Theorem: An infinity of primes. Proof: Assume the contrary, that there are only P primes. Consider the set of integers up to N that are representible as products of the P primes (allowing repeated prime factors). The size of this set is O((logN)^P), which grows more slowly than N. QED. Restatement: If there were only a finite number of primes, the set of fully-factorable D-digit numbers would only be polynomial in D. I'm amused that the argument doesn't depend on properties of divisibility, but rather on the assoc-comm property of multiplication, and that the product XY is much bigger than X and Y. [extra credit: find more words with 5 I's. serious XC: turn this into a lower bound on pi(N).] This argument would have been perfectly comprehensible to Fermat, though he might have had trouble formalizing it. But the idea of arguing based on rate-of-growth, comparing polynomial(D) to exponential(D), seems modern. So I wonder when this proof was discovered. Rich