Lehmer numbers and primitive roots modulo a prime

Cohen, S. D. and Trudgian, T. (2019) Lehmer numbers and primitive roots modulo a prime. Journal of Number Theory, 203, pp. 68-79. (doi: 10.1016/j.jnt.2019.03.004)

[img] Text
190306.pdf - Accepted Version
Available under License Creative Commons Attribution Non-commercial No Derivatives.



A Lehmer number modulo a prime p is an integer a with 1 ≤ a ≤ p − 1 whose inverse a¯ within the same range has opposite parity. Lehmer numbers that are also primitive roots have been discussed by Wang and Wang in an endeavour to count the number of ways 1 can be expressed as the sum of two primitive roots that are also Lehmer numbers (an extension of a question of Golomb). In this paper we give an explicit estimate for the number of Lehmer primitive roots modulo p and prove that, for all primes p �= 2, 3, 7, Lehmer primitive roots exist. We also make explicit the known expression for the number of Lehmer numbers modulo p and improve the estimate for the number of solutions to the Golomb–Lehmer primitive root problem.

Item Type:Articles
Glasgow Author(s) Enlighten ID:Cohen, Professor Stephen
Authors: Cohen, S. D., and Trudgian, T.
College/School:College of Science and Engineering > School of Mathematics and Statistics
Journal Name:Journal of Number Theory
ISSN (Online):1096-1658
Published Online:16 April 2019
Copyright Holders:Copyright © 2019 Elsevier Inc.
First Published:First published in Journal of Number Theory 203:68-79
Publisher Policy:Reproduced in accordance with the copyright policy of the publisher

University Staff: Request a correction | Enlighten Editors: Update this record