On the congruence $x^x\equiv \lambda (\mathrm{mod}p)$

Abstract

In the present paper we obtain several new results related to the problem of upper bound estimates for the number of solutions of the congruence $$x^x\equiv \lambda (\mathrm{mod}p);x\in \mathbb{N},x\le p-1,$$ where $p$ is a large prime number, $\lambda$ is an integer corpime to $p$. Our arguments are based on recent estimates of trigonometric sums over subgroups due to Shkredov and Shteinikov.