Manjul Bhargava

Abstract:   The famous "Four Squares Theorem" of Lagrange asserts that any positive integer can be expressed as the sum of four square numbers. That is, the quadratic form a^2 + b^2 + c^2 + d^2 represents all (positive) integers. When does a general quadratic form represent all integers? When does it represent all odd integers? When does it represent all primes? We show how all these questions turn out to have very simple and surprising answers. In particular, we describe the recent work (joint with Jonathan Hanke, Duke University) in proving Conway's "290-Conjecture".