Perhaps everyone knows how to prove “there exist infinitely many primes” by one or more ways. Here I just want to share a proof using topology and prime factorization theorem. It is also (perhaps highly) possible that most of the readers know this proof except me, but I just think that it is interesting so I want to go through it here. Of course, the prerequisite is some point set topology.

**Theorem.** There exist infinitely many primes.

*Proof.* (by H. Furstenberg, 1955) Define a topology on by setting the basic open sets to be sets in the form

where is a nonzero integer and is an integer. This is a set of an arithmetic sequence. We call the collection . To show that this is a basis for some topology, we only need to show:

(1) ;

(2) The intersection of finitely many members of is in .

(1) is clear because . For (2), suppose is a basic set for each . The intersection is either empty, or contains some integer , and also

where is the lowest common multiple of . So (2) is justified and is a basis of some topology.

Notice that each is closed because its complement

is an open set.

By the prime factorization theorem, every integer other than 1 or -1 appears in some where is a positive prime. Also . Thus

,

which is a union of closed sets. If there are only finitely many primes, then the right hand side is a finite union of closed sets, which is closed. Then is closed so the set is open, which is impossible, because every open set is infinite, as each is so.

very funny proof!

So very elegant and abstract at the same time…