A Quaternionic Proof of the Universality of Some Quadratic Forms

Abstract

Quadratic forms over Z that represent all positive integers are called universal. Starting with Ramanujan, 54 universal quaternary quadratic forms without cross product terms were discovered. The form that is the sum of four squares was proved universal by Hurwitz using a special ring of quaternions. Here seven other quaternary quadratic forms are shown universal by investigation of appropriate rings of quaternions.