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.