Goldilocks and The Three Proofs of The Infinitude of The Primes
A fellow PhD student recently asked me "Have you seen the topological proof that there are infinitely many primes?" and I'm possibly in sparse company being excited by this sentence. It reminded me that I'd seen another insanely overpowered proof that there are infinitely many primes a while ago and so I'd like to present both of these along with Euclid's megaclassic proof. Paul Erdos, the most published mathematician of all time, though agnostic, referred to god as the Supreme Fascist and believed that he held a book of the perfect proof of every theorem in maths. He would exclaim "This one's from The Book" when he saw a particularly elegant proof and the first of our proofs is undoubtedly one of these. Suppose, for contradiction, that there are a finite number of primes. I'm going to consider the resulting number when I multiply all these finitely many primes together and add 1. This new number is not divisible by any of the primes. Indeed...