On Grosswald's conjecture on primitive roots

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 $p>3.67\times 10^{71}$.Comment: 7 page

On consecutive primitive elements in a finite field

For $q$ an odd prime power with $q>169$ we prove that there are always three consecutive primitive elements in the finite field $\mathbb{F}_{q}$. Indeed, there are precisely eleven values of $q \leq 169$ for which this is false. For $4\leq n \leq 8$ we present conjectures on the size of $q_{0}(n)$ such that $q>q_{0}(n)$ guarantees the existence of $n$ consecutive primitive elements in $\mathbb{F}_{q}$, provided that $\mathbb{F}_{q}$ has characteristic at least~$n$. Finally, we improve the upper bound on $q_{0}(n)$ for all $n\geq 3$.Comment: 10 pages, 2 table

A proof of the conjecture of Cohen and Mullen on sums of primitive roots

We prove that for all $q>61$, every non-zero element in the finite field $\mathbb{F}_{q}$ can be written as a linear combination of two primitive roots of $\mathbb{F}_{q}$. This resolves a conjecture posed by Cohen and Mullen.Comment: 8 pages; to appear in Mathematics of Computatio