Representations of squares by certain quinary quadratic forms

Published in Acta Arithmetica, 2013

Abstract: Hurwitz determined the number of ways a square can be expressed as a sum of five squares. In this work, we determine the number of solutions, in integers of each of the equations $x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}+2x_{5}^{2}=n^{2}$, $x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}+3x_{5}^{2}=n^{2}$ and $x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+2x_{4}^{2}+3x_{5}^{2}=n^{2}$ for any given positive integer $n$. Fifteen further results that were discovered by computer experimentation are presented.