2021/01/31 by Matthew Just, Paul Pollack, Just, Matthew +1
Mathematics · #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Combinatorics #FOS: Mathematics #Field (mathematics) #Mathematics #Meromorphic and Entire Functions #Number Theory (math.NT) #Order (exchange) #Prime (order theory) #Primitive root modulo n #Pure mathematics #math.NT
paper · pdf · doi:10.48550/arxiv.2102.00370
13 pages
arxiv created 2021/01/31 · openalex publication_date 2021/01/31 · arxiv updated 2021/02/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 \mathbbFp×. 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).