Prove an odd natural integer is composite if and only if it can be written in the form $n=x^2-y^2, y+1 < x$
I do not see why $y+1 < x$ and I do not really know where to start.
Consider $n= x^2-y^2$
We can factor this as:
$n=(x-y)(x+y)$ and thus we have a factorization for the composite integer $n$.
However, we must ensure that n is composite and not prime. Therefore for $n$ to be composite $(x-y)$ or $(x+y)$ must not equal $1$. Clearly, $(x-y)$ is the only factor that could equal $1$. So we set up the inequality:
$x-y > 1 \iff x > y+1$ Hence, we have shown this direction. (How do I show $n$ is odd)