On the Diophantine equation ( 2 x - 1 ) ( p y - 1 ) = 2 z 2

Ruizhou Tong

Czechoslovak Mathematical Journal (2021)

  • Volume: 71, Issue: 3, page 689-696
  • ISSN: 0011-4642

Abstract

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Let p be an odd prime. By using the elementary methods we prove that: (1) if 2 ∤ x , p ≡ ± 3 ( mod 8 ) , the Diophantine equation ( 2 x - 1 ) ( p y - 1 ) = 2 z 2 has no positive integer solution except when p = 3 or p is of the form p = 2 a 0 2 + 1 , where a 0 > 1 is an odd positive integer. (2) if 2 ∤ x , 2 ∣ y , y ≠ 2 , 4 , then the Diophantine equation ( 2 x - 1 ) ( p y - 1 ) = 2 z 2 has no positive integer solution.

How to cite

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Tong, Ruizhou. "On the Diophantine equation $(2^x-1)(p^y-1)=2z^2$." Czechoslovak Mathematical Journal 71.3 (2021): 689-696. <http://eudml.org/doc/297745>.

@article{Tong2021,
abstract = {Let $p$ be an odd prime. By using the elementary methods we prove that: (1) if $2\nmid x$, $p\equiv \pm 3\hspace\{4.44443pt\}(\@mod \; 8),$ 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 $2\nmid x$, $2\mid y$, $y\ne 2,4,$ then the Diophantine equation $(2^\{x\}-1)(p^\{y\}-1)=2z^\{2\}$ has no positive integer solution.},
author = {Tong, Ruizhou},
journal = {Czechoslovak Mathematical Journal},
keywords = {elementary method; Diophantine equation; positive integer solution},
language = {eng},
number = {3},
pages = {689-696},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On the Diophantine equation $(2^x-1)(p^y-1)=2z^2$},
url = {http://eudml.org/doc/297745},
volume = {71},
year = {2021},
}

TY - JOUR
AU - Tong, Ruizhou
TI - On the Diophantine equation $(2^x-1)(p^y-1)=2z^2$
JO - Czechoslovak Mathematical Journal
PY - 2021
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 71
IS - 3
SP - 689
EP - 696
AB - Let $p$ be an odd prime. By using the elementary methods we prove that: (1) if $2\nmid x$, $p\equiv \pm 3\hspace{4.44443pt}(\@mod \; 8),$ 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 $2\nmid x$, $2\mid y$, $y\ne 2,4,$ then the Diophantine equation $(2^{x}-1)(p^{y}-1)=2z^{2}$ has no positive integer solution.
LA - eng
KW - elementary method; Diophantine equation; positive integer solution
UR - http://eudml.org/doc/297745
ER -

References

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