Jeśmanowicz' conjecture with congruence relations

Yasutsugu Fujita, and Takafumi Miyazaki. "Jeśmanowicz' conjecture with congruence relations." Colloquium Mathematicae 128.2 (2012): 211-222. <http://eudml.org/doc/284026>.

@article{YasutsuguFujita2012,
abstract = {Let a,b and c be relatively prime positive integers such that a²+b² = c². We prove that if $b ≡ 0 (mod 2^\{r\})$ and $b ≡ ±2^\{r\} (mod a)$ for some non-negative integer r, then the Diophantine equation $a^\{x\} + b^\{y\} = c^z$ has only the positive solution (x,y,z) = (2,2,2). We also show that the same holds if c ≡ -1 (mod a).},
author = {Yasutsugu Fujita, Takafumi Miyazaki},
journal = {Colloquium Mathematicae},
keywords = {exponential Diophantine equations; Pythagorean triples; Pell equations},
language = {eng},
number = {2},
pages = {211-222},
title = {Jeśmanowicz' conjecture with congruence relations},
url = {http://eudml.org/doc/284026},
volume = {128},
year = {2012},
}

TY - JOUR
AU - Yasutsugu Fujita
AU - Takafumi Miyazaki
TI - Jeśmanowicz' conjecture with congruence relations
JO - Colloquium Mathematicae
PY - 2012
VL - 128
IS - 2
SP - 211
EP - 222
AB - Let a,b and c be relatively prime positive integers such that a²+b² = c². We prove that if $b ≡ 0 (mod 2^{r})$ and $b ≡ ±2^{r} (mod a)$ for some non-negative integer r, then the Diophantine equation $a^{x} + b^{y} = c^z$ has only the positive solution (x,y,z) = (2,2,2). We also show that the same holds if c ≡ -1 (mod a).
LA - eng
KW - exponential Diophantine equations; Pythagorean triples; Pell equations
UR - http://eudml.org/doc/284026
ER -