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

TL;DR

This paper introduces an algebraic method based on number field norms to count solutions of quadratic form equations like x^2 + ay^2 = n, especially for specific values of a such as Heegner numbers and 27.

Contribution

It provides a novel algebraic approach using ring of integers norms to determine the number of solutions for certain quadratic forms, extending previous methods.

Findings

Counts solutions for specific a values like Heegner numbers and 27

Uses algebraic number theory to relate solutions to norms in quadratic fields

Offers a new perspective on classical Diophantine equations

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 $a=27$.