 
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
