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

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

Research output: Contribution to journalArticlepeer-review

Abstract

We investigate the distribution of n M(n) where M(n) = max{|a - b| : 1 ≤ a,b ≤ n - 1 and ab ≡ 1 (mod n)}. 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.

Original languageEnglish (US)
Pages (from-to)3463-3468
Number of pages6
JournalProceedings of the American Mathematical Society
Volume133
Issue number12
DOIs
StatePublished - Dec 2005

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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