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.

Result: Disproved
Status: Resolved
AI contribution: AI co-developed
Method: Argument
Field: Number theory
Posed by: Ross
Year posed: 2024
Years open: 2y
Solved: 2026-07-13
Model: GPT-5.5 Pro
Vendor: OpenAI
Collaborators: John M. Campbell
Verification: Unreviewed
Publication: Preprint
Significance: 10 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What the AI did

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.

Verification

Single-author arXiv preprint; not yet peer-reviewed.

Source

arXiv:2607.11043 - Disproofs of two conjectures concerning nondeficient numbers

Ross's Two Conjectures on Nondeficient Numbers

What the AI did

Verification

Source

Discussion