Universal sums of polygonal numbers

Abstract

In this talk, we will consider certain “finiteness theorems”. In a celebrated result of Conway and Schneeberger, it was shown that every positive-definite integral quadratic form is universal if and only if it represents every integer up to 15 (the proof was later simplified and generalized by Bhargava). Applying this result to arbitrary repeated sums of squares (i.e., diagonal quadratic forms), we consider generalizations where the quadratic form is replaced with sums of m-gonal numbers (the m = 4 case is sums of squares). One finds that for each m there exists a finiteness result of the above type, and the main result in this talk is a bound on the growth of the constant up to which one must check for universality. This is joint work with Jingbo Liu.