Comparing multiplicative orders mod p, as p varies

摘要

Schinzel and Wójcik have shown that if α, β are rational numbers not 0 or ± 1, then ordp(α)=ordp(β) for infinitely many primes p, where ordp() denotes the order in Fpx. We begin by asking: When are there infinitely many primes p with ordp(α) > ordp(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 ∈ OK are nonzero and neither is a root of unity, then there are infinitely many maximal ideals P of OK for which ordP(α) = ordP(β).