2017/12/11 by Cohen, Stephen D., Trudgian, Tim
#11A07 #11L05 #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1712.03990
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 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 ≠ 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.