# 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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topWang 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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