What was already known
Erdős and Rényi proved an upper bound of log₂N+O(log log N). Erdős and Hall ruled out an o(log log log N) error but left the wider o(log log N) proposal open.
AI involvement
The authors attribute the qualitative proof and its repair to GPT-5.2 Pro after human task setting and a request to fix the gap. Tang checked and rewrote the argument; Ma and Tang developed the quantitative strengthening.
The v1 paper explicitly attributes the original qualitative proof and its repair to GPT-5.2 Pro, with no human mathematical step beyond posing the task and requesting a fix. This label does not extend to the human quantitative refinement.
Importance
The qualitative model-origin disproof settles Erdős problem 543; the sharper constant belongs to Ma and Tang.
Limitations
The first model proof had a serious gap that was identified and repaired in a second prompted session. Original chat links expired; attribution rests on the authors’ process account and Tang’s corrected note.
Technical verification
Tang's corrected note reports a step-by-step human repair and check of the qualitative proof, and Ma and Tang report independent verification of the final paper's statements and proofs. This is a source-reported completed human proof check, not a check reproduced here.
Formal checking
Step-by-step correction and independent author proof verification
- Checker
- Quanyu Tang; Jie Ma and Quanyu Tang
- Scope
- Qualitative prime-cyclic counterexample to the o(log log N) bound, as retained after correction
- Basis
- Reported by the linked checker
Checking a formal statement does not establish that it matches the original problem.
Human understanding
Tang's corrected note and Ma–Tang's proof overview explain the missed-subset-sum second-moment mechanism and distinguish it from their stronger quantitative argument.
Correction and publication history
The qualitative model-origin disproof settles Erdős problem 543; the sharper constant belongs to Ma and Tang.
Read this published revision