The Banks-Martin Conjecture on Primitive Sets

Precise statement

Banks and Martin conjectured in 2013 that for a primitive set $A$ and any set $Q$ of primes, the Erdos sum of the members of $A$ composed only of primes in $Q$ is at most the corresponding sum over $Q$ itself. The unrestricted form turned out to be false once $Q$ is allowed to contain $2$; 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.

The source statement is reproduced for indexing with attribution. Mathematical correctness requires domain-expert or mechanical review. VibeMath has not independently audited statement fidelity, correctness, priority, or novelty.

What AI did

GPT-5.5 Pro (early version)

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.

Provider: OpenAI · Prompt public: unknown · Independence: unknown

Verification boundary

unreviewed

arXiv preprint, not peer-reviewed.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.

Correctness: unknown · statement fidelity: unaudited · peer review: none

Timeline

1. 2026-05-01

   arXiv:2605.00301 - Primitive sets and von Mangoldt chains: Erdos Problem #1196 and beyond

   VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.

Known method families

argument (source-reported)

Source-reported tools: argument.

Independent: unknown · difference confidence: 0

What remains uncertain

VibeMath has not independently audited the mathematical statement, proof, or novelty claim.

• VibeMath has not independently verified the mathematical claim.
• AI-attempt independence and training-data exposure are unknown unless explicitly documented.
• VibeMath has not independently audited the mathematical statement, proof, or novelty claim.