THE EXPONENTIAL DIOPHANTINE EQUATION AX2 + BY 2 = λkZ AND ITS APPLICATIONS

Abstract

Let $ A,B \in \mathbb N$ with $A\gt 1, B\gt 1$ and $\gcd(A,B)=1$, $k\geq 2$ be an integer coprime with $AB$, and let $\lambda\in\{1,2,4\}$ be such that if $\lambda=4$, then $A\ne 4$ and $B\ne 4$; and if $k$ is even, then $\lambda=4$. In this paper, we shall describe all solutions of the equation $$ AX^2+BY^2=\lambda k^Z,\,\, X,Y,Z\in \mathbb Z,\,\gcd(X,Y)=1,\,\,Z\gt 0 $$ with $X|^*A$ or $Y|^*B$, where the symbol $X|^*A$ means that every prime divisor of $X$ divides $A$. Then, using this result, we give some more general results on the number of solutions of the equation $la^x+mb^y=\lambda c^z$, $x\gt 1$, $y\gt 1$, $z\gt 1$. In addition, using Cao's result on Pell equation, we obtain some im- provement of Terai's results on the equations $a^x+2=c^z, a^x+4=c^z$ and $a^x+2^y=c^z$.