Filip Saidak's Proof

By Chris Caldwell

Euclid may have been the first to give a proofthat there are infintely many primes.  Below we give another proof by Filip Saidak [Saidak2005], similar to Goldbach's argument, but in a way even simpler.

Theorem.

There are infinitely many primes.

Proof.

Let n > 1 be a positive integer.  Since n and n+1 are consecutive integers, they must be coprime, and 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 3 different prime factors.  This can be continued indefinitely.∎