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PERFECT POWERS THAT ARE SUMS OF TWO POWERS OF FIBONACCI NUMBERS

Part of: Diophantine equations

Published online by Cambridge University Press:  30 August 2018

ZHONGFENG ZHANG Open the ORCID record for ZHONGFENG ZHANG [Opens in a new window]  and
ALAIN TOGBÉ
ZHONGFENG ZHANG
Affiliation:
School of Mathematics and Statistics, Zhaoqing University, Zhaoqing 526061, China email bee2357@163.com
ALAIN TOGBÉ*
Affiliation:
Department of Mathematics, Statistics and Computer Science, Purdue University Northwest, 1401 S. U.S. 421 Westville, IN 46391, USA email atogbe@pnw.edu
*
✉atogbe@pnw.edu
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Abstract

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In this paper, we consider the Diophantine equations

$$\begin{eqnarray}\displaystyle F_{n}^{q}\pm F_{m}^{q}=y^{p} & & \displaystyle \nonumber\end{eqnarray}$$
with positive integers $q,p\geq 2$ and $\gcd (F_{n},F_{m})=1$, where $F_{k}$ is a Fibonacci number. We obtain results for $q=2$ or $q$ an odd prime with $q\equiv 3\;(\text{mod}\;4),3<q<1087$, and complete solutions for $q=3$.

Keywords

exponential equationFibonacci number

MSC classification

Primary: 11D61: Exponential equations
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 99 , Issue 1 , February 2019 , pp. 34 - 41
Copyright
© 2018 Australian Mathematical Publishing Association Inc. 

Footnotes

The first author was supported by NSF of China (No. 11601476) and the Guangdong Provincial Natural Science Foundation (No. 2016A030313013 ) and Foundation for Distinguished Young Teacher in Higher Education of Guangdong, China (YQ2015167). The second author thanks Purdue University Northwest for support.

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