Proofs from THE BOOK

Proofs from THE BOOK (Summer 2020, Session II) is about extremely elegant proofs from many areas of mathematics. We will loosely follow Aigner and Ziegler’s book by the same title, going through the beautiful proofs they present and putting them into a broader mathematical context.

The title of the book and the class comes from the great 20th-century mathematician Paul Erdős, who liked to talk about a deity who had a book of perfect proofs to all theorems. We will glimpse into that book and study those proofs and where they come from, as well as where else the proof techniques lead elsewhere in mathematics.

We will start by proving that there are infinitely many primes. While you have likely already seen Euclid’s famous and wonderful proof of this theorem, there are also many others that have their own strengths and weaknesses. From a proof of the infinitude of primes, one can usually extract an upper bound on the size of the nth prime. We will look at whether these proofs give realistic bounds.

Here is a sampling of other questions we will look at:

1) For which positive integers n is it possible to divide a square into n triangles of equal area?

2) How many labeled rooted trees are there with n vertices? (See the diagram below for a hint.)

3) Given two polyhedra with the same volume, is it possible to cut one of them up into finitely many pieces and rearrange them to make the other one? (Hilbert’s Third Problem)