(**19/10/22 **In a comment below, Bernard has noted the proof should be properly attributed to Paul Erdős, not Thue. Thue’s counting argument is similar in spirit, but not as quick.)

This post has no deeper meaning.^{1} Its purpose is just to present a very nice argument, which we gave in our talk last week, and which we feel should be better known.

One of the beautiful theorems that every school student should see is the infinity of primes.^{2} The standard Euclid proof tends to be difficult for students to appreciate, however, since, although the arithmetic is trivial, the argument is typically clouded with the unsettling concepts of infinity and contradiction.^{3 }

The following proof by Norwegian mathematician Axel Thue involves counting and a little more arithmetic, but avoids a head-on confrontation with infinity. Instead, Thue provides a direct guarantee of the number of primes up to a certain point. Versions of Thue’s argument can also be found at *cut-the-knot* and in *Hardy and Wright*,^{4} but the following seems cleaner to us. Continue reading “Erdős’s (not Thue’s) Proof of the Infinity of Primes”