Integers of the form $ax^2+bxy+cy^2$

Abstract

We provide a characterization of integers represented by the positive definite binary quadratic form $ax^2+bxy+cy^2$. Suppose that $D=b^2-4ac$ and $d_K$ is the discriminant of the imaginary quadratic field $K={\mathbb {Q}}(\sqrt{D})$. We call $f=\sqrt{D/d_K}$ the conductor of $ax^2+bxy+cy^2$. In order to prove the main results, we define the “relative conductor” of two orders in an imaginary quadratic field. We provide a characterization of decomposition of proper ideals of orders in imaginary quadratic fields. Next, we provide characterizations of prime powers $l^h$, where l divides the conductor, represented by the positive definite binary quadratic form $ax^2+bxy+cy^2$. Some interesting applications of the main results are also presented. For example, we provide an equivalent condition for when the equation $m=4x^2+2xy+7y^2$ has an integer solution. Note that its discriminant and conductor are $-\,108$ and 6 and we do not assume that m is prime to 2 or 3.