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

Ford, Kevin; Khan, Mizan; Shparlinski, Igor; Yankov, Christian

doi:10.1090/S0002-9939-05-07962-1

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.

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