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

TL;DR

This paper introduces a conjecture akin to Skolem's for exponential diophantine equations, demonstrates its near-universal validity, and proposes a systematic method for solving such equations based on a generalized number-theoretic result.

Contribution

It formulates a new conjecture for exponential diophantine equations, proves its validity for most cases, and develops a systematic solution method utilizing advanced number theory.

Findings

The conjecture holds for almost all exponential diophantine equations.

A systematic solution method is proposed based on the conjecture.

Generalization of Erdős, Pomerance, and Schmutz's result aids in the method's implementation.

Abstract

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.