Linear combinations of primitive elements of a finite field

TL;DR

This paper investigates properties of primitive roots in finite fields, demonstrating that for primes greater than 13, there exist pairs of primitive roots with sums and inverse sums also primitive, refining previous results.

Contribution

It refines a previous result by Li and Han, proving the existence of primitive root pairs with primitive sums and inverse sums for all primes greater than 13.

Findings

For all primes p > 13, there exist primitive roots a, b with a + b and a^{-1} + b^{-1} also primitive.

Improves upon previous bounds for primitive root combinations in finite fields.

Provides new insights into the structure of primitive roots and their linear combinations.

Abstract

We examine linear sums of primitive roots and their inverses in finite fields. In particular, we refine a result by Li and Han, and show that every $p> 13$ has a pair of primitive roots $a$ and $b$ such that $a+ b$ and $a^{-1} + b^{-1}$ are also primitive roots mod $p$.