Geodesic
On the Diophantine equation $(2^x-1)(p^y-1)=2z^2$
Czechoslovak Mathematical Journal, Tome 71 (2021) no. 3, pp. 689-696
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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.
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.
DOI : 10.21136/CMJ.2021.0057-20
Classification : 11B39, 11D61
Keywords: elementary method; Diophantine equation; positive integer solution 
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Tong, Ruizhou. On the Diophantine equation $(2^x-1)(p^y-1)=2z^2$. Czechoslovak Mathematical Journal, Tome 71 (2021) no. 3, pp. 689-696. doi: 10.21136/CMJ.2021.0057-20

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