On the maximal difference between an element and its inverse in residue rings
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- by Kevin Ford, Mizan R. Khan, Igor E. Shparlinski and Christian L. Yankov PDF
- Proc. Amer. Math. Soc. 133 (2005), 3463-3468 Request permission
Abstract:
We investigate the distribution of $n - M(n)$ where \[ M(n)=\max \left \{ \left | a-b\right |\ :\ 1 \leq a,b\leq n-1 \textrm {\ and\ } ab \equiv 1\pmod n\right \}.\] 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.References
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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
- MR Author ID: 325647
- ORCID: 0000-0001-9650-725X
- 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
- MR Author ID: 192194
- 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
- Received by editor(s): July 16, 2004
- Published electronically: June 8, 2005
- Communicated by: David E. Rohrlich
- © Copyright 2005 American Mathematical Society
- 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
- MathSciNet review: 2163580