Combinatorics and Graph Theory

    

Circuit Theoretic Proof of the Non-Existence of Odd Perfect Numbers & Their Generalization

Authors: Christopher C. Mbakwe

This paper presents a novel proof of the non-existence of odd perfect numbers using the framework of algebraic circuit theory and spectral graph theory. We construct a specialized resistive network, Γm(n), where the topology is uniquely determined by the divisor structure of an integer n. By embedding the arithmetic properties of the sum-of-divisors function σ(n) into the Kirchhoff Laplacian L(Γ),we demonstrate that the potential distribution of the network satisfies a discrete harmonic extension if and only if n satisfies specific divisor identities. We then generalize the result to odd k perfect numbers for k > 1.

Comments: 32 Pages.

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Submission history

[v1] 2026-01-13 23:03:40
[v2] 2026-03-13 17:56:54
[v3] 2026-04-04 00:24:39

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