## Equality of orders of a set of integers modulo a prime

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- by Olli Järviniemi PDF
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## Abstract:

For finitely generated subgroups $W_1, \ldots , W_t$ of $\mathbb {Q}^{\times }$, integers $k_1, \ldots , k_t$, a Galois extension $F$ of $\mathbb {Q}$ and a union of conjugacy classes $C \subset \mathrm {Gal}(F/\mathbb {Q})$, we develop methods for determining if there exist infinitely many primes $p$ such that the index of the reduction of $W_i$ modulo $p$ divides $k_i$ and the Artin symbol of $p$ on $F$ is contained in $C$. The results are a multivariable generalization of H.W. Lenstra’s work. We present several applications, including a characterization of all integers $a_1, \ldots , a_n$ such that $\mathrm {ord}_p(a_1) = \ldots = \mathrm {ord}_p(a_n)$ for infinitely many primes $p$. The obtained results are conditional to a generalization of the Riemann hypothesis.## References

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## Additional Information

**Olli Järviniemi**- Affiliation: Department of Mathematics and Statistics, P.O. Box 68, 00014 Helsinki, Finland
- Email: olli.jarviniemi@helsinki.fi
- Received by editor(s): December 12, 2019
- Received by editor(s) in revised form: June 12, 2020, October 14, 2020, and January 8, 2021
- Published electronically: June 25, 2021
- Communicated by: Matthew Papanikolas
- © Copyright 2021 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**149**(2021), 3651-3668 - MSC (2020): Primary 11R45, 11R32
- DOI: https://doi.org/10.1090/proc/15498
- MathSciNet review: 4291567