Seeking proofs that depend on the notion of even-vs-odd parity to prove their points

Question

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.

Answer 1

In even-dimensional Euclidean spaces, spherical waves have trailing edges; in odd-dimensional spaces they don't.

Answer 2

The case of even perfect numbers is settled since long ($P=M(M+1)/2$ for $M$ a Mersenne prime). That of odd perfect numbers is still open.

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The diophantine equation

$$x^p+y^p=z^p$$ where $p$ is a prime only has solutions for even $p$.

Answer 3

The parity of the order of a finite group plays an enormous role in finite group theory!

The Feit-Thompson Theorem settled Burnside's conjecture that every non-Abelian simple group has even order.

There were numerous stepping-stones along the way, and a big part of the classification of finite simple groups was devoted to studying how involutions (non-identity elements of order two, guaranteed to exist in groups of even order, by Cauchy's theorem) affect the structure of a group; in particular looking at what's possible for the centralizer of an involution (apparently this is a consequence of the Brauer-Fowler Theorem, although I don't see the connection).

One such stepping stone was Suzuki's CA paper, which studied groups (now called CA groups) in which the centralizer of any non-identity element is Abelian. In this paper, Suzuki showed that any CA group with odd order is solvable, and hence not simple.