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 < ... <
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!