Prove that there are infinitely many primes

Show that the set of prime numbers is infinite.

Euclid settled this in Book IX, in about five lines, and as close to perfectly as an argument gets. That has stopped no one. There are proofs by analysis, by topology, by counting, and by explicitly constructing coprime numbers.

The interesting question is therefore not which proof is shortest. Euclid tells you only that the primes do not run out; Euler's proof tells you something much sharper about how fast they thin; Furstenberg's tells you almost nothing extra but is unforgettable.

Post a proof, and say what it gives you beyond the bare statement.