Definitions

infinitude

Furstenberg's proof of the infinitude of primes

In number theory, Hillel Furstenberg's proof of the infinitude of primes is a celebrated topological proof that the integers contain infinitely many prime numbers. When examined closely, the proof is less a statement about topology than a statement about certain properties of arithmetic sequences. Like Euclid's classical proof, Furstenberg's is a proof by contradiction. The proof was published in 1955 in the American Mathematical Monthly while Furstenberg was still an undergraduate student at Yeshiva University.

Furstenberg's proof

Define a topology on the integers Z by declaring a subset U ⊆ Z to be an open set if and only if it is either the empty set, ∅, or it is a union of doubly infinite arithmetic sequences S(ab), where

S(a, b) = { a n + b | n in mathbf{Z} } = a mathbf{Z} + b.

In other words, U is open if and only if every x ∈ U admits some non-zero integer a such that S(ax) ⊆ U. The axioms for a topology are easily verified:

  • By definition, ∅ is open; Z is just the sequence S(1, 0), and so is open as well.
  • Any union of open sets is open: for any collection of open sets Ui and x in their union, any of the numbers ai for which S(aix) ⊆ Ui also shows that S(aix) ⊆ U.
  • The intersection of two (and hence finitely many) open sets is open: let U1 and U2 be open sets and let x ∈ U1 ∩ U2 (with numbers a1 and a2 establishing membership). Set a to be the lowest common multiple of a1 and a2. Then S(ax) ⊆ S(aix) ⊆ U1 ∩ U2.

The topology is quite different from the usual Euclidean one, and has two notable properties:

  1. Since any non-empty open set contains a doubly infinite sequence, no finite set can be open; put another way, the complement of a finite set cannot be a closed set.
  2. The "simple sets" S(ab) are both open and closed: we can write S(ab) as the complement of an open set as follows:

S(a, b) = mathbf{Z} setminus bigcup_{j = 1}^{a - 1} S(a, b + j).

The only integers that are not integer multiples of prime numbers are −1 and +1, i.e.

mathbf{Z} setminus { -1, + 1 } = bigcup_{p mathrm{, prime}} S(p, 0).

By the first property, the set on the left-hand side cannot be closed. On the other hand, by the second property, the sets S(p, 0) are closed. So, if there were only finitely many prime numbers, then the set on the right-hand side would be a finite union of closed sets, and hence closed. This would be a contradiction, so there must be infinitely many prime numbers.

References

  • Furstenberg, Harry (1955). "On the infinitude of primes". Amer. Math. Monthly 62 353.

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