On the fixed points of the map \(x \mapsto x^x\) modulo a prime, II

• Published in: arXiv.org 2017-07
• Main Authors: Adam Tyler Felix, Kurlberg, Pär
• Format: Article
• Language: English
• Subjects: Integers; Upper bounds
• ISSN: 2331-8422

We study number theoretic properties of the map \(x \mapsto x^{x} \mod{p}\), where \(x \in \{1,2,\ldots,p-1\}\), and improve on some recent upper bounds, due to Kurlberg, Luca, and Shparlinski, on the number of primes \(p < N\) for which the map only has the trivial fixed point \(x=1\). A key technical result, possibly of independent interest, is the existence of subsets \(\mathscr{N}_{q} \subset \{2,3,\ldots,q-1\}\) such that almost all \(k\)-tuples of distinct integers \(n_{1}, n_{2},\ldots,n_{k} \in \mathscr{N}_q\) are multiplicatively independent (if \(k\) is not too large), and \(|\mathscr{N}_q| = q \cdot (1+o(1))\) as \(q \to \infty\). For \(q\) a large prime, this is used to show that the number of solutions to a certain large and sparse system of \(\mathbb{F}_q\)-linear forms \(\{ \mathscr{L}_{n} \}_{n=2}^{q-1}\) "behaves randomly" in the sense that \(|\{ \mathbf{v} \in \mathbb{F}_{q}^{d} : \mathscr{L}_{n}(\mathbf{v}) =1, n = 2,3, \ldots, q-1 \}| \sim q^{d}(1-1/q)^{q} \sim q^{d}/e\). (Here \(d=\pi(q-1)\) and the coefficents of \(\mathscr{L}_{n}\) are given by the exponents in the prime power factorization of \(n\).)