Improvement of an estimate of H. Müller 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 Müller that were subsequently sharpened into an asymptotic estimate by the present author. Müller 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. We also generalize to other base numbers than 2. A further analysis of Müller’s method leads us to study and solve a certain Diophantine equation.