  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.
