On a conjecture involving twin practical numbers

Runbo Li
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 32, 2026, Number 3, Pages 544–555
DOI: 10.7546/nntdm.2026.32.3.544-555
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Authors and affiliations

Runbo Li
International Curriculum Center
Beijing, China

Abstract

In this paper, we state a conjecture that for any positive integer p, there are infinitely many practical numbers q such that both q^p and q^p + 2 are practical numbers. We prove the conjecture when p is a prime number between 7 and 100, refining a result involving twin practical numbers of Wang and Sun. We also prove several results related to this conjecture.

Keywords

  • Practical numbers
  • Cyclotomic polynomials

2020 Mathematics Subject Classification

  • 11A25
  • 11B83

References

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Manuscript history

  • Received: 29 January 2026
  • Revised: 19 October 2026
  • Accepted: 20 October 2026
  • Online First: 21 October 2026

Copyright information

Ⓒ 2026 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

Li, R. (2026). On a conjecture involving twin practical numbers. Notes on Number Theory and Discrete Mathematics, 32(3), 544-555, DOI: 10.7546/nntdm.2026.32.3.544-555.

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