Schinzel and Wójcik have shown that if α, β are rational numbers not 0 or ± 1, then ord_p(α)=ord_p(β) for infinitely many primes p, where ord_p() denotes the order in F_p^x. We begin by asking: When are there infinitely many primes p with ord_p(α) > ord_p(β)? We write down several families of pairs α,β 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 α, β ∈ O_K are nonzero and neither is a root of unity, then there are infinitely many maximal ideals P of O_K for which ord_P(α) = ord_P(β).

The first author (M.J.) was supported by the UGA Algebraic Geometry, Algebra, and Number Theory RTG grant, NSF award DMS-1344994. The second author (P.P.) was supported by NSF award DMS-2001581. We thank Michael Filaseta, Pieter Moree, Carl Pomerance, Enrique Trevino, and the referee for helpful comments. We are also grateful to MathOverflow user "Hhhhhhhhhhh" for the post which brought this question to our attention [Hhh20].