Kummer's Restatement of Euclid's Proof 
(From the Prime Pages' list of proofs)


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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 < ... < 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!
The Prime Pages
Another prime page by Chris K. Caldwell <caldwell@utm.edu>