Kummer's Restatement of Euclid's Proof
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.
- There are infinitely many primes.
- Suppose that there exist only finitely many primes p1 < p2 < ... < pr. Let N = p1.p2.....pr. The integer N-1, being a product of primes, has a prime divisor pi in common with N; so, pi divides N - (N-1) =1, which is absurd! ∎