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# 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

#### 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.