A Quaternionic Proof of the Universality of Some Quadratic Forms

Abstract

The problem of finding all quadratic forms over Z that represent each positive integer received significant attention in a paper of Ramanujan in 1917. Exactly fifty four quaternary quadratic forms of this type without cross product terms were shown to represent all positive integers. The classical case of the quadratic form that is just the sum of four squares received an alternate proof by Hurwitz using a special ring of quaternions. Here we prove that seven other quaternary quadratic forms can be shown to represent all positive integers by investigation of the corresponding quaternion rings.

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Cite this paper

@inproceedings{Deutsch2008AQP, title={A Quaternionic Proof of the Universality of Some Quadratic Forms}, author={Jesse Ira Deutsch}, year={2008} }