On the equation F(n^k - 1) = D

I. Kátai, B. M. M. Khanh, B. M. Phong
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 31, 2025, Number 4, Pages 718–727
DOI: 10.7546/nntdm.2025.31.4.718-727
Full paper (PDF, 225 Kb)

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Authors and affiliations

I. Kátai
Department of Computer Algebra, University of Eötvös Loránd
1117 Budapest, Hungary

B. M. M. Khanh
Department of Computer Algebra, University of Eötvös Loránd
1117 Budapest, Hungary

B. M. Phong
Department of Computer Algebra, University of Eötvös Loránd
1117 Budapest, Hungary

Abstract

We prove that if F is a completely multiplicative function and k \in \{2, 3\} such that the equation F(n^k - 1) = 1 holds for every n \in \mathbb{N}, n > 1, then F is the identity function. A similar result is proved for the equation F(n^4 - 1) = 1 assuming a suitable conjecture concerning prime numbers. The equation F(n^3 + 1) = 1 is also studied.

Keywords

  • Completely multiplicative function
  • Identity function
  • Functional equation
  • Dirichlet character

2020 Mathematics Subject Classification

  • 11A07
  • 11A25
  • 11N25
  • 11N64

References

  1. Kátai, I., Khanh, B. M. M., & Phong, B. M. (2023). On the equation F(n^3)=F(n^3-1)+D and some conjectures. Publicationes Mathematicae Debrecen, 103(1–2), 257–267.
  2. Kátai, I., Khanh, B. M. M., & Phong, B. M. (2025). On the equation F(n^2 + 1) = bF(n^2 - 1)+ c. Annales Universitatis Scientiarum Budapestinensis de Rolando Eötvös Nominatae. Sectio Computatorica, 58, 177–189.

Manuscript history

  • Received: 12 June 2025
  • Revised: 28 September 2025
  • Accepted: 1 October 2025
  • Online First: 12 October 2025

Copyright information

Ⓒ 2025 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

Kátai, I., Khanh, B. M. M., & Phong, B. M. (2025). On the equation F(n^k - 1) = D. Notes on Number Theory and Discrete Mathematics, 31(4), 718-727, DOI: 10.7546/nntdm.2025.31.4.718-727.

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