

Euclid may have been the first to give
a proof that there are infintely many primes, but his proof has been followed
by many others. Below we give Goldbach's clever proof using the Fermat numbers
(written in a letter to Euler, July 1730), plus a few variations. See the
page "There are Infintely Many Primes" for several more
proofs.
First we need a lemma.
Note that any sequence that is pairwise relatively prime will work in this proof. This type of sequence is easy to construct. For example, choose relatively prime integers a and b, then define a_{n} as follows. a_{1}=a,This includes both the Fermat numbers (with a=1,b=2) and Sylvester's sequence (with a=1,b=2): a_{1}=2 and a_{n+1}= a_{n}^{2}a_{n}+1.In fact, the proof really only needs a sequence which has a subsequence which is pairwise relatively prime, e.g., the Mersenne numbers. 
Another prime page by Chris K. Caldwell <caldwell@utm.edu> 