Equidistribution of Diophantine pairs among the equivalence classes of quadratic forms
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866912791964680192 |
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| author | Dražić, Goran Kazalicki, Matija Mrazović, Rudi |
| author_facet | Dražić, Goran Kazalicki, Matija Mrazović, Rudi |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_22902 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Equidistribution of Diophantine pairs among the equivalence classes of quadratic forms Dražić, Goran Kazalicki, Matija Mrazović, Rudi Number Theory Dynamical Systems 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. |
| title | Equidistribution of Diophantine pairs among the equivalence classes of quadratic forms |
| topic | Number Theory Dynamical Systems |
| url | https://arxiv.org/abs/2512.22902 |