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 -