The notion of the parity is very important in a variety of branches of mathematics. Specifically, I am looking for proofs that use parity in the even-vs-odd sense to prove their points.
For example, it is instructive to show that $\sqrt{2}$ is irrational by contradiction and assuming there exist relatively prime integers $m,n$ where $n \neq 0$ such that $\frac{m^2}{n^2} = 2$. By assuming that $m^2$ is even, you can arrive at a contradiction. A similar argument applies when assuming that $m^2$ is odd$.
Another example is showing that the alternating group of even permutations ($A_n$ within $S_n$) is a subgroup under composition of permutations. That claim can be extended to show that the odd permutations of $S_n$ do not form a subgroup.
I'm not looking for explanations of even/odd functions or how to partition $n$ into distinct odd/even parts, but rather my question is: What proofs exists that depend on the notion of even-vs-odd parity to prove their points? Ideally an answer will supply a description of the theorem/proof or a reference or a full proof itself.
