Primitive root producing quadratics

TL;DR

This paper explores quadratic polynomials that generate primitive roots for primes, linking the problem to class number one and prime-producing quadratics, and introduces an algorithm to find larger examples, achieving a new record with 31082.

Contribution

It presents a novel algorithm for discovering quadratic polynomials that produce primitive roots for many primes, extending previous records significantly.

Findings

D.H. Lehmer's polynomial produces 206 primes with 326 as a primitive root

An algorithm can find polynomials generating larger primitive root sequences

Y. Gallot's application set a new record with 31082 primes

Abstract

D.H. Lehmer found a quadratic polynomial such that 326 is a primitive root for the first 206 primes represented by this polynomial. It is shown that this is related to the class number one problem and prime producing quadratics. An algorithm is described to find more impressive examples in the same spirit. Y. Gallot used it to establish the current record in which 206 is being replaced by 31082.