On consecutive primitive elements in a finite field

TL;DR

This paper proves that for large enough finite fields with odd prime power order, there are always three consecutive primitive elements, and it explores bounds and conjectures for longer sequences of consecutive primitive elements.

Contribution

The paper establishes the existence of three consecutive primitive elements in finite fields for all sufficiently large q and improves bounds on q0(n) for longer sequences.

Findings

Existence of three consecutive primitive elements for all q > 169

Precisely eleven q values ≤ 169 lack three consecutive primitive elements

Improved upper bounds on q0(n) for n ≥ 3

Abstract

For $q$ an odd prime power with $q>169$ we prove that there are always three consecutive primitive elements in the finite field $\mathbb{F}_{q}$. Indeed, there are precisely eleven values of $q \leq 169$ for which this is false. For $4\leq n \leq 8$ we present conjectures on the size of $q_{0}(n)$ such that $q>q_{0}(n)$ guarantees the existence of $n$ consecutive primitive elements in $\mathbb{F}_{q}$, provided that $\mathbb{F}_{q}$ has characteristic at least~$n$. Finally, we improve the upper bound on $q_{0}(n)$ for all $n\geq 3$.