On the solutions of the exponential Diophantine equationa^x+b^y= (m²+ 1)^z

In this paper, we consider the Diophantine equation a x + b y = c z . Combining Laurent's result on lower bounds for linear forms in two logarithms, Bugeaud's result on upper bounds for the p-adic logarithms, and Bilu-Hanrot-Voutier's result on primitive divisors of Lucas numbers, we obtain a sharper computable upper bound for the number of positive integer solutions of the equation ∣v r ∣ x + ∣u r∣ y = c z , where and r is an odd number....