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

TL;DR

This paper investigates the fixed points of the map x ↦ x^x modulo primes, improves bounds on primes with only trivial fixed points, and introduces subsets of integers with multiplicative independence properties, revealing probabilistic behavior of related linear systems.

Contribution

It provides new bounds on primes with trivial fixed points and constructs large subsets of integers with multiplicative independence, advancing understanding of the map's fixed points and related algebraic structures.

Findings

Improved upper bounds on the number of primes with only trivial fixed points.

Existence of large subsets with multiplicative independence properties.

Demonstration that certain linear systems behave randomly, with solutions approximating a Poisson distribution.

Abstract

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 $n$.)