The Banks-Martin Conjecture on Primitive Sets
Banks and Martin conjectured in 2013 that for a primitive set and any set of primes, the Erdos sum of the members of composed only of primes in is at most the corresponding sum over itself. The unrestricted form turned out to be false once is allowed to contain ; Lichtman proposed a revised form restricted to odd primes. That revised conjecture, long viewed as a unifying master theorem for the area, is proved here.
- Result
- Proved
- Status
- Resolved
- AI contribution
- AI co-developed
- Method
- Argument
- Field
- Number theory
- Posed by
- William D. Banks, Greg Martin; revised form proposed by Jared Duker Lichtman
- Year posed
- 2013
- Years open
- 13y
- Solved
- 2026-05-01
- Model
- GPT-5.5 Pro (early version)
- Vendor
- OpenAI
- Collaborators
- Boris Alexeev, Kevin Barreto, Yanyang Li, Jared Duker Lichtman, Liam Price, Jibran Iqbal Shah, Quanyu Tang, Terence Tao
- Verification
- Unreviewed
- Publication
- Preprint
- Significance
- 25 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What the AI did
This paper discloses per theorem rather than in a blanket statement, and this theorem is one of the more modest entries: an early version of GPT-5.5 Pro was used to assist with the initial proof. Elsewhere in the same paper the model's role is larger, with the proof of the Erdos #1196 theorem generated by an autonomous GPT-5.4 Pro run whose transcript is public. Across all of it the authors state that the final proofs were generated and reviewed by them, using the AI-generated proofs as starting points where appropriate. The whole method, Markov chains with von Mangoldt weights, was itself suggested by model output.
Verification
arXiv preprint, not peer-reviewed. Other results in the same paper were formalized in Lean using Codex and Gauss, but this theorem was not among them.
Source
arXiv:2605.00301 - Primitive sets and von Mangoldt chains: Erdos Problem #1196 and beyond
Submitted by Curator34