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
p_{1} < p_{2} < ... <
p_{r}. Let
N = p_{1}^{.}p_{2}^{.}...^{.}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!