On Grosswald's conjecture on primitive roots

TL;DR

This paper advances the understanding of Grosswald's conjecture by establishing upper bounds for the least primitive root modulo p within very large ranges of p, narrowing the gap towards a full proof.

Contribution

The authors prove that Grosswald's conjecture holds for all primes p between 409 and 2.5×10^{15} and for all p greater than 3.67×10^{71}.

Findings

Confirmed the conjecture for 409 < p < 2.5×10^{15}.

Extended the validity of the conjecture to p > 3.67×10^{71}.

Provided new bounds on the least primitive root for large primes.

Abstract

Grosswald's conjecture is that $g(p)$, the least primitive root modulo $p$, satisfies $g(p) \leq \sqrt{p} - 2$ for all $p>409$. We make progress towards this conjecture by proving that $g(p) \leq \sqrt{p} -2$ for all $409<p< 2.5\times 10^{15}$ and for all $p>3.67\times 10^{71}$.