On equation Xn − 1 = BZn

Abstract

We consider the Diophantine equation [Formula: see text], where [Formula: see text] is understood as a parameter. We prove that if the equation has a solution, then either the Euler totient of the radical, [Formula: see text], has a common divisor with the exponent [Formula: see text], or the exponent is a prime and the solution stems from a solution to the diagonal case of the Nagell–Ljunggren equation: [Formula: see text]. This allows us to apply recent results on this equation to the binary Thue equation in question. In particular, we can then display parametrized families for which the Thue equation has no solution. The first such family was proved by Bennett in his seminal paper on binary Thue equations (see [M. A. Bennet, Rational approximation to algebraic numbers of small height: the Diophantine equation [Formula: see text], J. Reine Angew. Math. 535 (2001) 1–49]).