An algebraic approach to count the number of representations of an integer by the quadratic form $x^2+ay^2$ for certain values of $a$

Abstract

By considering the norm of elements in the ring of integers in $\mathbb{Q}(\sqrt{-a})$, we give an algebraic approach to count the number of integral solutions of diophantine equations of the form $x^2+ay^2=n$ where $a$ is a Heegner number or $a=27$.