On the Maximal Difference between an Element and its Inverse in Residue Rings
by
Mizan R Khan
Eastern Connecticut State University
Coauthors: Igor E. Shparlinski and Christian L. Yankov

Let M(n) = max{|a-b|: 1 <= a, b <= n-1 and ab \equiv 1 (mod n) }. We will discuss the following 2 asymptotic results.

(a) n-M(n) = O(n^.75+o(1)).

(b) limsup_{n --> \infty} \fracn-M(n)\surdn = \infty.

We use exponential sums to prove (a) and we prove (b) by invoking results on the distribution of integers with a divisor in a given interval.

Date received: April 29, 2004