# Furstenberg’s proof of the infinitude of the primes

Another very classical proof that’s just too beautiful to ignore. This time, the theorem is millennia old, but it’s really the proof that I’m interested in.

Lemma. There are infinitely many primes.

Proof. Define the sets for with . Note that if ; say it contains some , then Hence, the intersection of two sets of this form is again of this form (or empty). Thus, they form the basis for a topology on . Notice that the sets are also open.

Now suppose that there were finitely many primes. Then the intersection is a finite intersection of opens, hence open. On the other hand, it equals the set of integers divisible by no prime number, which is . But all basic open sets are infinite, so this set can never be open. Remark. This proof is in essence the usual proof, where we consider and conclude that some prime has to divide it and this prime can be none of the . Indeed, when we showed that the intersection of basic opens is open, we went to a set whose period is (which for gives period ). Since 1 is in , so should be.

That said, it doesn’t seem possible to adopt this proof to reprove other theorems like ‘there are infinitely many primes congruent to 3 modulo 4’. The problem is that the above proof seems to rely on a global analysis of the set : it has to be an infinite set. So the method is too crude to prove theorems with congruence restrictions.

# Cauchy’s theorem

This is a really classical result, but this proof is just so magical that I have to include it.

Lemma. Let be a finite group, and let be a prime dividing . Then has an element of order .

Proof. Consider the action of on given by Then , since the orbits of size all have size (and the fixed points are exactly the union of the orbits of size 1). The right hand side is as . Thus, But there is a bijection The former trivially contains the element 1, hence (since its size is divisible by ) it has to contain some other element . This element will then have (exact) order . Remark. Of course, the result also follows from the much stronger Sylow theorems (of which we only need the existence statement).