Partitions of numbers and the algebraic principle of Mersenne, Fermat and even perfect numbers

A. M. S. Ramasamy
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 30, 2024, Number 4, Pages 755–775
DOI: 10.7546/nntdm.2024.30.4.755-775
Full paper (PDF, 305 Kb)

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Authors and affiliations

A. M. S. Ramasamy
Department of Mathematics, Pondicherry University
Pondicherry – 605014, India

Abstract

Let ρ be an odd prime greater than or equal to 11. In a previous work, starting from an M-cycle in a finite field 𝔽_ρ, it has been established how the divisors of Mersenne, Fermat and Lehmer numbers arise. The converse question has been taken up in a succeeding work and starting with a factor of these numbers, a method has been provided to find an odd prime ρ and the M-cycle in 𝔽_ρ contributing the factor under consideration. Continuing the study of the two previous works, a certain type of partition of a natural number is considered in the present paper. Concerning the Mersenne, Fermat and even perfect numbers, the algebraic principle is established.

Keywords

• Partition
• Different kinds of M-cycles
• The functions T and U
• Invariants of a natural number
• Tests of primality of Mersenne and Fermat numbers

2020 Mathematics Subject Classification

• 11A51
• 11B50
• 11P81
• 11T06

References

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Manuscript history

• Received: 16 June 2024
• Revised: 13 November 2024
• Accepted: 14 November 2024
• Online First: 14 November 2024

Copyright information

Ⓒ 2024 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).