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| Format: | Preprint |
| Published: |
2021
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2105.06574 |
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| _version_ | 1866912792054857728 |
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| author | Dražić, Goran |
| author_facet | Dražić, Goran |
| contents | For a nonzero rational number $q$, a rational $D(q)$-$n$-tuple is a set of $n$ distinct nonzero rationals $\{a_1, a_2, \dots, a_n\}$ such that $a_ia_j+q$ is a square for all $1 \leqslant i < j \leqslant n$. We investigate for which $q$ there exist infinitely many rational $D(q)$-quintuples. We show that assuming the Parity Conjecture for the twists of several explicitly given elliptic curves, the density of such $q$ is at least $295026/296010\approx 99.5\%$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2105_06574 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Rational $D(q)$-quintuples Dražić, Goran Number Theory For a nonzero rational number $q$, a rational $D(q)$-$n$-tuple is a set of $n$ distinct nonzero rationals $\{a_1, a_2, \dots, a_n\}$ such that $a_ia_j+q$ is a square for all $1 \leqslant i < j \leqslant n$. We investigate for which $q$ there exist infinitely many rational $D(q)$-quintuples. We show that assuming the Parity Conjecture for the twists of several explicitly given elliptic curves, the density of such $q$ is at least $295026/296010\approx 99.5\%$. |
| title | Rational $D(q)$-quintuples |
| topic | Number Theory |
| url | https://arxiv.org/abs/2105.06574 |