On some results of Hurwitz and Deutsch about certain quadratic forms

Abstract

In this paper we introduce some results of Hurwitz and Deutsch about the number of representations of integers by quadratic forms x2+y2+z2+w2 and x2+y2+2z2+2w2 with certain parity conditions on the variables x, y, z and w. The purpose of this paper is to provide another proof of these results by the use of Liouville type identities, and see how many more similar results can be shown with this technique.