On the existence of certain Lehmer numbers modulo a prime

Abstract

A Lehmer number modulo an odd prime number $p$ is a residue class $a\in F_p^{\times }$ whose multiplicative inverse $\bar{a}$ has opposite parity. Lehmer numbers that are also primitive roots are called Lehmer primitive roots. Analogously, in this article we say that a residue class $a\in F_p^{\times }$ is a Lehmer non-primitive root modulo $p$ if $a$ is Lehmer number modulo $p$ which is not a primitive root. We provide explicit estimates for the difference between the number of Lehmer non-primitive roots modulo a prime $p$ and their “expected number”, which is $\frac{p-1-\varphi (p-1)}{2}$. Similar explicit bounds are also provided for the number of $k$-consecutive Lehmer numbers modulo a prime, and $k$-consecutive Lehmer primitive roots We also prove that for any prime number $p>3.05\times 10^{14}$, there exists a Lehmer non-primitive root modulo $p$. Moreover, we show that for any positive integer $k\ge 2$ (respectively, $k\ge 5$) and for all primes $p\ge exp\left(12^2{k}^3\right)$ (respectively, $p\ge exp(6.87^k)$), there exist $k$ consecutive Lehmer numbers modulo $p$ (respectively, $k$ consecutive Lehmer primitive roots modulo $p$). For large primes $p$, these theorems generalize two results which were proven in a paper by Cohen and Trudgian appeared in the Journal of Number Theory in 2019.