Variations on a theme of Schinzel and Wójcik

Abstract

Schinzel and Wójcik have shown that if $\hat{I}\pm ,{\hat{I}}^2$ are rational numbers not $0$ or $\pm 1$, then ${\mathrm{ord}}_p(\hat{I}\pm )={\mathrm{ord}}_p({\hat{I}}^2)$ 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 ${\mathrm{ord}}_p(\hat{I}\pm )>{\mathrm{ord}}_p({\hat{I}}^2)$? We write down several families of pairs $\hat{I}\pm ,{\hat{I}}^2$ 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 Wójcik's theorem for the integers of an imaginary quadratic field $K$: If $\hat{I}\pm ,{\hat{I}}^2\in {\mathcal{O}}_K$ are nonzero and neither is a root of unity, then there are infinitely many maximal ideals $P$ of ${\mathcal{O}}_K$ for which ${\mathrm{ord}}_P(\hat{I}\pm )={\mathrm{ord}}_P({\hat{I}}^2)$.

13 pages