Fürstenberg’s proof of the infinitude of primes
Consider the arithmetic progression topology on the positive integers, where a basis of open sets is given by subsets of the form . Arithmetic progressions themselves are by definition open, and in fact clopen, since
Consider the set , where the union runs over all primes . Then the complement of in is the single element , which is clearly not an open set (every open set is infinite in this topology). Thus is not closed, but since we have written 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 Number Records. Springer, 1996. p. 10
|Title||Fürstenberg’s proof of the infinitude of primes|
|Date of creation||2013-03-22 14:42:10|
|Last modified on||2013-03-22 14:42:10|
|Last modified by||mathcam (2727)|