Skolem's conjecture confirmed for a family of exponential equations, II

TL;DR

This paper proves Skolem's conjecture for a specific family of exponential equations, extending previous results and demonstrating that non-solvability implies non-solvability modulo some modulus for these equations.

Contribution

It establishes the conjecture for equations of a particular form involving fixed integers and variable exponents, expanding the class of equations for which the conjecture holds.

Findings

Proved Skolem's conjecture for equations of the form x^n - b y_1^{k_1} ... y_ell^{k_ell} = ±1.

Extended the recent theorem of Hajdu and Tijdeman to a broader class of exponential equations.

Confirmed the conjecture for a new family of equations, supporting its general validity.

Abstract

According to Skolem's conjecture, if an exponential Diophantine equation is not solvable, then it is not solvable modulo an appropriately chosen modulus. Besides several concrete equations, the conjecture has only been proved for rather special cases. In this paper we prove the conjecture for equations of the form $x^n-by_1^{k_1}\dots y_\ell^{k_\ell}=\pm 1$, where $b,x,y_1,\dots,y_\ell$ are fixed integers and $n,k_1,\dots,k_\ell$ are non-negative integral unknowns. This result extends a recent theorem of Hajdu and Tijdeman.