On the Nonexistence of Odd Perfect Numbers

We present a proof demonstrating the nonexistence of odd perfect numbers. By applying modular arithmetic contradictions, structural factorizations, and survivor elimination arguments based on Euler's form we rigorously eliminate all possibilities for the existence of an odd perfect number. This result resolves a longstanding open problem in number theory, first posed over 2000 years ago. The method is elementary but comprehensive, providing a complete contradiction without assuming unproven conjectures.