Improvement of an estimate of H. Mueller involving the order of 2(mod u) II

Abstract

Let m>=1 be an arbitrary fixed integer and let N_m(x) count the number of odd integers u<=x such that the order of 2 modulo u is not divisible by m. In case m is prime estimates for N_m(x) were given by H. Mueller that were subsequently sharpened into an asymptotic estimate by the present author. Mueller on his turn extended the author's result to the case where m is a prime power and gave bounds in the case m is not a prime power. Here an asymptotic for N_m(x) is derived that is valid for all integers m. This asymptotic would easily have followed from Mueller's approach were it not for the fact that a certain Diophantine equation has non-trivial solutions. All solutions of this equation are determined. We also generalize to other base numbers than 2. For a very sparse set of these numbers Mueller's approach does work.