📝 SummarXiv

Abstract

An ideal setting to exhibit infinite sets of primes $p$ relative to which an integer is a primitive root $\pmod p$ is provided by the ultraproduct ring $\widetilde{\mathbb{Z}}=\prod_{\mathfrak{U}} \mathbb{Z}_p$ with respect to a nonprincipal ultrafilter $\mathfrak{U}$ on $\wp(\mathbb{P})$, extant via Tarski's Ultrafilter Theorem and a Chebotarev Density Theorem construction realising $\mathbb{L}\deff\mathbb{M}(\sqrt{-2})$, for a maximal radical-free subfield $\mathbb{M}\subseteq\overline{\Q}$ with $\mathbb{L}\cap\Q(\mu_{\infty})=\Q(\sqrt{-2})$, as the relative algebraic closure $\mathrm{Abs}(\widetilde{\mathbb{K}})$ of the prime field of $\widetilde{\mathbb{K}}=\prod_{\mathfrak{U}} \mathbb{F}_p$. Results include positive resolutions of the conjectured infinitude of primes $p$ for which $\bullet$ $\frac{p-1}{2}$ is prime; $\bullet$ a non-perfect-square $-1\neq m\in\mathbb{Z}$ is a primitive root $\pmod p$; establishing as manifest the efficacy of ultraproduct treatments in resolving number theory problems requiring authentication of countably infinite conforming sets.