A New Proof of Euclid's Theorem

New proof. Let n be an arbitrary positive integer greater than 1. Since n and n + 1 are consecutive integers, they must be coprime. Hence the number N2 = n(n + 1) must have at least two different prime factors. Similarly, since the integers n(n+1) and n(n+1)+1 are consecutive, and therefore coprime, the number N3 = n(n + 1)[n(n + 1) + 1] must have at least three different prime factors. This process can be continued indefinitely, so the number of primes must be infinite.