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The congruence $ x^{x}\equiv\lambda\pmod p$


Authors: J. Cilleruelo and M. Z. Garaev
Journal: Proc. Amer. Math. Soc. 144 (2016), 2411-2418
MSC (2010): Primary 11A07
DOI: https://doi.org/10.1090/proc/12919
Published electronically: October 21, 2015
MathSciNet review: 3477057
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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

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

where $ p$ is a large prime number and $ \lambda $ is an integer coprime to $ p$. Our arguments are based on recent estimates of trigonometric sums over subgroups due to Shkredov and Shteinikov.

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J. Cilleruelo
Affiliation: Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM) and Departamento de Matemáticas, Universidad Autónoma de Madrid, Madrid-28049, Spain
Email: franciscojavier.cilleruelo@uam.es

M. Z. Garaev
Affiliation: Centro de Ciencias Matemáticas, Universidad Nacional Autónoma de México, C.P. 58089, Morelia, Michoacán, México
Email: garaev@matmor.unam.mx

DOI: https://doi.org/10.1090/proc/12919
Received by editor(s): March 23, 2015
Received by editor(s) in revised form: July 31, 2015
Published electronically: October 21, 2015
Communicated by: Ken Ono
Article copyright: © Copyright 2015 American Mathematical Society