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 ${U}_{a,b}=\{n\in {\mathbb{Z}}^{+}|n\equiv bmoda\}$. Arithmetic progressions themselves are by definition open, and in fact clopen, since
${U}_{a,b}^{c}={\displaystyle \bigcup _{c\ne b}}{U}_{a,c}$ |
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 |
---|---|
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 |