New Proofs of Euclid's and Euler's Theorems

In this note we give a new proof of the existence of infinitely many prime numbers. There are several different proofs with many variants, and some of them can be found in [1, 3, 4, 5, 6]. This proof is based on a simple counting argument using the inclusionexclusion principle combined with an explicit formula. A different proof based on counting arguments is due to Thue (1897) and can be found in [6] together with several generalizations, and a remarkable variant of it was given by Chaitin [2] using algorithmic information theory. Moreover, we prove that the series of reciprocals of the primes diverges. Our proofs arise from a connection between the inclusion-exclusion principle and the infinite product of Euler. Let {pi }i be the sequence of prime numbers, and let us define the following recurrence: a0 = 0, ak+1 = ak + 1 − ak pk+1 .