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Note for the Perfect Numbers
Revision notes
We improved the last theorem. Certainly, it was wrong the argument that supports that we can take the number 2 as upper bound at the end of the last theorem. However, we found a beauty and elegant way to fix this issue and so, we believe this is the closest version to the final solution.
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.
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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]