Let m, n and p be any integers such that m and n are consecutive and

Let \(\displaystyle m^2 - n^2 = p\). if m > n, then by definition m=n+1. Because m and n are consecutive integers neither are both even nor odd.

Case (i)

Let m be even and n be odd

because m is even \(\displaystyle m=2k_{1}; k_{1}\in \mathbb{Z}\) and \(\displaystyle n=2k_{2}+1; k_{2} \in \mathbb{Z}\)

\(\displaystyle m^2-n^2=p\)

\(\displaystyle \Rightarrow (2k_{1})^2 - (2k_{2} + 1)^2 = p\)

\(\displaystyle \Rightarrow 4k_{1}^{2} - 4k_{2}^2 - 4k_{2} - 1 = p\)

is this on the right track?