Skolem’s conjecture confirmed for a family of exponential equations, II
Volume 197 / 2021
Acta Arithmetica 197 (2021), 129-136
MSC: 11D61, 11D79.
DOI: 10.4064/aa191116-1-6
Published online: 8 October 2020
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.
