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

TL;DR

This paper investigates upper bounds for the number of solutions to the congruence x^x ≡ λ mod p, utilizing recent trigonometric sum estimates over subgroups to advance understanding of this number-theoretic problem.

Contribution

It introduces new upper bound estimates for solutions of the congruence x^x ≡ λ mod p, based on recent developments in exponential sum bounds.

Findings

Derived new upper bounds for solution counts

Applied recent trigonometric sum estimates to congruence problems

Enhanced understanding of the distribution of solutions

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 $$ x^{x}\equiv \lambda\pmod p;\quad x\in \mathbb{N},\quad x\le p-1, $$ 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.