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Note for the Perfect Numbers

EasyChair Preprint 13131, version 4

Versions: 1234history
6 pagesDate: July 15, 2024

Abstract

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.

Keyphrases: Mersenne primes, divisor sum function, perfect numbers, prime numbers

BibTeX entry
BibTeX does not have the right entry for preprints. This is a hack for producing the correct reference:
@booklet{EasyChair:13131,
  author    = {Frank Vega},
  title     = {Note for the Perfect Numbers},
  howpublished = {EasyChair Preprint 13131},
  year      = {EasyChair, 2024}}
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