On the representation of integers by binary quadratic forms

TL;DR

This paper proves that for any irreducible binary quadratic form with integer coefficients, any integer solutions to the form's equation are related by a rational automorphism, revealing a symmetry property of such forms.

Contribution

It establishes that all integer solutions to an irreducible binary quadratic form are connected via rational automorphisms, highlighting a symmetry in the solution set.

Findings

Any solutions to the form are related by a rational automorphism.

The automorphism maps one solution to another within the form.

The result applies to all irreducible binary quadratic forms.

Abstract

In this note we show that for a given irreducible binary quadratic form $f(x,y)$ with integer coefficients, whenever we have $f(x,y) = f(u,v)$ for integers $x,y,u,v$, there exists a rational automorphism of $f$ which sends $(x,y)$ to $(u,v)$.