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2 Bulletin of the Polish Academy of Sciences. Mathematics

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On the Euler function on differences between the coordinates of points on modular hyperbolas

Shparlinski I. E.

Bulletin of the Polish Academy of Sciences. Mathematics

2008

Vol. 56, no 1

1-7

For a prime p > 2, an integer a with gcd(a,p) = 1 and real 1 ≤ X,Y < p, we consider the set of points on the modular hyperbola Ηa,p(X,Y) = {(x,y) : x,y ≡ a (mod p), 1 ≤ x ≤ X, 1 ≤ y ≤ Y}. We give asymptotic formulas for the average values (x,y)∈ ... [wzór] with the Euler function φ(k) on the difference between the components of points of Ηa,p(X,Y).

Primitive points on a modular hyperbola

Shparlinski I. G.

Bulletin of the Polish Academy of Sciences. Mathematics

2006

Vol. 54, no 3-4

193-200

For positive integers m, U and V, we obtain an asymptotic formula for the number of integer points (u, v) ∈ [1, U] x [1, V] which belong to the modular hyperbola uv ≡ 1 (mod m) and also have god(u, v) = 1, which are also known as primitive points. Such points have a nice geometric interpretation as points on the modular hyperbola which are "visible" from the origin.