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 resolves this ancient conjecture through a rigorous proof by contradiction, demonstrating that odd perfect numbers are impossible. 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 a smallest odd perfect number $N$, with $I(N) = 2$, we apply a novel lemma to express $I(N)$ as a product over its prime factors. Constraints from established results, including the requirement of at least 10 distinct prime factors and a bound on their reciprocal sum, enable us to derive a scaled inequality. By meticulously bounding the terms of this product expansion, we show that the sum falls short of the necessary threshold for $k \geq 10$ prime factors, yielding a contradiction. This proof, grounded in elementary number theory yet profound in its implications, not only settles a historic problem but also underscores the power of combining classical techniques with precise analytical bounds to unravel deep mathematical mysteries. Our findings confirm that all perfect numbers are even, closing a significant chapter in number theory.