Fürstenberg's proof of the infinity of prime numbers

Fürstenberg's proof of the infinity of prime numbers is an extraordinary proof, published in 1955 , of the well-known fact, already proven by Euclid , that there are infinitely many prime numbers . He was discovered by Hillel Fürstenberg when he was still an undergraduate student at Yeshiva University . The proof came as a surprise to the mathematical community because it uses topological methods to prove a well-known number theory proposition. The proof was published in the American Mathematical Monthly in 1955 and included as a beautiful and extraordinary proof in the collection Das BUCH der Proofs by Martin Aigner and Günter M. Ziegler .

The proof

On closer inspection, the proof is the consideration of certain properties of arithmetic sequences . Like Euclid's classical proof of the infinity of prime numbers , Fürstenberg's proof is a contradiction proof .

A topological space is specified by specifying a base of the open sets . As the open sets of the topology of the arithmetic progression , those subsets of the integers are defined, which are the union of two-sided arithmetic sequences ${\displaystyle \mathbb {Z} }$

${\displaystyle S(a,b)=\{an+b\mid n\in \mathbb {Z} \}=a\mathbb {Z} +b}$

can be written, where and are natural numbers. The axioms of the topology must be used to check that what is defined is actually a topology . ${\displaystyle a}$ ${\displaystyle b}$

This topology has the following properties:

1. Because every nonempty open set contains an arithmetic sequence, no nonempty finite set can be open; the complement of a nonempty finite set cannot be a closed set.
2. The simple sets are both open and closed sets, because one can write as a complement of open sets: ${\displaystyle S(a,b)}$ ${\displaystyle S(a,b)}$

${\displaystyle S(a,b)=\mathbb {Z} \setminus \bigcup _{j=1}^{a-1}S(a,b+j).}$

The numbers 1 and −1 are the only whole numbers that are not multiples of prime numbers, so

${\displaystyle \mathbb {Z} \setminus \{-1,+1\}=\bigcup _{p{\text{ ist Primzahl}}}S(p,0).}$

Because of the first property, the set on the left side of the equation cannot be closed. Because of the second property, the sets are closed. If there were only finitely many prime numbers, then the (then finite) union of the closed sets on the right-hand side would be a closed set. This leads to a contradiction, and the following applies: There are infinitely many prime numbers. ${\displaystyle S(a,0)}$

literature

• Martin Aigner , Günter M. Ziegler : The book of evidence. Springer, Berlin et al. 2002, ISBN 3-540-42535-7 .
• Harry Fürstenberg : On the infinitude of primes . In: American Mathematical Monthly . Vol. 62, No. 5, 1955, p. 353. doi : 10.2307 / 2307043 .

Web links

• Furstenberg's proof that there are infinitely many prime numbers at Everything2 (English)
• Cam McLeman: Fürstenberg's proof of the infinitude of primes . In: PlanetMath . (English)