A Hasse-type principle for exponential diophantine equations and its applications

We propose a conjecture, similar to Skolem's conjecture, on a Hasse-type principle for exponential diophantine equations. We prove that in a sense the principle is valid for "almost all" equations. Based upon this we propose a general method for the solution of exponential diophantine equations. Using a generalization of a result of Erd\H{o}s, Pomerance and Schmutz concerning Carmichael's $\lambda$ function, we can make our search systematic for certain moduli needed in the method.