A short note on universality of some quadratic forms

Abstract

In a paper by J. Deutsch [1], a quaternionic proof of the universality of seven quaternary quadratic forms was given. The proof relies on a construction very similar to that of Hurwitz quaternions, and its associated division algorithm. Of course, these results are evident, if one uses the Conway-Schneeberger Fifteen Theorem [2], as the author also mentioned, however it is interesting to give a direct proof for some specific quadratic forms based on simple argument. It is the purpose of this short note to prove five of the seven quadratic forms mentioned and proven by Deutsch, using the universality of the classical quadratic form associated to the celebrated Lagrange's Theorem of Four Squares and Euler's trick. Mathematics Subject Classification: 11A67, 11R52, 11E25