# On the Golomb’s conjecture and Lehmer’s numbers

Open Mathematics (2017)

• Volume: 15, Issue: 1, page 1003-1009
• ISSN: 2391-5455

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## Abstract

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Let p be an odd prime. For each integer a with 1 ≤ a ≤ p − 1, it is clear that there exists one and only one ā with 1 ≤ ā ≤ p − 1 such that a · ā ≡ 1 mod p. Let N(p) denote the set of all primitive roots a mod p with 1 ≤ a ≤ p − 1 in which a and ā are of opposite parity. The main purpose of this paper is using the analytic method and the estimate for the hybrid exponential sums to study the solvability of the congruence a + b ≡ 1 mod p with a, b ∈ N(p), and give a sharper asymptotic formula for the number of the solutions of the congruence equation.

## How to cite

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Wang Tingting, and Wang Xiaonan. "On the Golomb’s conjecture and Lehmer’s numbers." Open Mathematics 15.1 (2017): 1003-1009. <http://eudml.org/doc/288553>.

@article{WangTingting2017,
abstract = {Let p be an odd prime. For each integer a with 1 ≤ a ≤ p − 1, it is clear that there exists one and only one ā with 1 ≤ ā ≤ p − 1 such that a · ā ≡ 1 mod p. Let N(p) denote the set of all primitive roots a mod p with 1 ≤ a ≤ p − 1 in which a and ā are of opposite parity. The main purpose of this paper is using the analytic method and the estimate for the hybrid exponential sums to study the solvability of the congruence a + b ≡ 1 mod p with a, b ∈ N(p), and give a sharper asymptotic formula for the number of the solutions of the congruence equation.},
author = {Wang Tingting, Wang Xiaonan},
journal = {Open Mathematics},
keywords = {D. H. Lehmer’s numbers; Golomb’s conjecture; The hybrid exponential sums; Congruence equation; Primitive roots},
language = {eng},
number = {1},
pages = {1003-1009},
title = {On the Golomb’s conjecture and Lehmer’s numbers},
url = {http://eudml.org/doc/288553},
volume = {15},
year = {2017},
}

TY - JOUR
AU - Wang Tingting
AU - Wang Xiaonan
TI - On the Golomb’s conjecture and Lehmer’s numbers
JO - Open Mathematics
PY - 2017
VL - 15
IS - 1
SP - 1003
EP - 1009
AB - Let p be an odd prime. For each integer a with 1 ≤ a ≤ p − 1, it is clear that there exists one and only one ā with 1 ≤ ā ≤ p − 1 such that a · ā ≡ 1 mod p. Let N(p) denote the set of all primitive roots a mod p with 1 ≤ a ≤ p − 1 in which a and ā are of opposite parity. The main purpose of this paper is using the analytic method and the estimate for the hybrid exponential sums to study the solvability of the congruence a + b ≡ 1 mod p with a, b ∈ N(p), and give a sharper asymptotic formula for the number of the solutions of the congruence equation.
LA - eng
KW - D. H. Lehmer’s numbers; Golomb’s conjecture; The hybrid exponential sums; Congruence equation; Primitive roots
UR - http://eudml.org/doc/288553
ER -

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