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

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 divis...

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Bibliographic Details
Published in:Proceedings of the American Mathematical Society 2005-12, Vol.133 (12), p.3463-3468
Main Authors: Ford, Kevin, Khan, Mizan R., Shparlinski, Igor E., Christian L. Yankov
Format: Article
Language:English
Subjects:
Algebra
Exact sciences and technology
Heuristics
Integers
Mathematical congruence
Mathematical constants
Mathematical intervals
Mathematics
Number theory
Polynomials
Sciences and techniques of general use
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Summary: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.
ISSN:0002-9939
1088-6826
DOI:10.1090/S0002-9939-05-07962-1