Here is a really neat proof of the infinitude of primes! It's so simple that it makes me say: "Why didn't I think of it?". In fact, it was made just in 2005, by Filip Saidak, and I was really surprised to know that. So here is the proof:
Let \(n > 1\) be a positive integer. Since \(n\) and \(n+1\) are consecutive integers, they must be coprime, and hence the number \(N_2 = n(n + 1)\) 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 \(N_3 = n(n + 1)[n(n + 1) + 1]\) must have at least 3 different prime factors. This can be continued indefinitely.