The congruence xn ≡ – an (mod m): Solvability and related OEIS sequences

Jorma K. Merikoski, Pentti Haukkanen, Timo Tossavainen
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 30, 2024, Number 3, Pages 516–529
DOI: 10.7546/nntdm.2024.30.3.516-529
Full paper (PDF, 223 Kb)

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Authors and affiliations

Jorma K. Merikoski
Faculty of Information Technology and Communication Sciences, Tampere University
FI-33014 Tampere, Finland

Pentti Haukkanen
Faculty of Information Technology and Communication Sciences, Tampere University
FI-33014 Tampere, Finland

Timo Tossavainen
Department of Arts, Communication and Education, Lulea University of Technology
SE-97187 Lulea, Sweden

Abstract

We study the solvability of the congruence x^n\equiv -a^n\pmod{m}, where n,m\in\mathbb{Z}_+, a\in\mathbb{Z}, and \gcd{(a,m)}=1. Our motivation arises from computer experiments concerning a geometric property of the roots of the congruence x^n+y^n\equiv 0\pmod{p}, where n\in\mathbb{Z}_+ and p\in\mathbb{P}. We encounter several OEIS sequences. We also make new observations on some of them.

Keywords

  • Congruence of powers
  • Integer sequence
  • Experimental geometry

2020 Mathematics Subject Classification

  • 11A07
  • 11B83
  • 11Y99
  • 51M04

References

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Manuscript history

  • Received: 24 May 2023
  • Revised: 15 September 2024
  • Accepted: 29 September 2024
  • Online First: 30 September 2024

Copyright information

Ⓒ 2024 by the Authors.
This is an Open Access paper distributed under the terms and conditions of the Creative Commons Attribution 4.0 International License (CC BY 4.0).

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Cite this paper

Merikoski, J. K., Haukkanen, P., & Tossavainen, T. (2024). The congruence xn ≡ – an (mod m): Solvability and related OEIS sequences. Notes on Number Theory and Discrete Mathematics, 30(3), 516-529, DOI: 10.7546/nntdm.2024.30.3.516-529.

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