Improvement of an estimate of H. Mueller involving the order of 2(mod u) II
Pieter Moree
Abstract
Let m>=1 be an arbitrary fixed integer and let Nm(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 Nm(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 Nm(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.
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