A note on solitary numbers

Sagar Mandal
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 31, 2025, Number 3, Pages 617–623
DOI: 10.7546/nntdm.2025.31.3.617-623
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Authors and affiliations

Sagar Mandal
Department of Mathematics and Statistics, Indian Institute of Technology Kanpur
Kalyanpur, Kanpur, Uttar Pradesh 208016, India

Abstract

Does 14 have a friend? Until now, this has been an open question. In this note, we prove that a potential friend F of 14 is an odd, non-square positive integer. 7 appears in the prime factorization of F with an even exponent while at most two prime divisors of F can have odd exponents in the prime factorization of F. If p | F such that p is congruent to 7 modulo 8, then p^{2a} || F, for some positive integer a. Further, no prime divisor of F has an exponent congruent to 7 modulo 8 and no prime divisor can exceed 1.4\sqrt{F}. The primes 3, 5 cannot appear simultaneously in the prime factorization of F. If (3,F) > 1 or (5,F) > 1, then \omega(F) \geq 4, otherwise \omega(F)\geq 8.

Keywords

  • Abundancy index
  • Sum of divisors
  • Friendly numbers
  • Solitary numbers

2020 Mathematics Subject Classification

  • 11A25

References

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Manuscript history

  • Received: 21 February 2024
  • Revised: 23 August 2025
  • Accepted: 4 September 2025
  • Online First: 16 September 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

Mandal, S. (2025). A note on solitary numbers. Notes on Number Theory and Discrete Mathematics, 31(3), 617-623, DOI: 10.7546/nntdm.2025.31.3.617-623.

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