Original scientific paper

https://doi.org/10.3336/gm.55.2.03

A note on the exponential Diophantine equation \((A^2n)^x+(B^2n)^y=((A^2+B^2)n)^z\)

Maohua Le ; Institute of Mathematics, Lingnan Normal College, Guangdong, 524048 Zhangjiang, China
Gökhan Soydan orcid.org/0000-0002-6321-4132 ; Department of Mathematics, Bursa Uludağ University, 16059 Bursa, Turkey

Abstract

Let \(A\) $B$ be positive integers such that $\min\{A,B\}>1$, $\gcd(A,B) = 1$ and $2|B.$ In this paper, using an upper bound for solutions of ternary purely exponential Diophantine equations due to R. Scott and R. Styer, we prove that, for any positive integer $n$, if $A >B^3/8$, then the equation $(A^2 n)^x + (B^2 n)^y = ((A^2 + B^2)n)^z$ has no positive integer solutions $(x,y,z)$ with $x > z > y$; if $B>A^3/6$, then it has no solutions $(x,y,z)$ with $y>z>x$. Thus, combining the above conclusion with some existing results, we can deduce that, for any positive integer $n$, if $B\equiv 2 \pmod{4}$ and $A >B^3/8$, then this equation has only the positive integer solution $(x,y,z)=(1,1,1)$.

Keywords

Ternary purely exponential Diophantine equation

Hrčak ID:

248663

URI

https://hrcak.srce.hr/248663

Publication date:

23.12.2020.

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