Furstenberg's proof of the infinitude of primes

Fürstenberg's proof of the infinitude of primes is a 1955 published exceptional proof of Euclid already proven, well-known fact that there are infinitely many prime numbers. He was discovered by Hillel Furstenberg, when he was still studying as an undergraduate student at Yeshiva University. The evidence presented for the mathematical community is a surprise, because it uses topological methods to prove a known number-theoretic statement. The proof was published in 1955 in the American Mathematical Monthly and included as a beautiful and unusual evidence in the collection Proofs from THE BOOK by Martin Aigner and Günter M. Ziegler.

The proof

On closer inspection, it is in the proof to the consideration of certain properties of arithmetic sequences. As the classical proof of the infinitude of primes of Euclid is Fürstenbergs proof a proof by contradiction.

A topological space is specified by specifying a basis of open sets. When the open sets of the topology of the arithmetic progression are defined those subsets of the integers that two-sided as the union of arithmetic progressions

Can be written, and wherein said integers are. That there is indeed a topology in which Defined, must be checked at the axioms of topology.

This topology has the following properties:

The numbers 1 and -1 are the only integers which are not multiples of prime numbers, ie

Because of the first characteristic amount can not be completed on the left side of the equation. Because of the second property, the amounts have been completed. If there were only finitely many primes, then would the ( then finite) union of closed sets on the right side is a closed set. This results in a contradiction, and we have: There are infinitely many prime numbers.

Swell

• Martin Aigner, Günter M. Ziegler: The Book of Evidence. Springer, Berlin et al 2002, ISBN 3-540-42535-7.
• Harry Furstenberg: On the infinitude of primes. In: American Mathematical Monthly. 62, 1955, pp. 353, doi: 10.2307/2307043.