H. Fürstenberg published in 1955 this proof that there are infinitely many prime numbers. It uses (of all things) topology, and is probably one of the most novel proofs of this fact. Furstenberg's work often involves using objects of one mathematical field in a very different one; this proof is an almost elementary example of this tendency. And it's almost completely unlike any other proof that there is no largest prime number you may have seen.

Define a topology on the integers Z by defining the open neighborhoods to be all arithmetic sequences Sa,b={an+b: n∈Z}. So a set B⊆Z is open iff it is empty or every x∈B has some a for which Sa,x⊆B.

This does actually define a topology on Z:

  1. ∅ and Z are open sets.
  2. Any union of open sets is trivially open.
  3. The intersection of 2 open sets B1,B2 is open. Indeed, if x∈B1∩B2, then there exist a1,a2 such that Sai,x⊆Bi (i=1,2). Let a=lcm(a1,a2) be the least common multiple of a1,a2 (or take a=a1a2 if you prefer to be "wasteful" but get an even simpler proof). Then a gives a subsequence of our arithmetical progressions: Sa,x⊆Sai,x. So x∈Sa,x⊆B1∩B2, and any finite intersection of open sets is open.

We only need 2 notes on this topology:

  1. Note that any non-empty open set contains an arithmetical progressions, hence it must be infinite.
  2. And note that any progressions Sa,b is not only an open set but also a closed set, because Sa,b=Z\(Sa,b+1∪Sa,b+2∪...∪Sa,b+a-1) gives it as the complement of an open set.
Now any number apart from -1 and +1 appears in some arithmetical progression Sp,0 for some prime p. So
Z\{-1,+1} = p primeSp,0.
By note "A", {-1,+1} isn't open, so the LHS isn't a closed set. But if there were only finitely many primes, then the RHS would be a finite union of closed (by note "B") sets, hence closed. Thus, there must be infinitely many primes.

An interesting exercise is to try to determine just how primes and divisibility are so important here. My take is that we only use multiplication when proving that the arithmetical sequences actually do give a topology...

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