Lehmer numbers and primitive roots modulo a prime

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.