 

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, SchinzelWójcik problem, antielite prime, antielite number, orderdominant pair. 
Subject: 
Primary 11A07, 11R11; Secondary 11A15. 


Abstract
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).


Acknowledgements
The first author (M.J.) was supported by the UGA Algebraic Geometry, Algebra, and Number Theory RTG grant, NSF award DMS1344994. The second author (P.P.) was supported by NSF award DMS2001581. 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
justmatt@uga.edu
Paul Pollack:
Department of Mathematics
University of Georgia
Athens, GA 30602, USA
pollack@uga.edu

