Note for the Perfect Numbers

This paper tackles a longstanding problem in number theory: the infinitude of perfect numbers. A perfect number is defined as a positive integer whose sum of all its divisors is equal to twice the number itself. While Euclid's method provides a framework for constructing even perfect numbers using Mersenne primes, the infinitude of Mersenne primes remains an open question. If there are finitely many Mersenne primes, then there would also be a finite number of even perfect numbers. In this note, showing that there are finitely many Mersenne primes, we provide a partial answer by proving that is false the infinitude of even perfect numbers. The proof utilizes elementary techniques and relies on properties of the divisor sum function (sigma function) and the inherent structure of prime numbers.

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Title: ScienceOpen Preprints

Publisher: ScienceOpen

Publication date (Electronic preprint): 20 April 2024

Affiliations

[1 ] GROUPS PLUS TOURS INC., 9611 Fontainebleau Blvd, Miami, FL, 33172, USA;

Author notes

[* ]Email: vega.frank@ 123456gmail.com .

Author information

Frank Vega https://orcid.org/0000-0001-8210-4126

Article

DOI: 10.14293/PR2199.000821.v1

SO-VID: bdcd3b02-3608-4697-ae32-9f41e05df3c8

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This work has been published open access under Creative Commons Attribution License CC BY 4.0 , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Conditions, terms of use and publishing policy can be found at www.scienceopen.com .

History

Date received : 20 April 2024

References

Bateman P. T., Selfridge J. L., Wagstaff S. S.. The Editor's Corner: The New Mersenne Conjecture. The American Mathematical Monthly. Vol. 96(2)1989. JSTOR. [Cross Ref]

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[1] Bateman P. T., Selfridge J. L., Wagstaff S. S.. The Editor's Corner: The New Mersenne Conjecture. The American Mathematical Monthly. Vol. 96(2)1989. JSTOR. [Cross Ref]