I'm working on a homework problem that is as follows:
Suppose that $n$ is a positive even integer with $n/2$ odd. Prove that there do not exist positive integers $x$ and $y$ with $x^2 - y^2 = n$.
It looked like a good candidate for proof by contradiction. So I know that I still assume the argument "$n$ is a positive even integer with $n/2$ odd" but will try to show that there exists positive integers $x$ and $y$ with $x^2 - y^2 = n$, and if this reaches a contradiction then I have proven the original conjecture.
So I started with the $n/2$ is odd and rewrote it as $n/2 = 2k+1$, for some $k$ in the integers.
Then I knew that $n = 4k+2$, and then tried so equate that with $x^2 - y^2 = 4k+2$.
EDIT: I then recognized that $x^2 - y^2$ is equivalent to $(x + y)(x - y)$ but that doesn't seem to be very helpful, because if you divide one or the other out you get a term on the RHS in terms of k and x and y.
I'm going to keep playing with it but I don't really have any good strategies going forward. Any help is appreciated!