Fürstenberg’s proof of the infinitude of primes

Fürstenberg’s proof ([1], [2]) that there are infinitely many primes is an amusing and beautiful blend of elementary number theory and point-set topology.

Consider the arithmetic progressionPlanetmathPlanetmath topologyPlanetmathPlanetmath on the positive integers, where a basis of open sets is given by subsets of the form Ua,b={n+|nbmoda}. Arithmetic progressions themselves are by definition open, and in fact clopen, since


where the union is taken over a set of distinct residue classesMathworldPlanetmath modulo a. Hence the complement of Ua,b is a union of open sets and so is open, so Ua,b itself is closed (and hence clopen).

Consider the set U=pUp,0, where the union runs over all primes p. Then the complement of U in + is the single elementMathworldMathworld {1}, which is clearly not an open set (every open set is infiniteMathworldPlanetmath in this topology). Thus U is not closed, but since we have written U as a union of closed sets and a union of closed sets is again closed, this implies that there must be infinitely many terms appearing in that union, i.e. that there must be infinitely many distinct primes.


  • 1 Furstenberg, Harry, On the infinitude of primes, American Mathematical Monthly, Vol. 62, 1955, p. 353.
  • 2 Ribenboim, Paulo. The New Book of Prime NumberMathworldPlanetmath Records. Springer, 1996. p. 10
Title Fürstenberg’s proof of the infinitude of primes
Canonical name FurstenbergsProofOfTheInfinitudeOfPrimes
Date of creation 2013-03-22 14:42:10
Last modified on 2013-03-22 14:42:10
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 8
Author mathcam (2727)
Entry type Proof
Classification msc 11A41
Related topic HausdorffSpaceNotCompletelyHausdorff