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

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

2005-07-28

arxiv.org/abs/math/0507596v2

Arch. Math. 87 (2006), 129-140

Mathematics

Number Theory

11 pages, 2 Tables; Proposition 3 has now been corrected along with a few typos

Scientific paper

10.1007/s00013-006-1704-z

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.

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