m^2 - 4n^2 = 4(m^2 - n^2)$ and $s = 2(m^2 - n^2)$, meaning that it's even.
If both $x$ and $y$ are odd, let's say $x = 2m + 1$ and $y = 2n + 1$. Then $x^2 - y^2 = (4m^2 + 4m + 1) - (4n^2 + 4n + 1) = 4(m^2 + m - n^2 - n)$ and $s = 2(m^2 + m - n^2 - n)$, meaning that it's even again.
