On the solutions of the exponential Diophantine equation a ^x + b ^y = (m ² + 1) ^z

Abstract

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. Moreover, if r is a prime and r ≡ 5 (mod 8) or r ≡ 19 (mod 24), , we prove that the equation has only the solution (x, y, z) = (2, 2, r).