Furstenberg's proof of the infinitude of primes


In mathematics, particularly in number theory, Hillel Furstenberg's proof of the infinitude of primes is a topological proof that the integers contain infinitely many prime numbers. When examined closely, the proof is less a statement about topology than a statement about certain properties of arithmetic sequences. Unlike Euclid's classical proof, Furstenberg's proof is a proof by contradiction. The proof was published in 1955 in the American Mathematical Monthly while Furstenberg was still an undergraduate student at Yeshiva University.

Furstenberg's proof

Define a topology on the integers Z, called the evenly spaced integer topology, by declaring a subset U ⊆ Z to be an open set if and only if it is either the empty set, ∅, or it is a union of arithmetic sequences S, where
In other words, U is open if and only if every x ∈ U admits some non-zero integer a such that S ⊆ U. The axioms for a topology are easily verified:
This topology has two notable properties:
  1. Since any non-empty open set contains an infinite sequence, a finite set cannot be open; put another way, the complement of a finite set cannot be a closed set.
  2. The basis sets S are both open and closed: they are open by definition, and we can write S as the complement of an open set as follows:
The only integers that are not integer multiples of prime numbers are −1 and +1, i.e.
By the first property, the set on the left-hand side cannot be closed. On the other hand, by the second property, the sets S are closed. So, if there were only finitely many prime numbers, then the set on the right-hand side would be a finite union of closed sets, and hence closed. This would be a contradiction, so there must be infinitely many prime numbers.