A problem in comparative order theory

Abstract: Write

m a t h r m {o r d}_{p} (c d o t)

for the multiplicative order in

m a t h b b F_{p}^{i m e s}

. Recently, Matthew Just and the second author investigated the problem of classifying pairs

for which

holds for infinitely many primes

p

. They called such pairs order-dominant. We describe an easily-checkable sufficient condition for

to be order-dominant. Via the large sieve, we show that almost all integer pairs

satisfy our condition, with a power savings on the size of the exceptional set.

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