How DeepMind’s AI Solved Nine Erdős Problems for Only a Few Hundred Dollars Each

Method

DeepMind introduced the AlphaProof Nexus framework, which combines the creative reasoning of large language models with the strict verification of the Lean theorem‑proving compiler.

Human mathematicians provide a sketch of a proof containing placeholders (marked with tags such as EVOLVE‑BLOCK or EVOLVE‑VALUE). The AI then takes over the entire proof development, handling high‑level strategy, detailed logical deductions, lemma generation, and parameter tuning.

Baseline agent: think‑try loop

The baseline architecture launches multiple stateless sub‑agents that run a multi‑turn interaction loop. The underlying model (Gemini 3.1 Pro) generates a reasoning chain, invokes search and replace tools to modify the proof sketch, and the Lean compiler instantly validates each change. Errors are fed back to the model for self‑correction, repeating until all proof gaps are filled.

Full‑featured agent: AlphaProof integration

The full‑featured design adds a second model (Gemini 3.0 Flash) that acts as a judge, assigning Elo scores for clarity, plausibility, and novelty to generated proof sketches. This evolutionary process selects the most promising candidates, and an auxiliary tool called AlphaProof, trained on reinforcement‑learning for competition‑level problems, assists the solving.

Although the full system was expected to dominate, analysis showed that the simple baseline agent also succeeded on all nine Erdős problems, suggesting that as base models become more capable, complex scaffolding may become unnecessary.

The Nine Erdős Problems Solved

Problem 12(i) – Dense set avoiding divisibility (1970) : AI constructed an infinite set where no element divides the sum of two larger elements while maintaining a prescribed lower density, using the Chinese Remainder Theorem and avoidance of specific arithmetic progressions.

Problem 12(ii) – Higher‑density limit (1970) : Extending the previous result, AI employed a Behrend‑style construction to achieve an ultra‑dense infinite set satisfying the non‑divisibility condition.

Problem 125 – Sum‑density of mixed‑base digit sets (1996) : By applying Diophantine approximation, AI proved that the lower density of the sum set of ternary {0,1} numbers and quaternary {0,1} numbers is zero.

Problem 138 (variant) – Gaps in Van der Waerden numbers (1981) : Using a greedy coloring extension algorithm with local conflict analysis, AI showed that required gaps between Van der Waerden numbers grow without bound as the target arithmetic‑progression length increases.

Problem 152 – Isolated points in Sidon sets (1994) : AI performed a detailed boundary analysis of internal points and shifted neighbors to prove the existence of many isolated points in the sum set of a large Sidon set.

Problem 741(i) – Sum‑density after set splitting (1994) : AI demonstrated that a set whose self‑sum has positive upper density can be partitioned into two subsets each retaining positive upper density in their self‑sums.

Problem 741(ii) – Splitting and gap limits (1994) : AI identified a special “second‑order base” set with a forbidden‑zone structure; any bipartition forces at least one part’s sum set to contain arbitrarily large gaps.

Problem 846 – Geometric paradox of planar point sets (1992) : AI proved the existence of an infinite planar point set where any finite subset contains a large proportion of non‑collinear points, yet the whole set cannot be partitioned into finitely many subsets with no three collinear points.

Problem 26 (extension) – Extremal density of integer multiples (1995) : Through an iterative construction using an increasing prime sequence, AI showed that for a certain integer sequence, adding any fixed offset yields multiples whose natural‑number density is bounded above by three‑quarters.

The computational cost varied widely; most problems cost dozens to a few hundred dollars, while the cheapest required only $7.5–$15.

These results illustrate that AI can now tackle genuine open research‑level mathematics, turning large‑model reasoning into a practical tool for formal proof discovery.