A divisibility question about shifted powers — the first Erdős problem genuinely resolved end-to-end by an LLM pipeline, certified by a second AI and checked by humans.
The Problems
Not schoolwork — the questions that resisted Erdős, Hilbert, and everyone since. The club keeps three shelves: what is still open, what AI recently settled, and what took humanity centuries.
Multicolour Triangle Ramsey Numbers (Erdős #183)
AI-resolved
How many colours force a monochromatic triangle? Erdős's fifty-year-old lower-bound challenge, resolved superexponentially by Astra.
Connes's Embedding/Rigidity Conjecture
AI-resolved
Are the hyperfinite II₁ factor's matrix approximations unique — must certain groups be visible in their von Neumann algebras? Disproved by OpenAI's Astra model, ending a fifty-year expectation.
Existence of Non-Sofic Groups
AI-resolved
Are all finitely generated groups approximable by finite ones? OpenAI's Astra model announces a construction proving not — resolving a central question of group theory.
Feige's Anticoncentration Conjecture
AI-resolved
Sums of small independent random variables resist concentrating too sharply. A decade-plus-old conjecture, initially proved by ChatGPT — and by two independent teams the same day.
Must a finite set of reals grow fast under addition or multiplication? Over the reals: no — a cascade begun by AI's unit-distance breakthrough. Over the integers the conjecture lives on.
The Cycle Double Cover Conjecture
AI-resolved
Can every bridgeless graph be covered by circuits traversing each edge exactly twice? Yes — with at most eight loops. Settled by GPT-5.6 Sol after fifty years.
The Jacobian Conjecture (three variables)
AI-resolved
If a polynomial self-map of space has everywhere-invertible derivative, must it be globally invertible? False in three variables: a 216-character counterexample ended an 87-year hunt.
Erdős's Unit Distance Problem
AI-resolved
How many pairs of n plane points can sit exactly one unit apart? Erdős guessed his grid was unbeatable. An AI model proved him wrong — the first prominent open problem resolved autonomously by AI.