On the exponential diophantine equation (n-1)(x) + (n+2)(y) = n(z)

Abstract

Suppose that n is a positive integer. We show that the only positive integer solutions (n, x, y, z) of the exponential Diophantine equation

(n - 1)(x) + (n + 2)(y) = nz, n >= 2, xyz not equal 0,

are (3, 2, 1, 2), (3,1, 2, 3). The main tools in the proofs are Baker's theory and Bilu-Hanrot-Voutier's result on primitive divisors of Lucas numbers.