Configurations Of Consecutive Primitive Roots

TL;DR

This paper proves the existence of specific configurations of consecutive and quasi-consecutive primitive roots in finite fields for large primes, extending understanding of primitive root distributions.

Contribution

It establishes the existence of various configurations of consecutive primitive roots in finite fields, generalizing previous results for large primes.

Findings

Existence of configurations of primitive roots in finite fields

Results hold for large primes p and small integer k

Configurations involve fixed tuples of distinct integers

Abstract

Let $p \geq 2$ be a large prime, and let $k \ll \log p $ be a small integer. This note proves the existence of various configurations of $(k+1)$-tuples of consecutive and quasi consecutive primitive roots $n+a_0, n+a_1, n+a_2, \ldots, n+a_k$ in the finite field $\mathbb{F}_p$, where $a_0,a_1, \ldots, a_k$ is a fixed $(k+1)$-tuples of distinct integers.