On the number of representations of a positive integer as a sum of two binary quadratic forms


Let $\mathbb{N}$ denote the set of positive integers and let $\mathbb{N}_0 = \mathbb{N} \cup \lbrace0\rbrace$. Let $a_1x^2+b_1xy+c_1y^2$ and $a_2z^2+b_2zt+c_2t^2$ be two positive-definite, integral, binary quadratic forms. The number of representations of $n \in \mathbb{N}_0$ as a sum of these two binary quadratic forms is \begin{align} & N(a_1,b_1,c_1,a_2,b_2,c_2;n):= \mbox{ card} \lbrace (x,y,z,t) \in \mathbb{Z}^4 | \cr & n= a_1x^2+b_1xy + c_1y^2+a_2z^2+b_2zt+c_2t^2 \rbrace. \end{align} When $(b_1,b_2) \neq (0,0)$ we prove under certain conditions on $a_1, b_1, c_1, a_2, b_2$ and $c_2$ that $N(a_1,b_1,c_1,a_2,b_2,c_2;n)$ can be expressed as a finite linear combination of quantities of the type $N(a,0,b,c,0,d;n)$ with $a, b, c$ and $d$ positive integers, see Theorem 1.1. Thus, when the quantities $N(a,0,b,c,0,d;n)$ are known, we can determine $N(a_1,b_1,c_1,a_2,b_2,c_2;n)$. This determination is carried out for a number of quaternary quadratic forms $a_1x^2+b_1xy+c_1y^2+a_2z^2+b_2zt+c_2t^2$.

In International Journal of Number Theory, Volume 10, Issue 06, September 2014