A Note on Odd Perfect Numbers

For over two millennia, the question of whether odd perfect numbers---positive integers whose divisors sum to twice the number itself---exist has intrigued mathematicians, from Euclid's construction of even perfect numbers using Mersenne primes to Euler's exploration of potential odd counterparts. This paper resolves this enduring conjecture by proving, through a rigorous proof by contradiction, that odd perfect numbers do not exist. We utilize the abundancy index, defined as $I(n) = \frac{\sigma(n)}{n}$, where $\sigma(n)$ is the sum of the divisors of $n$, and the Euler totient function $\varphi(n)$. Assuming an odd perfect number $N$ exists with $I(N) = 2$, we employ the inequality $\frac{\sigma(N) \cdot \varphi(N)}{N^2} > \frac{8}{\pi^2}$ for odd $N$ and establish that $\frac{N}{\varphi(N)} \geq 3$ for odd perfect numbers with at least 10 distinct prime factors. This leads to a contradiction, as $\frac{N}{\varphi(N)}$ cannot be less than $\frac{\pi^2}{4} \approx 2.4674$ while being at least 3. Rooted in elementary number theory, this proof combines classical techniques with precise analytical bounds to confirm that all perfect numbers are even, resolving a historic problem in number theory.