Number Theory

    

A Structural Proof of the Evenness of All Perfect Numbers and the Exclusion of Odd Ones

Authors: Younghwan Yun

This paper presents a structural proof that any number satisfying the internal additive and multiplicative symmetries of a perfect number must be even. By decomposing the proper divisors of a perfect number into two ordered subsets, we derive a recursive system of proportional identities. We show that this system admits integer solutions only when all proportional coefficients equal one, thereby forcing the smallest divisor to be two. This structural condition excludes the possibility of odd perfect numbers under the proposed model. Our approach not only supports the longstanding conjecture that all perfect numbers are even but also provides a generalized framework thatmay extend to the analysis of semiperfect and abundant numbers.

Comments: 7 Pages.

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Submission history

[v1] 2025-05-31 20:18:48
[v2] 2025-06-08 21:28:17

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