The Infinitude of the Primes and Some Coloring Theorems

We first prove the infinitude of the primes via a special case of Rado's theorem whose proof is based on the infinite Ramsey theorem. In the proof, we use the colorings of the positive integers introduced by Levent Alpoge [1] and Andrew Granville [2]. Finally, using Rado's theorem for integral domains, we will give another proof for the infinitude of nonassociated prime elements in any unique factorization domain R with a few units.