# 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 progression topology on the positive integers, where a basis of open sets is given by subsets of the form $U_{a,b}=\{n\in\mathbb{Z}^{+}|n\equiv b\mod a\}$. Arithmetic progressions themselves are by definition open, and in fact clopen, since

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

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

• 1 Furstenberg, Harry, On the infinitude of primes, American Mathematical Monthly, Vol. 62, 1955, p. 353.
• 2 Ribenboim, Paulo. The New Book of Prime Number Records. Springer, 1996. p. 10
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