The fundamental theorem of arithmetic states that every integer can be uniquely written as a product of primes (i.e. is a unique factorisation domain).

The usual proof proceeds through the Euclidean algorithm. Yesterday at lunch I was surprised to learn (thanks to Ole Warnaar) of a different proof bypassing the Euclidean algorithm which I reproduce below. Its primary attraction is its cuteness, as it provides a weaker result than the usual proof (i.e. doesn’t prove that is a Euclidean domain, or even a principal ideal domain).

Suppose that where are distinct primes. Consider the finite cyclic group . It has a composition series where the group appears times as a simple subquotient (and no other simple factors appear). Therefore by the Jordan-Holder theorem, the primes together with their multiplicities are unique. QED.