  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.
