The problem
Erdős #728 asks for which k the expression n(n+1)⋯(n+k−1) divides a related product of shifted factorials for all n — one of hundreds of concrete number-theoretic questions Erdős scattered through his papers.
Erdős Problem #728 — the first fully autonomous LLM resolution
AI-resolvedReferences
History & significance
Kevin Barreto (Cambridge undergraduate) and Liam Price spent December 2025 throwing batches of Bloom's Erdős Problems database at GPT-5.2, having learned that models succeed when prompted as though the problem is easier than advertised. Their Christmas-day "first" on #333 turned out to be already solved in the literature — the cautionary tale of the entire field.
The resolution (solved by AI)
Solved 4 January 2026 by Barreto and Price with GPT-5.2 Pro, then logically certified by Harmonic's Aristotle system and formalised by ChatGPT. Nobody could find prior art: the first verified fully-autonomous LLM resolution of a genuinely-open Erdős problem. Within weeks Google DeepMind's Aletheia agent had swept 700 open Erdős problems (December 2025 evaluation, published January 2026), autonomously solving several including #1051 — and by May 2026 DeepMind reported nine of 353 formalised open problems resolved at a few hundred dollars each. The dam had broken.