Primitive points on a modular hyperbola

Igor E. Shparlinski

doi:10.4064/ba54-3-1

Primitive points on a modular hyperbola

Igor E. Shparlinski

School of Computing

Research output: Contribution to journal › Article › peer-review

Abstract

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 gcd(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.

Original language	English
Pages (from-to)	193-200
Number of pages	8
Journal	Bulletin of the Polish Academy of Sciences. Mathematics
Volume	54
Issue number	3
DOIs	https://doi.org/10.4064/ba54-3-1
Publication status	Published - 2006

Keywords

congruences
primitive roots
irregularities of distribution
estimates on character sums

Access to Document

10.4064/ba54-3-1

Cite this

@article{e320b6b2eb364c4b924e3de5e0bc2808,

title = "Primitive points on a modular hyperbola",

abstract = "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 gcd(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.",

keywords = "congruences, primitive roots, irregularities of distribution, estimates on character sums",

author = "Shparlinski, {Igor E.}",

year = "2006",

doi = "10.4064/ba54-3-1",

language = "English",

volume = "54",

pages = "193--200",

journal = "Bulletin of the Polish Academy of Sciences. Mathematics",

issn = "0239-7269",

publisher = "Polska Akademia Nauk, Instytut Matematyczny",

number = "3",

}

TY - JOUR

T1 - Primitive points on a modular hyperbola

AU - Shparlinski, Igor E.

PY - 2006

Y1 - 2006

N2 - 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 gcd(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.

AB - 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 gcd(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.

KW - congruences

KW - primitive roots

KW - irregularities of distribution

KW - estimates on character sums

U2 - 10.4064/ba54-3-1

DO - 10.4064/ba54-3-1

M3 - Article

SN - 0239-7269

VL - 54

SP - 193

EP - 200

JO - Bulletin of the Polish Academy of Sciences. Mathematics

JF - Bulletin of the Polish Academy of Sciences. Mathematics

IS - 3

ER -

Primitive points on a modular hyperbola

Abstract

Keywords

Access to Document

Fingerprint

Cite this