| Title:
|
On the Diophantine equation $(2^x-1)(p^y-1)=2z^2$ (English) |
| Author:
|
Tong, Ruizhou |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
71 |
| Issue:
|
3 |
| Year:
|
2021 |
| Pages:
|
689-696 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $p$ be an odd prime. By using the elementary methods we prove that: (1) if $2\nmid x$, $p\equiv \pm 3\pmod 8,$ the Diophantine equation $(2^{x}-1)(p^{y}-1)=2z^{2}$ has no positive integer solution except when $p=3$ or $p$ is of the form $p=2a_{0}^{2}+1$, where $a_{0}>1$ is an odd positive integer. (2) if $2\nmid x$, $2\mid y$, $y\neq 2,4,$ then the Diophantine equation $(2^{x}-1)(p^{y}-1)=2z^{2}$ has no positive integer solution. (English) |
| Keyword:
|
elementary method |
| Keyword:
|
Diophantine equation |
| Keyword:
|
positive integer solution |
| MSC:
|
11B39 |
| MSC:
|
11D61 |
| idZBL:
|
07396191 |
| idMR:
|
MR4295239 |
| DOI:
|
10.21136/CMJ.2021.0057-20 |
| . |
| Date available:
|
2021-08-02T08:02:57Z |
| Last updated:
|
2023-10-02 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/149050 |
| . |
| Reference:
|
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| Reference:
|
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| Reference:
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| Reference:
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| Reference:
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