The congruence $x^{x}\equiv \lambda \pmod p$
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- by J. Cilleruelo and M. Z. Garaev
- Proc. Amer. Math. Soc. 144 (2016), 2411-2418
- DOI: https://doi.org/10.1090/proc/12919
- Published electronically: October 21, 2015
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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 \[ x^{x}\equiv \lambda \pmod p;\quad x\in \mathbb {N},\quad x\le p-1, \] 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.References
- Antal Balog, Kevin A. Broughan, and Igor E. Shparlinski, On the number of solutions of exponential congruences, Acta Arith. 148 (2011), no. 1, 93–103. MR 2784012, DOI 10.4064/aa148-1-7
- Antal Balog, Kevin A. Broughan, and Igor E. Shparlinski, Sum-products estimates with several sets and applications, Integers 12 (2012), no. 5, 895–906. MR 2988554, DOI 10.1515/integers-2012-0012
- J. Cilleruelo and M. Z. Garaev, Congruences involving product of intervals and sets with small multiplicative doubling modulo a prime and applications, Preprint (2014).
- Roger Crocker, On residues of $n^{n}$, Amer. Math. Monthly 76 (1969), 1028–1029. MR 248072, DOI 10.2307/2317129
- Joshua Holden and Pieter Moree, Some heuristics and results for small cycles of the discrete logarithm, Math. Comp. 75 (2006), no. 253, 419–449. MR 2176407, DOI 10.1090/S0025-5718-05-01768-0
- I. D. Shkredov, On exponential sums over multiplicative subgroups of medium size, Finite Fields Appl. 30 (2014), 72–87. MR 3249821, DOI 10.1016/j.ffa.2014.06.002
- Yu. N. Shteinikov, Estimates of trigonometric sums modulo a prime, Preprint, 2014.
- Lawrence Somer, The residues of $n^{n}$ modulo $p$, Fibonacci Quart. 19 (1981), no. 2, 110–117. MR 614045
- I. M. Vinogradov, Elements of number theory, Dover Publications, Inc., New York, 1954. Translated by S. Kravetz. MR 0062138
Bibliographic Information
- J. Cilleruelo
- Affiliation: Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM) and Departamento de Matemáticas, Universidad Autónoma de Madrid, Madrid-28049, Spain
- MR Author ID: 292544
- Email: franciscojavier.cilleruelo@uam.es
- M. Z. Garaev
- Affiliation: Centro de Ciencias Matemáticas, Universidad Nacional Autónoma de México, C.P. 58089, Morelia, Michoacán, México
- MR Author ID: 632163
- Email: garaev@matmor.unam.mx
- 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
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 144 (2016), 2411-2418
- MSC (2010): Primary 11A07
- DOI: https://doi.org/10.1090/proc/12919
- MathSciNet review: 3477057