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

Javier Cilleruelo; Moubariz Z. Garaev

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

Number Theory 2015-03-11 v1

Authors: Javier Cilleruelo , Moubariz Z. Garaev

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.

Keywords

congruences and supercongruences upper bounds prime number theory

Cite

@article{arxiv.1503.02730,
  title  = {On the congruence $x^x\equiv \lambda \pmod p$},
  author = {Javier Cilleruelo and Moubariz Z. Garaev},
  journal= {arXiv preprint arXiv:1503.02730},
  year   = {2015}
}

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