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

Authors:
Kevin Ford, Mizan R. Khan, Igor E. Shparlinski and Christian L. Yankov

Journal:
Proc. Amer. Math. Soc. **133** (2005), 3463-3468

MSC (2000):
Primary 11A07, 11N25

DOI:
https://doi.org/10.1090/S0002-9939-05-07962-1

Published electronically:
June 8, 2005

MathSciNet review:
2163580

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Abstract | References | Similar Articles | Additional Information

Abstract: We investigate the distribution of where

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 . 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 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:
https://doi.org/10.1090/S0002-9939-05-07962-1

Received by editor(s):
July 16, 2004

Published electronically:
June 8, 2005

Communicated by:
David E. Rohrlich

Article copyright:
© Copyright 2005
American Mathematical Society