TY - JOUR

T1 - On the number of distances between the coordinates of points on modular hyperbolas

AU - Shparlinski, Igor E.

AU - Winterhof, Arne

PY - 2008/5

Y1 - 2008/5

N2 - For a prime p > 2, an integer a with gcd (a, p) = 1 and 1 ≤ X, Y < p we give an asymptotic formula for the number of different Euclidean distances | x - y | defined by the points on the modular hyperbola {(x, y) : x y ≡ a (mod p), 1 ≤ x ≤ X, 1 ≤ y ≤ Y}. Furthermore, in the case X = Y = p - 1 we determine the exact number of different distances.

AB - For a prime p > 2, an integer a with gcd (a, p) = 1 and 1 ≤ X, Y < p we give an asymptotic formula for the number of different Euclidean distances | x - y | defined by the points on the modular hyperbola {(x, y) : x y ≡ a (mod p), 1 ≤ x ≤ X, 1 ≤ y ≤ Y}. Furthermore, in the case X = Y = p - 1 we determine the exact number of different distances.

UR - http://www.scopus.com/inward/record.url?scp=41149099607&partnerID=8YFLogxK

U2 - 10.1016/j.jnt.2007.04.016

DO - 10.1016/j.jnt.2007.04.016

M3 - Article

AN - SCOPUS:41149099607

VL - 128

SP - 1224

EP - 1230

JO - Journal of Number Theory

JF - Journal of Number Theory

SN - 0022-314X

IS - 5

ER -