Note for the Mersenne Primes

This paper tackles a longstanding problem in number theory: the existence of odd perfect numbers. A perfect number is defined as a positive integer whose sum of all its proper divisors (excluding itself) is equal to twice the number itself. While Euclid demonstrated a method to construct even perfect numbers using Mersenne primes (primes of the form $2^{n} - 1$), the existence of odd perfect numbers remained an open question. In this note, under the assumption that there are infinitely many Mersenne primes, we provide an intuitive answer by proving the non-existence of odd perfect numbers. The proof utilizes elementary techniques and relies on properties of the divisor sum function (sigma function) and the inherent structure of odd perfect numbers.