Skolem’s conjecture confirmed for a family of exponential equations
Skolem’s conjecture states that if an exponential Diophantine equation is not solvable, then it is not solvable modulo an appropriately chosen modulus. Apart from many concrete equations, the conjecture has been proved only for rather special classes of equations. Here we show that the conjecture is valid for the Catalan equation $u^x-v^y=1$ provided that one of $u,v$ is a prime. This is the first instance where the conjecture is proved for a family of equations with more than one term on the left-hand side, of which the bases are multiplicatively independent.