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Ross's Two Conjectures on Nondeficient Numbers

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

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number-theoryJul 13, 2026Significance 7/100Registry: unreviewed

Ross's Two Conjectures on Nondeficient Numbers

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

Source authenticatedRoss's Two Conjectures on Nondeficient Numbers →
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Ross introduced $\mathcal{S}$-perfect numbers, integers expressible as $1 + \sum \lambda_j d_j$ over their proper divisors with coefficients in $\mathcal{S}$, and conjectured that they have the same density as the nondeficient numbers, plus a second conjecture relating odd nondeficient numbers to $\mathcal{S}$-perfection. Both are false.

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Ross's Two Conjectures on Nondeficient Numbers — Mathematical Frontier Network