In class next week, we will begin by discussing the material in section 2.1 of the text. Especially, we will prove Theorem 2.15, which characterises those positive integers n that can be expressed in the form a^2 + b^2, where a and b are integers. (For example, 8 is of this form because it is 2^2 + 2^2; so is 16, which is 4^2 + 0^2. The number 7 is not the sum of two squares.)

In preparation for our time together, you might have a look at the recent review of the beautiful "Proofs from the Book," a volume that brings together especially beautiful proofs of interesting results from diverse parts of mathematics. The "Book" in the title is God's notebook of mathematicial proofs, in which the best possible proofs of all theorems are written down. (Paul Erdös invented this concept.) The author of the review lists several highlights of "Proofs from the Book"; one is Don Zagier's proof, first published in 1990, of our Theorem 2.15. The reviewer says that the first published proof of the theorem is that of Euler; he adds that Zagier's proof is so beautiful that it will certainly be the last one to be published! I'll try to understand Don Zagier's argument and then explain it to you. Before discussing Zagier's proof, however, I'll talk about the one in our textbook. Don Zagier is a mathematician in Bonn, Germany who went to high school in Stockton. When you come to my office hours, ask me to point out Zagier in the photos on my wall.

September 11, 1999