Yasutsugu Fujita, Maohua Le, Nobuhiro Terai: Some exponential Diophantine equations attached to Pythagorean triples, 359-365

Abstract:

Let $p$ be an odd prime and $t$ a positive integer. We show that if $(u,v)\in\{(2p^t,1),(p^t,2)\}$, then the equation $x^2+(2uv)^m=(u^2+v^2)^n$ has only the positive integer solutions $(x,m,n)=(u-v,1,1),(u^2-v^2,2,2)$.

Key Words: Exponential Diophantine equations, Pellian equations.

2010 Mathematics Subject Classification: Primary 11D61; Secondary 11D09.