A Note on Odd Perfect Numbers

For over two millennia, the question of whether odd perfect numbers---positive integers whose proper divisors sum to the number itself---exist has captivated mathematicians, from Euclid's elegant construction of even perfect numbers via Mersenne primes to Euler's probing of their odd counterparts. This paper makes significant progress on this ancient conjecture by presenting a rigorous proof by contradiction that odd perfect numbers not divisible by 3 cannot exist. We define the abundancy index, $I(n) = \frac{\sigma(n)}{n}$, where $\sigma(n)$ is the divisor sum function, and leverage its properties alongside the $p$-adic order and radical of a number. Assuming the existence of an odd perfect number $N$ not divisible by $3$, with $I(N) = 2$, we apply a novel lemma to express $I(N)$ as a product over its prime factors. The proof leverages deep connections between analytic number theory (zeta function bounds) and multiplicative properties of divisors (abundancy indices), demonstrating the power of combining these tools to resolve classical conjectures. The assumption that an odd perfect number avoids divisibility by 3 yields a contradiction.