Mathematical Research Letters

Volume 22 (2015)

Number 1

On the fixed points of the map $x \mapsto x^x$ modulo a prime

Pages: 141 – 168

DOI: http://dx.doi.org/10.4310/MRL.2015.v22.n1.a8

Authors

Pär Kurlberg (Department of Mathematics, Royal Institute of Technology, Stockholm, Sweden)

Florian Luca (School of Mathematics, University of the Witwatersrand, Johannesburg, South Africa)

Igor E. Shparlinski (Department of Pure Mathematics, University of New South Wales, Sydney, Australia)

Abstract

In this paper, we show that for almost all primes $p$ there is an integer solution $x \in [2, p-1]$ to the congruence $x^x \equiv x (\mathrm{mod \;} p)$. The solutions can be interpretated as fixed points of the map $x \mapsto x^x (\mathrm{mod \;} p)$, and we study numerically and discuss some unexpected properties of the dynamical system associated with this map.

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