Equidistribution of Diophantine pairs among the equivalence classes of quadratic forms

TL;DR

This paper proves that D(n)-pairs, defined by a specific quadratic relation, are evenly distributed among quadratic form classes, simplifying the proof of their asymptotic count.

Contribution

It establishes the asymptotic equidistribution of D(n)-pairs among quadratic form classes and provides a simpler proof for their asymptotic enumeration.

Findings

D(n)-pairs are asymptotically equidistributed among quadratic form classes.

A streamlined proof of Badesa's asymptotic formula is provided.

The distribution relates to proper SL_2(Z)-equivalence classes of quadratic forms.

Abstract

For a fixed integer n, a pair of nonzero integers {a, c} is called a D(n)-pair if the product ac plus n is a perfect square. In this short note we prove that D(n)-pairs are asymptotically equidistributed (via their associated quadratic forms) among proper SL_2(Z)-equivalence classes of binary quadratic forms of discriminant 4n with fixed content. As a consequence, we obtain a more streamlined and simpler proof of Badesa's asymptotic formula for the number of D(n)-pairs.