Lehmer numbers and primitive roots modulo a prime

TL;DR

This paper investigates Lehmer numbers that are primitive roots modulo a prime, providing explicit estimates for their count, proving their existence for most primes, and improving previous bounds on related problems.

Contribution

It offers an explicit estimate for Lehmer primitive roots, proves their existence for all primes except 2, 3, and 7, and refines bounds on the Golomb-Lehmer primitive root problem.

Findings

Lehmer primitive roots exist for all primes except 2, 3, and 7.

An explicit formula for the number of Lehmer numbers modulo p is provided.

The estimate for solutions to the Golomb-Lehmer problem is improved.

Abstract

A Lehmer number modulo a prime $p$ is an integer $a$ with $1 \leq a \leq p-1$ whose inverse $\bar{a}$ 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 S. Golomb). In this paper we give an explicit estimate for the number of Lehmer primitive roots modulo $p$ and prove that, for all primes $p \neq 2,3,7$, Lehmer primitive roots exist. We also make explicit the known expression for the number of Lehmer numbers modulo $p$ and improve the Wang--Wang estimate for the number of solutions to the Golomb--Lehmer primitive root problem.