Mathematics: number theorist Don Zagier’s one-line lesson

Every year, the mathematics department of the Ecole normale supérieure de Lyon organizes a weekend for its students. The recipe is always the same: about forty students, a dozen researchers or teacher-researchers from Lyon and a single outside guest. The guest is a renowned mathematician, often forty years older than the students, with the mission of presenting his or her personal vision of mathematics. The idea is to bring together an accomplished researcher with future researchers who know almost nothing about mathematics.

The place is favorable to informal meetings: the magnificent castle of Goutelas, 80 kilometers from Lyon. For twenty years, the guests have represented the diversity of mathematics: geometry, combinatorics, number theory, analysis, etc. On arrival, the students are often impressed by the reputation of the guest, but they do not know that he himself is worried about his own responsibility. He has most of his career behind him and is facing young people who have not yet started it. But this mutual anxiety does not last long, the magic works, and we often see exchanges of great richness.

The most recent guest, from October 7 to 9, was Don Zagier, 71 years old, a number theorist and a bit eccentric. He has dual American and German citizenship and is fluent in a number of languages (including French). He currently teaches in Bonn and Trieste, after having been a professor at the Collège de France. He is fascinated by numbers and formulas of all kinds, which is not so common in mathematics, contrary to what the general public thinks. He made some fundamental discoveries, but anecdotally he published a one-line proof of Fermat’s two-square theorem: a prime number other than 2 is the sum of two integer squares if and only if the remainder of its division by 4 is 1. For example, the remainder of the division of 41 by 4 is 1, and indeed 41 is the sum of 16 and 25. A one-line proof? Maybe, but a line that is probably incomprehensible to many readers of Le Monde.

A one-line proof

Zagier set the scene for the students with an assumed bad faith, by placing number theory far above geometry and therefore, according to him, even higher than topology. Many of his predecessors had expressed radically different opinions. A central theme of his lectures concerned the so-called “modular forms”, unknown to students. He began with a quip that there are five operations in arithmetic: addition, subtraction, multiplication, division and… modular forms!
His lectures consisted in showing how number theory allows us to take a fresh look at the theory of knots. A knot is the topological object one imagines: a string tied in space. Zagier reminded us of what all professionals know: that a mathematical result is all the more interesting when it builds bridges between theories that were thought to be independent. The presentations were not always easy to follow for the students, but the important point is that they saw a mathematician at work, with his passion, his personal history, his anecdotes, his vision and his conjectures.

On Saturday evening, Zagier proposed an exercise to the students. The next morning, he announced that five students had found the solution, by different ways. He admitted that he himself had searched for three years before finding a solution. Let us rejoice: the succession is assured.