A Quaternionic Proof of the Universality of Some Quadratic Forms

摘要

Quadratic forms over Z that represent all positive integers are calleduniversal. Starting with Ramanujan, 54 universal quaternary quadratic formswithout cross product terms were discovered. The form that is the sum of foursquares was proved universal by Hurwitz using a special ring of quaternions.Here seven other quaternary quadratic forms are shown universal byinvestigation of appropriate rings of quaternions.