Topological proof of the infinitude of primes

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 \mathbb{Z} by setting the basic open sets to be sets in the form

S_{a,b}=\{am+b:m\in \mathbb{Z}\}

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

(1) \displaystyle \mathbb{Z}=\bigcup_{a,b} S_{a,b};

(2) The intersection of finitely many members of \mathcal{C} is in \mathcal{C}.

(1) is clear because \mathbb{Z}=S_{1,0}. For (2), suppose S_{a_i,b_i} is a basic set for each i=1,\cdots,n. The intersection is either empty, or contains some integer c, and also

\displaystyle \bigcap_{i=1}^n S_{a_i,b_i}=S_{k,c}

where k is the lowest common multiple of a_1,\cdots,a_n.  So (2) is justified and \mathcal{C} is a basis of some topology.

Notice that each S_{a,b} is closed because its complement

\mathbb{Z}\setminus S_{a,b}=S_{a,b+1}\cup S_{a,b+2}\cup \cdots \cup S_{a,b+a-1}

is an open set.

By the prime factorization theorem, every integer other than 1 or -1 appears in some S_{p,0} where p is a positive prime. Also 0\in S_{2,0}. Thus

\displaystyle \mathbb{Z}\setminus\{1,-1\}=\bigcup_{p \text{ prime }} S_{p,0},

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 \mathbb{Z}\setminus\{1,-1\} is closed so the set \{1,-1\} is open, which is impossible, because every open set is infinite, as each S_{a,b} is so. \square

This entry was posted in Number Theory, Topology. Bookmark the permalink.

2 Responses to Topological proof of the infinitude of primes

  1. lamwk says:

    very funny proof!

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