On the unique solvability of the simultaneous Pell equations $x^2-ay^2 = 1$ and $z^2-bx^2 = 1$
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866917689270730752 |
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| author | Hilgart, Tobias Ziegler, Volker |
| author_facet | Hilgart, Tobias Ziegler, Volker |
| contents | We consider the simultaneous Pell equations $$x^2 - ay^2 = 1, \qquad z^2 - bx^2 = 1,$$ where $a > b\geq 2$ are positive integers. We describe a procedure which, for any fixed $b$, either confirms that the simultaneous Pell equations have at most one solution in positive integers, or finds all exceptions for which we have proved that there are at most finitely many. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_06191 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the unique solvability of the simultaneous Pell equations $x^2-ay^2 = 1$ and $z^2-bx^2 = 1$ Hilgart, Tobias Ziegler, Volker Number Theory 11D09, 11Y50 We consider the simultaneous Pell equations $$x^2 - ay^2 = 1, \qquad z^2 - bx^2 = 1,$$ where $a > b\geq 2$ are positive integers. We describe a procedure which, for any fixed $b$, either confirms that the simultaneous Pell equations have at most one solution in positive integers, or finds all exceptions for which we have proved that there are at most finitely many. |
| title | On the unique solvability of the simultaneous Pell equations $x^2-ay^2 = 1$ and $z^2-bx^2 = 1$ |
| topic | Number Theory 11D09, 11Y50 |
| url | https://arxiv.org/abs/2406.06191 |