Claims

../

2026_01_06_barreto: For every C and small epsilon, infinitely many triples with a and b between epsilon n and (1 minus epsilon) n have a plus b minus n of order C log n and a! b! dividing n! (a+b-n)!; an argument of GPT-5.2 formalized by Aristotle.

2026_01_14_pomerance: For almost all m the binomial coefficient m+k choose k divides 2m choose m for every k up to exp(0.8 sqrt(log m)), which gives the triples the problem asks for with a gap far beyond C log n; published in Integers (2026).

2026_07_13_pickhardt: Two manuscripts by the Paratelligent Research Agent and Jeff Pickhardt, posted to the site's thread after the problem was settled, each claiming a deterministic proof of the affirmative answer with any constant times log n.

---