A problem in comparative order theory

Abstract

Write ${\mathrm{ord}}_p(\cdot )$ for the multiplicative order in ${\mathbb{F}}_p^{\times }$. Recently, Matthew Just and the second author investigated the problem of classifying pairs $\hat{I}\pm ,{\hat{I}}^2\in {\mathbb{Q}}^{\times }\setminus \{\pm 1\}$ for which ${\mathrm{ord}}_p(\hat{I}\pm )>{\mathrm{ord}}_p({\hat{I}}^2)$ holds for infinitely many primes $p$. They called such pairs order-dominant. We describe an easily-checkable sufficient condition for $\hat{I}\pm ,{\hat{I}}^2$ to be order-dominant. Via the large sieve, we show that almost all integer pairs $\hat{I}\pm ,{\hat{I}}^2$ satisfy our condition, with a power savings on the size of the exceptional set.

12 pages; accepted version incorporating minor edits