Variations on a theme of Schinzel and W\'ojcik

Abstract

Schinzel and W\'ojcik have shown that if $\alpha, \beta$ are rational numbers not $0$ or $\pm 1$, then $\mathrm{ord}_p(\alpha)=\mathrm{ord}_p(\beta)$ for infinitely many primes $p$, where $\mathrm{ord}_p(\cdot)$ denotes the order in $\mathbb{F}_p^{\times}$. We begin by asking: When are there infinitely many primes $p$ with $\mathrm{ord}_p(\alpha) > \mathrm{ord}_p(\beta)$? We write down several families of pairs $\alpha,\beta$ for which we can prove this to be the case. In particular, we show this happens for "100\%" of pairs $A,2$, as $A$ runs through the positive integers. We end on a different note, proving a version of Schinzel and W\'{o}jcik's theorem for the integers of an imaginary quadratic field $K$: If $\alpha, \beta \in \mathcal{O}_K$ are nonzero and neither is a root of unity, then there are infinitely many maximal ideals $P$ of $\mathcal{O}_K$ for which $\mathrm{ord}_P(\alpha) = \mathrm{ord}_P(\beta)$.