A note on the exponential Diophantine equation ( 4 m ² + 1 ) x + ( 5 m ² - 1 ) y = ( 3 m ) z

Jianping Wang; Tingting Wang; Wenpeng Zhang

Colloquium Mathematicae (2015)

  • Volume: 139, Issue: 1, page 121-126
  • ISSN: 0010-1354

Abstract

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Let m be a positive integer. Using an upper bound for the solutions of generalized Ramanujan-Nagell equations given by Y. Bugeaud and T. N. Shorey, we prove that if 3 ∤ m, then the equation ( 4 m ² + 1 ) x + ( 5 m ² - 1 ) y = ( 3 m ) z has only the positive integer solution (x,y,z) = (1,1,2).

How to cite

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Jianping Wang, Tingting Wang, and Wenpeng Zhang. "A note on the exponential Diophantine equation $(4m²+1)^x + (5m²-1)^y = (3m)^z$." Colloquium Mathematicae 139.1 (2015): 121-126. <http://eudml.org/doc/283767>.

@article{JianpingWang2015,
abstract = {Let m be a positive integer. Using an upper bound for the solutions of generalized Ramanujan-Nagell equations given by Y. Bugeaud and T. N. Shorey, we prove that if 3 ∤ m, then the equation $(4m²+1)^x + (5m²-1)^y = (3m)^z$ has only the positive integer solution (x,y,z) = (1,1,2).},
author = {Jianping Wang, Tingting Wang, Wenpeng Zhang},
journal = {Colloquium Mathematicae},
keywords = {exponential Diophantine equation; generalized Ramanujan-Nagell equations; positive integer solution},
language = {eng},
number = {1},
pages = {121-126},
title = {A note on the exponential Diophantine equation $(4m²+1)^x + (5m²-1)^y = (3m)^z$},
url = {http://eudml.org/doc/283767},
volume = {139},
year = {2015},
}

TY - JOUR
AU - Jianping Wang
AU - Tingting Wang
AU - Wenpeng Zhang
TI - A note on the exponential Diophantine equation $(4m²+1)^x + (5m²-1)^y = (3m)^z$
JO - Colloquium Mathematicae
PY - 2015
VL - 139
IS - 1
SP - 121
EP - 126
AB - Let m be a positive integer. Using an upper bound for the solutions of generalized Ramanujan-Nagell equations given by Y. Bugeaud and T. N. Shorey, we prove that if 3 ∤ m, then the equation $(4m²+1)^x + (5m²-1)^y = (3m)^z$ has only the positive integer solution (x,y,z) = (1,1,2).
LA - eng
KW - exponential Diophantine equation; generalized Ramanujan-Nagell equations; positive integer solution
UR - http://eudml.org/doc/283767
ER -

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