New York Journal of Mathematics
Volume 27 (2021), 600-614


Matthew Just and Paul Pollack

Comparing multiplicative orders mod p, as p varies

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Published: April 12, 2021.
Keywords: multiplicative order, support problem, Schinzel--Wójcik problem, anti-elite prime, anti-elite number, order-dominant pair.
Subject: Primary 11A07, 11R11; Secondary 11A15.

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).


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].

Author information

Matthew Just:
Department of Mathematics
University of Georgia
Athens, GA 30602, USA


Paul Pollack:
Department of Mathematics
University of Georgia
Athens, GA 30602, USA