Nonstandard proofs of the fundamental theorem of arithmetic

Thirty or so years ago, someone showed me a clever proof of the Fundamental Theorem of Arithmetic that did not make use of the lemma "If $p\mid ab$ then $p\mid a$ or $p\mid b$". I'm unable to reconstruct the argument; all I remember is that it used induction and that it didn't generalize to other number rings. Can anyone provide such a proof, or provide other offbeat elementary proofs of unique factorization of natural numbers into primes?