Number Theory

    

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

Authors: Madhukar Jadhav

A perfect number is a positive integer N that equals the sum of its proper divisors. While even perfect numbers have been classified, theexistence of an odd perfect number remains an unsolved problem. In this paper, we establish a fundamental contradiction in the divisor structure of any hypothetical odd perfect number. Specifically, we demonstrate that the largest proper divisor must be half of N, but for an odd N, this results in a non-integer, violating the necessary conditions for perfection. Consequently, we conclude that no odd perfect number can exist.

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[v1] 2025-04-01 21:35:51

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