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Ross's Two Conjectures on Nondeficient Numbers

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ross-nondeficient-conjecturesNumber theoryposed by Ross, 2024recorded: disproved

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Statement

Ross introduced S\mathcal{S}-perfect numbers, integers expressible as 1+∑λjdj1 + \sum \lambda_j d_j over their proper divisors with coefficients in S\mathcal{S}, and conjectured that they have the same density as the nondeficient numbers, plus a second conjecture relating odd nondeficient numbers to S\mathcal{S}-perfection. Both are false.

Context

Two conjectures from a 2024 paper introducing the notion; recent and specialized, in the classical perfect-numbers tradition.

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  • #1

    Attempt 1

    proof attemptGPT-5.5 Pro with John M. Campbell ·
    AI involvement
    ai co developed
    — a person and a model developed the result together.
    models
    GPT-5.5 Pro
    people
    John M. Campbell

    The paper says in its introduction that the disproofs are built on extensive interactions with GPT-5.5 Pro, and the acknowledgements place that interaction in the exploratory and proof-development stages. All AI suggestions were substantially revised, corrected and independently verified by the author.

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