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.
@article{bwmeta1.element.doi-10_1515_math-2017-0083,
author = {Wang Tingting and Wang Xiaonan},
title = {On the Golomb's conjecture and Lehmer's numbers},
journal = {Open Mathematics},
volume = {15},
year = {2017},
pages = {1003-1009},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_1515_math-2017-0083}
}
Wang Tingting; Wang Xiaonan. On the Golomb’s conjecture and Lehmer’s numbers. Open Mathematics, Tome 15 (2017) pp. 1003-1009. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_1515_math-2017-0083/