A Fundamental Contradiction Proving the Non-Existence of Odd Perfect Numbers

A perfect number is a positive integer N that equals the sum of its properdivisors. While even perfect numbers have been classified, the existence of an oddperfect number remains an open question in number theory. In this paper, wepresent a contradiction that arises in the divisor structure of any hypothetical oddperfect number. Specifically, we demonstrate that the largest proper divisor mustbe N/2, which cannot be an integer for an odd N. This contradiction invalidatesthe necessary conditions for a perfect number, thereby proving that no odd perfect number can exist.