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 $alpha, beta in mathbb{Q}^{times}setminus{pm 1}$ for which $mathrm{ord}_p(alpha) > mathrm{ord}_p(beta)$ holds for infinitely many primes $p$. They called such pairs order-dominant. We describe an easily-checkable sufficient condition for $alpha,beta$ to be order-dominant. Via the large sieve, we show that almost all integer pairs $alpha,beta$ satisfy our condition, with a power savings on the size of the exceptional set.
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