Fürstenberg's Proof of the Infinitude of Primes  (From the Prime Pages' list of proofs) Home Search Site Largest The 5000 Top 20 Finding How Many? Mersenne Glossary Prime Curios! Prime Lists FAQ e-mail list Titans Submit primes 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. Another prime page by Chris K. Caldwell