# Fürstenberg’s proof of the infinitude of primes

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

 $\displaystyle U_{a,b}^{c}=\bigcup_{c\neq b}U_{a,c}$

where the union is taken over a set of distinct residue classes  modulo $a$. Hence the complement of $U_{a,b}$ is a union of open sets and so is open, so $U_{a,b}$ itself is closed (and hence clopen).

Consider the set $U=\cup_{p}U_{p,0}$, where the union runs over all primes $p$. Then the complement of $U$ in $\mathbb{Z}^{+}$ is the single element  $\{1\}$, which is clearly not an open set (every open set is infinite  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.

## References

Title Fürstenberg’s proof of the infinitude of primes FurstenbergsProofOfTheInfinitudeOfPrimes 2013-03-22 14:42:10 2013-03-22 14:42:10 mathcam (2727) mathcam (2727) 8 mathcam (2727) Proof msc 11A41 HausdorffSpaceNotCompletelyHausdorff