Kummer's Restatement of Euclid's Proof

By Chris Caldwell

Euclid may have been the first to give a proof that there are infintely many primes.  Even after 2000 years it stands as an excellent model of reasoning.  Kummer gave a more elegant version of this proof which we give below (following Ribenboim [Ribenboim95, p. 4]).  See the page "There are Infinitely Many Primes" for several other proofs.

Theorem.

There are infinitely many primes.

Proof.

Suppose that there exist only finitely many primes p₁ < p₂ < ... < p_r. Let N = p₁^.p₂^....^.p_r.   The integer N-1, being a product of primes, has a prime divisor p_i in common with N; so, p_i divides N - (N-1) =1, which is absurd! ∎