A class of solutions of the equation d(n2) = d(φ(n))

Zahra Amroune, Djamel Bellaouar and Abdelmadjid Boudaoud
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 29, 2023, Number 2, Pages 284–309
DOI: 10.7546/nntdm.2023.29.2.284-309
Full paper (PDF, 309 Kb)

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Authors and affiliations

Zahra Amroune
Laboratory of Pure and Applied Mathematics (LMPA),
University of M’sila, B.P. 166, Ichbilia, 28000 M’sila, Algeria

Djamel Bellaouar
Department of Mathematics, University 08 Mai 1945 Guelma,
B.P. 401 Guelma 24000, Algeria

Abdelmadjid Boudaoud
Laboratory of Pure and Applied Mathematics (LMPA),
University of M’sila, B.P. 166, Ichbilia, 28000 M’sila, Algeria

Abstract

For any positive integer n let d\left( n\right) and \varphi \left( n\right) be the number of divisors of n and the Euler’s phi function of n, respectively. In this paper we present some notes on the equation d\left( n^{2}\right) =d\left( \varphi \left( n\right) \right). In fact, we characterize a class of solutions that have at most three distinct prime factors. Moreover, we show that Dickson’s conjecture implies that d\left( n^{2}\right) =d\left( \varphi \left( n\right) \right) infinitely often.

Keywords

  • Diophantine equations
  • Euler’s phi function
  • Divisor function

2020 Mathematics Subject Classification

  • 11A25
  • 11A41
  • 11D99

References

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Manuscript history

  • Received: 27 July 2022
  • Revised: 29 March 2023
  • Accepted: 26 April 2023
  • Online First: 2 May 2023

Copyright information

Ⓒ 2023 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

Amroune, Z., Bellaouar, D., & Boudaoud, A. (2023). A class of solutions of the equation d(n2) = d(φ(n)). Notes on Number Theory and Discrete Mathematics, 29(2), 284-309, DOI: 10.7546/nntdm.2023.29.2.284-309.

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