Here’s a proof of the infinitude of primes that occurred to me when for some reason Delta upgraded me to First Class on a flight April 2. Can anyone provide a reference?
Suppose that the set P of primes and 1 has just n+1 elements. Now every number at most 2k can be obtained by choosing k elements of P with replacement, which can be done in (k+n choose n) ways. Therefore
2k ≤ (k+n choose n) ≤ (k+n)n ,
which fails for k large.