From Dmitri Fon-Der-Flaass:

There are infinitely many prime numbers.

Proof: Assume that there are only prime numbers. Then the number of all ways to multiply of them (perhaps with repetitions) is a polynomial in of degree and is therefore . On the other hand, all natural numbers up to can be written as a product of at most primes. Therefore — a contradiction.

Reminds me of the proof using the Euler product for the zeta function. It has a pole at 0, which can not be true if there are only finite number of primes.

By:

mishaon April 15, 2010at 4:15 am

This proof occurred to me and I posted it on April 3 at http://blogs.williams.edu/Morgan/2010/04/03/infinitely-many-primes-by-combinatorics/ Earlier references?

By:

Frank Morganon April 19, 2010at 12:13 pm