On the fixed points of the map x→xx modulo a prime

Pär Kurlberg, Florian Luca, Igor E. Shparlinski

Research output: Contribution to journalArticlepeer-review

7 Citations (Scopus)


In this paper, we show that for almost all primes p there is an integer solution xε [2,p-1] to the congruence xx ≡ x (mod p). The solutions can be interpretated as fixed points of the map x→xx (mod p), and we study numerically and discuss some unexpected properties of the dynamical system associated with this map.

Original languageEnglish
Pages (from-to)141-168
Number of pages28
JournalMathematical Research Letters
Issue number1
Publication statusPublished - 2015
Externally publishedYes


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