TY - JOUR

T1 - On the maximal difference between an element and its inverse in residue rings

AU - Ford, Kevin

AU - Khan, Mizan R.

AU - Shparlinski, Igor E.

AU - Yankov, Christian L.

PY - 2005/12

Y1 - 2005/12

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=29144515632&partnerID=8YFLogxK

U2 - 10.1090/S0002-9939-05-07962-1

DO - 10.1090/S0002-9939-05-07962-1

M3 - Article

AN - SCOPUS:29144515632

SN - 0002-9939

VL - 133

SP - 3463

EP - 3468

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

IS - 12

ER -