Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2018
|
| Soggetti: | |
| Accesso online: | https://arxiv.org/abs/1807.04116 |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866908968091123712 |
|---|---|
| author | Voutier, Paul M |
| author_facet | Voutier, Paul M |
| contents | We obtain best possible results for the number of coprime positive integer solutions of the equation in the title when $a$ is a positive integer, $b=p^{m}$, $2p^{m}$ or $4p^{m}$, where $m$ is a non-negative integer, $p$ is prime, $\gcd \left( a^{2}, b \right)$ is squarefree and $X^{2}- \left( a^{2}+b \right) Y^{2}=-4$ has a solution in positive integers. We prove our results by establishing best possible bounds for the number of distinct squares in certain binary recurrence sequences, including those associated with such equations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1807_04116 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Sharp bounds on the number of squares in recurrence sequences and solutions of $X^{2}-\left( a^{2}+b \right) Y^{4}=-b$ Voutier, Paul M Number Theory 11D25, 11J82 We obtain best possible results for the number of coprime positive integer solutions of the equation in the title when $a$ is a positive integer, $b=p^{m}$, $2p^{m}$ or $4p^{m}$, where $m$ is a non-negative integer, $p$ is prime, $\gcd \left( a^{2}, b \right)$ is squarefree and $X^{2}- \left( a^{2}+b \right) Y^{2}=-4$ has a solution in positive integers. We prove our results by establishing best possible bounds for the number of distinct squares in certain binary recurrence sequences, including those associated with such equations. |
| title | Sharp bounds on the number of squares in recurrence sequences and solutions of $X^{2}-\left( a^{2}+b \right) Y^{4}=-b$ |
| topic | Number Theory 11D25, 11J82 |
| url | https://arxiv.org/abs/1807.04116 |