Equidistribution of Diophantine pairs among the equivalence classes of quadratic forms

2025/12/28 by Goran Dražić, Matija Kazalicki, Dražić, Goran +3
Mathematics · Computer Science · #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Coding theory and cryptography

paper · doi:10.48550/arxiv.2512.22902

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 SL₂(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.

Citations

• Rational D(q)-quintuples
• Rational D(q)-quadruples
• There is no Diophantine quintuple
• Effective equidistribution of S-integral points on symmetric varieties
• Some Rational Diophantine Sextuples

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• Binary quadratic forms that represent almost the same primes
• Diophantine approximation with sums of two squares II
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