Summary: Many proofs that the primes are in...nite
J. Marshall Ashy and T. Kyle Petersen
Theorem 1. There are in...nitely many prime numbers.
How many proofs do you know that there are in...nitely 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 di¤erent 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!1
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 1. Pick a known proof of the in...nitude of the primes and expand it into an in...nite
family of proofs.
We shall give several examples below. Our ...rst one converts a well known modernization of
Euclid's 2300 year old proof of Theorem 1 into an in...nite number of similar, but distinct, proofs.
Example 1. Assume that the number of primes is ...nite, and label them p1; : : : ; pn. Let k be any
positive integer. Here is the kth proof: Form
N = N(k) = k p1 pn + 1: