What was already known
Van Doorn, Weisenberg and Cambie had made substantial progress and supplied a preliminary sieve lemma. Their result did not reach Sawhney’s exact extremal density and structure.
AI involvement
ChatGPT 5 Pro suggested the retained sieve lemma and b-dependent exclusions. Sawhney checked the lemma, found counterexamples to stronger model claims and completed the extremal proof.
The model supplied a retained additional sieve lemma and a useful cross-sieving direction; Sawhney completed and corrected the argument.
Importance
The result settles the asymptotic extremal size and structure in this named number-theory problem.
Limitations
Sawhney rejected incorrect model attempts to rule out two positive-density residue classes; the final cross-sieving and numerical bounds are his work.
Technical verification
Sawhney gives a complete human proof of Lemma 2.2 and Proposition 1.1, explicitly tests and rejects the model's false branches, and records the numerical constants used in the final bound. This is source-reported author checking of the assessed AI lemma and its role, not an independent AIProofWatch reproduction.
Formal checking
Written proof, correction of model counterarguments using a modulo-169 example, and explicit numerical inequality review
- Checker
- Mehtaab Sawhney, author
- Scope
- The retained Lemma 2.2 sieve and the resulting extremal bound in Proposition 1.1
- Basis
- Reported by the linked checker
Checking a formal statement does not establish that it matches the original problem.
Human understanding
Sawhney explains why b-dependent exclusions make the sieve work and how a point in the alternate class bounds both residue classes, including why the model's stronger claim failed.
Correction and publication history
The result settles the asymptotic extremal size and structure in this named number-theory problem.
Read this published revision