Simultaneous Primitive Root Values Of Polynomials

TL;DR

This paper proves that for certain fixed integers and polynomials, there is a positive density of primes for which these values are simultaneous primitive roots, extending to multiple polynomial cases.

Contribution

It establishes the nonzero density of primes where fixed integers and polynomial values are simultaneous primitive roots, generalizing to admissible polynomial tuples.

Findings

Primes with fixed integers and polynomial values as simultaneous primitive roots have positive density.

The analysis extends to admissible polynomial tuples with size logarithmic in p.

The results apply to a broad class of polynomials beyond quadratic forms.

Abstract

Let $z\ne \pm1,w^2$ be a fixed integer, and let $f(t)\ne g(t)^2$ be a fixed polynomial over the integers. It is shown that the subset of primes $p\geq 2$ such that $z$ and $f(z)$ is a pair of simultaneous primitive roots modulo $p$ has nonzero density in the set of primes. The same analysis generalizes to \textit{admissible} $k$-tuple of polynomials $z$, $f_1(z)$, $f_2(z), \ldots$, $f_k(z)$, such that $f_i(z)\ne g_i(z)^2$, and $k\ll \log p$ is a small integer.