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)

Abstract

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
• Status: Published
• Refereed: Yes
• 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
• Publisher: Elsevier
• ISSN: 0022-314X
• 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