Proofs of some geometric conjectures on the power sum congruence modulo a prime

Pentti Haukkanen
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 31, 2025, Number 4, Pages 776–784
DOI: 10.7546/nntdm.2025.31.4.776-784
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Authors and affiliations

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

Abstract

The main purpose of this paper is to verify the geometric conjectures of Mustonen (2022) concerning the solutions and the number of solutions of the congruence

    \[x^n+y^n \equiv 0 \pmod{p},\]

where p is a prime. For p>2, the nontrivial solutions lie on the “lines” y \equiv cx \pmod{p}, where c ranges over the n-th roots of -1 modulo p. The total number of solutions is 1+(p-1)d if d divides (p-1)/2, and 0 otherwise, where d=\gcd(n, p-1). For each c, the lines are equally spaced.

Keywords

  • Congruence of powers
  • Experimental geometry
  • Power residue
  • Primitive root
  • Cyclic group

2020 Mathematics Subject Classification

  • 11A07
  • 11A15
  • 11Y99
  • 51M04

References

  1. Apostol, T. M. (1986). Introduction to Analytic Number Theory (3rd printing), Springer.
  2. Lang, S. (2002). Algebra (Revised 3rd edition), Graduate Texts in Mathematics Series, Vol. 211, Springer.
  3. 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.
  4. Mustonen, S. (1992). Survo: An Integrated Environment for Statistical Computing and Related Areas. Survo Systems. Available online at: https://www.survo.fi/kirjat/index.html.
  5. Mustonen, S. (2022). Diophantine equations Xn + Yn ≡ 0 (mod P). Additional results and graphical presentations. Available online at: https://www.survo.fi/papers/Dioph2022.pdf.

Manuscript history

  • Received: 15 September 2025
  • Revised: 29 October 2025
  • Accepted: 2 November 2025
  • Online First: 5 November 2025

Copyright information

Ⓒ 2025 by the Author.
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

Haukkanen, P. (2025). Proofs of some geometric conjectures on the power sum congruence modulo a prime. Notes on Number Theory and Discrete Mathematics, 31(4), 776-784, DOI: 10.7546/nntdm.2025.31.4.776-784.

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