On the special cases of Carmichael’s totient conjecture

Anthony G. Shannon, Peter J.-S. Shiue, Tian-Xiao He, and Christopher Saito
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 31, 2025, Number 3, Pages 504–534
DOI: 10.7546/nntdm.2025.31.3.504-534
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Authors and affiliations

Anthony G. Shannon
Warrane College, The University of New South Wales
Kensington, NSW 2033, Australia

Peter J.-S. Shiue
Department of Mathematical Sciences, University of Nevada, Las Vegas
Las Vegas, Nevada, 89154-4020, USA

Tian-Xiao He
Department of Mathematics, Illinois Wesleyan University
Bloomington, Illinois 61702, USA

Christopher Saito
Department of Mathematical Sciences, University of Nevada, Las Vegas
Las Vegas, Nevada, 89154-4020, USA

Abstract

Euler’s totient function, \varphi(n), is the arithmetic function defined as the number of positive integers less than or equal to n that are relatively prime to n. In his 1922 paper [3], Professor R. D. Carmichael conjectured that for each positive integer n, there exists at least one positive integer m \neq n such that \varphi(m) = \varphi(n).

In this paper, we consider some relevant literature and explore Carmichael’s totient conjecture for particular values of \varphi(n)=k. Our main consideration will be the set X_k=\left\{n\in\mathbb{N}:\varphi(n)=k\right\}. In identifying X_k for k=2^t, 2p^s, 2^2p, and 2pq, where p and q are distinct prime numbers, we find that Carmichael’s conjecture holds for those select cases, provide an algorithm, and some related results. The conjecture remains an open problem in number theory [9].

Keywords

  • Carmichael Conjecture
  • Euler totient function
  • Fermat chain
  • Fermat primes
  • Fibonacci numbers
  • Germain primes
  • Integer components
  • Primitive prime divisors

2020 Mathematics Subject Classification

  • 11A07
  • 11Y11

References

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Manuscript history

  • Received: 19 May 2025
  • Revised: 28 July 2025
  • Accepted: 4 August 2025
  • Online First: 15 August 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

Shannon, A. G., Shiue, P. J.-S., He, T.-X., & Saito, C. (2025). On the special cases of Carmichael’s totient conjecture. Notes on Number Theory and Discrete Mathematics, 31(3), 504-534, DOI: 10.7546/nntdm.2025.31.3.504-534.

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