Euclid may have been the first to give
a proof that there are infintely many primes. Since then there have been
many other proofs given. Perhaps the strangest is the following topological
proof by Fürstenberg [Fürstenberg55]. See the page "There are Infinitely Many Primes"
for several other proofs.
Theorem.
There are infinitely many primes.
Proof.
Define a topology on the set of integers by using the arithmetic progressions
(from -infinity to +infinity) as a basis. It is easy to verify that this yields
a topological space. For each prime p let Ap
consists of all multiples of p. Ap is closed
since its complement is the union of all the other arithmetic progressions with
difference p. Now let A be the union of the progressions
Ap. If the number of primes is finite, then A is
a finite union of closed sets, hence closed. But all integers except -1 and 1
are multiples of some prime, so the complement of A is {-1, 1} which is
obviously not open. This shows A is not a finite union and there are
infinitely many primes.
