A Hasse-type principle for exponential Diophantine equations over number fields and its applications

Abstract

In this paper we extend a conjecture (which is a variant of a classical conjecture of Skolem) to exponential Diophantine equations over algebraic number fields. On the one hand, using a generalization of a result of Erdős, Pomerance and Schmutz concerning ‘small’ values of Carmichael’s $$\lambda $$ function, we give strong support for the validity of the conjecture, both theoretically and numerically. On the other hand, we demonstrate the applicability of our method by finding all representations of powers of 2, 3, 5, 7 as a sum of three balancing numbers. Note that this problem reduces to finding all solutions of certain exponential Diophantine equations over a number field.