I just returned from a 10-day trip to India. It was my first visit there. I gave a talk at the ICM Satellite Conference: Various Aspects of Dynamical Systems. The conference was hosted by the Department of Mathematics at the M. S. University, Baroda, which is in Vadodara (formerly Baroda) in the Indian state of Gujarat. Unfortunately, I wasn’t able to go early to attend the ICM (International Congress of Mathematicians) in Hyderabad.
My talk was on some joint work with my friend and long-time collaborator, Jim Wiseman. There were a lot of good speakers at the conference (including Jim, who spoke about the research on which we’re currently working). The most well-known mathematician at the conference was Hillel Furstenberg. He treated us to an hour-long talk, a presentation of some open problems, and the valedictory address at the end of the conference. I also had the pleasure of eating one dinner with him and his wife.
So I thought I’d take a moment to share one of Furstenberg’s coolest (and shortest) proofs. He published a one-paragraph note in the May 1955 issue of the American Mathematical Monthly, entitled “On the Infinitude of Primes.” It is a topological proof that there are infinitely many primes. (Note: apologies to my readers who have not take a course in point-set topology.)
Theorem. There are infinitely many primes.
Suppose, for the sake of contradiction, that there are finitely many primes.
How do we know that is a basis? We must check two things. First, it must cover , which it does. Second, if and are in and , then there must be an element such that . Indeed, suppose and are the differences between successive terms in the sequences and , respectively. Let be the least common multiple of and . Then is the required basis element.
Observe that every arithmetic progression is a closed set: the complement of each arithmetic progression is an open set since it is the union of (finitely many) arithmetic progressions, each having the same difference. For example,
For each prime , let and let . We assumed that there are finitely many primes, so is the union of finitely many closed sets. Thus is closed and its complement is open.
However, , which is clearly not open (every open set is either empty or infinite). This is a contradiction. So there are infinitely many primes.
[Update: here’s a reworked version of Furstenberg’s proof that avoids all topological language.]