Many proofs that the primes are infinite J. Marshall Ash1 and T. Kyle Petersen Summary: Many proofs that the primes are infinite J. Marshall Ash1 and T. Kyle Petersen Theorem 1. There are infinitely many prime numbers. How many proofs do you know that there are infinitely many primes? Nearly every student of mathematics encounters Euclid's classic proof at some point, and many working mathematicians could provide one or two more if asked. If you had to guess, how many different proofs of Theorem 1 do you think there are? A dozen? A hundred? Certainly many have taken joy in coming up with, and sharing, novel proofs of the theorem. The techniques used have drawn from virtually all parts of mathematics. There have been proofs using the tools of Algebra, Number Theory, Analysis, and even Topology!2 Our goal here is not to catalogue or classify the proofs that have appeared in the literature. Rather, we propose the following as exercise to enhance a number theory class, a history of math class, a senior capstone class, a math club meeting, et cetera: Exercise. Pick a known proof of the infinitude of the primes and expand it into an infinite family of proofs. We shall give several examples below. Our first one converts a well known modernization of Euclid's 2300 year old proof of Theorem 1 into an infinite number Collections: Mathematics