Finite sets containing near-primitive roots

TL;DR

This paper investigates the multiplicative order of integers modulo primes, showing that for almost all primes, certain elements or sets have orders exceeding specific bounds, advancing understanding of primitive roots and their distributions.

Contribution

The paper introduces new bounds for the multiplicative order of integers modulo primes and constructs explicit finite sets with elements of large order for almost all primes.

Findings

For almost all primes, one of a set of elements has order > p^{1/2 + 1/30}.

Explicit finite sets contain elements with order > p^{1-\epsilon} for almost all primes.

Results extend to orders modulo general integers n.

Abstract

Fix $a \in \mathbb{Z}$, $a\notin \{0,\pm 1\}$. A simple argument shows that for each $\epsilon > 0$, and almost all (asymptotically 100% of) primes $p$, the multiplicative order of $a$ modulo $p$ exceeds $p^{\frac12-\epsilon}$. It is an open problem to show the same result with $\frac12$ replaced by any larger constant. We show that if $a,b$ are multiplicatively independent, then for almost all primes $p$, one of $a,b,ab, a^2b, ab^2$ has order exceeding $p^{\frac{1}{2}+\frac{1}{30}}$. The same method allows one to produce, for each $\epsilon > 0$, explicit finite sets $\mathcal{A}$ with the property that for almost all primes $p$, some element of $\mathcal{A}$ has order exceeding $p^{1-\epsilon}$. Similar results hold for orders modulo general integers $n$ rather than primes $p$.