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On the maximal difference between an element and its inverse in residue rings

Author(s): Kevin Ford; Mizan R. Khan; Igor E. Shparlinski; Christian L. Yankov
Journal: Proc. Amer. Math. Soc. 133 (2005), 3463-3468.
MSC (2000): Primary 11A07, 11N25
Posted: June 8, 2005
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Abstract: We investigate the distribution of $n - M(n)$ where

\begin{displaymath}M(n)=\max\left\{ \, \left\vert a-b\right\vert : 1 \leq a,b\leq n-1 \textrm{ and } ab \equiv 1\pmod n\right\}.\end{displaymath}

Exponential sums provide a natural tool for obtaining upper bounds on this quantity. Here we use results about the distribution of integers with a divisor in a given interval to obtain lower bounds on $n - M(n)$. We also present some heuristic arguments showing that these lower bounds are probably tight, and thus our technique can be a more appropriate tool to study $n - M(n)$ than a more traditional way using exponential sums.

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Additional Information:

Kevin Ford
Affiliation: Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 West Green Street, Urbana, Illinois 61801
Email: ford@math.uiuc.edu

Mizan R. Khan
Affiliation: Department of Mathematics and Computer Science, Eastern Connecticut State University, Willimantic, Connecticut 06226
Email: khanm@easternct.edu

Igor E. Shparlinski
Affiliation: Department of Computing, Macquarie University, Sydney, NSW 2109, Australia
Email: igor@ics.mq.edu.au

Christian L. Yankov
Affiliation: Department of Mathematics and Computer Science, Eastern Connecticut State University, Willimantic, Connecticut 06226
Email: yankovc@easternct.edu

DOI: 10.1090/S0002-9939-05-07962-1
PII: S 0002-9939(05)07962-1
Received by editor(s): July 16, 2004
Posted: June 8, 2005
Communicated by: David E. Rohrlich
Copyright of article: Copyright 2005, American Mathematical Society

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