A note on the article by F. Luca “On the system of Diophantine equations $a²+b²={(m²+1)}^r$ and $a^x+b^y={(m²+1)}^z$” (Acta Arith. 153 (2012), 373-392)

Takafumi Miyazaki. "A note on the article by F. Luca “On the system of Diophantine equations $a²+b² = (m²+1)^r$ and $a^{x}+b^y = (m²+1)^z$” (Acta Arith. 153 (2012), 373-392)." Acta Arithmetica 164.1 (2014): 31-42. <http://eudml.org/doc/279513>.

@article{TakafumiMiyazaki2014,
abstract = {Let r,m be positive integers with r > 1, m even, and A,B be integers satisfying $A + B√(-1) = (m + √(-1))^\{r\}$. We prove that the Diophantine equation $|A|^x + |B|^y = (m² + 1)^z$ has no positive integer solutions in (x,y,z) other than (x,y,z) = (2,2,r), whenever $r > 10^\{74\}$ or $m > 10^\{34\}$. Our result is an explicit refinement of a theorem due to F. Luca.},
author = {Takafumi Miyazaki},
journal = {Acta Arithmetica},
language = {eng},
number = {1},
pages = {31-42},
title = {A note on the article by F. Luca “On the system of Diophantine equations $a²+b² = (m²+1)^r$ and $a^\{x\}+b^y = (m²+1)^z$” (Acta Arith. 153 (2012), 373-392)},
url = {http://eudml.org/doc/279513},
volume = {164},
year = {2014},
}

TY - JOUR
AU - Takafumi Miyazaki
TI - A note on the article by F. Luca “On the system of Diophantine equations $a²+b² = (m²+1)^r$ and $a^{x}+b^y = (m²+1)^z$” (Acta Arith. 153 (2012), 373-392)
JO - Acta Arithmetica
PY - 2014
VL - 164
IS - 1
SP - 31
EP - 42
AB - Let r,m be positive integers with r > 1, m even, and A,B be integers satisfying $A + B√(-1) = (m + √(-1))^{r}$. We prove that the Diophantine equation $|A|^x + |B|^y = (m² + 1)^z$ has no positive integer solutions in (x,y,z) other than (x,y,z) = (2,2,r), whenever $r > 10^{74}$ or $m > 10^{34}$. Our result is an explicit refinement of a theorem due to F. Luca.
LA - eng
UR - http://eudml.org/doc/279513
ER -