Szego Seminar: Proofs of Four-Square Theorem

Speaker: Mohao Yi, Washington University

Abstract: In 1770, Lagrange proved that every natural number is a sum of four squares (of natural numbers). This theorem was generalized in the next fifty years: In 1797-8, Legendre showed that a positive integer is a sum of three squares if and only if it is not $4^k(8m + 7)$ for any integers $k$ and $m$, and in 1834, Jacobi found an explicit formula for the number of ways to write a natural number as a sum of four squares. In this talk, I will sketch the proofs (but not necessarily the original ones) of these theorems with an application of quadratic forms and modular forms.

As always, pizza will be provided.