Glasnik Matematicki, Vol. 55, No. 2 (2020), 195-201.

A NOTE ON THE EXPONENTIAL DIOPHANTINE EQUATION (A2n)x+(B2n)y=((A2+B2)n)z

Maohua Le and Gökhan Soydan

Institute of Mathematics, Lingnan Normal College, Guangdong, 524048 Zhangjiang, China
e-mail: lemaohua2008@163.com

Department of Mathematics, Bursa Uludağ University, 16059 Bursa, Turkey
e-mail: gsoydan@uludag.edu.tr

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 >B3/8, then the equation (A2 n)x + (B2 n)y = ((A2 + B2)n)z has no positive integer solutions (x,y,z) with x > z > y; if B>A3/6, 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 ≡ 2 (mod 4) and A >B3/8, then this equation has only the positive integer solution (x,y,z)=(1,1,1).

2010 Mathematics Subject Classification.   11D61

Key words and phrases.   Ternary purely exponential Diophantine equation

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

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