Primitive values of rational functions at primitive elements of a finite field

TL;DR

This paper establishes conditions under which rational functions evaluated at primitive elements of finite fields produce primitive pairs, showing such pairs exist for large enough fields with only a few exceptions.

Contribution

It provides a new sufficient condition for the existence of primitive pairs involving rational functions over finite fields, extending previous results.

Findings

Primitive pairs exist for large finite fields with few exceptions.

For degree 2 rational functions, non-existence occurs only for 28 specific field sizes.

Almost all sufficiently large fields admit such primitive pairs.

Abstract

Given a prime power $q$ and an integer $n\geq2$, we establish a sufficient condition for the existence of a primitive pair $(\alpha,f(\alpha))$ where $\alpha \in \mathbb{F}_q$ and $f(x) \in \mathbb{F}_q(x)$ is a rational function of degree $n$. (Here $f=f_1/f_2$, where $f_1, f_2$ are coprime polynomials of degree $n_1,n_2$, respectively, and $n_1+n_2=n$.) For any $n$, such a pair is guaranteed to exist for sufficiently large $q$. Indeed, when $n=2$, such a pair definitely does {\em not} exist only for 28 values of $q$ and possibly (but unlikely) only for at most $3911$ other values of $q$.