Euclid's Theorem - Euclid's Proof

Euclid's Proof

Euclid offered the following proof published in his work Elements (Book IX, Proposition 20) and paraphrased here.

Take any finite list of prime numbers p₁, p₂, ..., p_n. It will be shown that at least one additional prime number not in this list exists. Let P be the product of all the prime numbers in the list: P = p₁p₂...p_n. Let q = P + 1. Then, q is either prime or not:

• If q is prime then there is at least one more prime than is listed.

• If q is not prime then some prime factor p divides q. If this factor p were on our list, then it would divide P (since P is the product of every number on the list); but as we know, p divides P + 1 = q. If p divides P and q then p would have to divide the difference of the two numbers, which is (P + 1) − P or just 1. But no prime number divides 1 so there would be a contradiction, and therefore p cannot be on the list. This means at least one more prime number exists beyond those in the list.

This proves that for every finite list of prime numbers, there is a prime number not on the list. Therefore there must be infinitely many prime numbers.

It is often erroneously reported that Euclid proved this result by contradiction, beginning with the assumption that the set initially considered contains all prime numbers, or that it contains precisely the n smallest primes, rather than any arbitrary finite set of primes. Although the proof as a whole is not by contradiction, in that it does not begin by assuming that only finitely many primes exist, there is a proof by contradiction within it: that is the proof that none of the initially considered primes can divide the number called q above.