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arXiv · 2512.22902

Equidistribution of Diophantine pairs among the equivalence classes of quadratic forms

Goran Dražić·Matija Kazalicki·Rudi Mrazović
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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.

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Goran Dražić, Matija Kazalicki, Rudi Mrazović. 2025-12-28. Equidistribution of Diophantine pairs among the equivalence classes of quadratic forms. https://arxiv.org/abs/2512.22902

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