Addendum to the paper: "Artin Prime Producing Quadratics" [Abh. Math. Sem. Univ. Hamburg 77 (2007), 109--127; MR2379332 (2008m:11194)] by P. Moree

TL;DR

This paper discusses the problem of finding integers and polynomials where a fixed number is a primitive root for many prime values of the polynomial, reviewing recent developments and correcting historical record discrepancies.

Contribution

It clarifies historical records related to prime-producing polynomials and highlights connections with recent work by Maynard and Pollack.

Findings

Corrects historical record of prime-producing polynomial records.

Connects the problem to recent advances in prime number research.

Emphasizes the importance of primitive roots in prime value problems.

Abstract

Can one find an integer $g$ and a polynomial $f$, such that $g$ is a primitive root for many consecutive (different) prime values assumed by $f$? Moree considered this problem in 2007 with computational assistence from Gallot and concentrated on the case where $f$ is quadratic. Recently Akbary and Scholten (Math. Comp., to appear) extended this work and also considered the case where $f$ is linear and cubic. In their paper they improved on a record of the authors in the quadratic case. However, mistakingly Moree had not mentioned in the 2007 paper the then record (from 2006), but only an older record. The 2006 record (due to Gallot) actually exceeds the 'record' from Akbary and Scholten. Aside from pointing this out, the goal of the note is to attend people to the problem mentioned in the first sentence of the abstract and point out some connections with recent spectacular work of Maynard and highly interesting work of Pollack.