On the congruence $x^x\equiv λ\pmod p$

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 $x^{x}\equiv λ\pmod p;\quad x\in \mathbb{N},\quad x\le p-1,$ where $p$ is a large prime number, $λ$ is an integer corpime to $p$. Our arguments are based on recent estimates of trigonometric sums over subgroups due to Shkredov and Shteinikov.

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