On Thu, Feb 26, 2015 at 2:45 PM, Bill Gosper <billgosper@gmail.com> wrote:
Hi William, you could simplify the p16 proof of the infinitude of the primes. Put any positive integer (e.g.1) in a bag. Repeat forever: Include in the bag the number formed by adding 1 to the product of all the numbers already in the bag.
Each new inclusion is relatively prime to all the rest, so no prime appears more than once among the factorizations. This requires an infinitude of primes.
Tweak: "... no prime appears in more than one of the factorizations." Primes might repeat within one of the factorizations. E.g. if we started with 4, vs 1. Man, I wonder what the first squareful Sylvester number is. --rwg
If we start with 1, it's sort of neat that the successive inclusions are 2, 3, 7, 43, ..., A000058, Sylvester's sequence, whose reciprocals sum to 1 by being the denominators of the greedy Egyptian expansion of 1-1/∞. --Bill Gosper