DeepMind’s AlphaProof Nexus Solves Nine Erdős Problems for a Few Hundred Dollars

AlphaProof Nexus solves nine Erdős problems

AlphaProof Nexus, a Gemini‑driven framework, produced Lean proofs for nine open Erdős problems that had remained unsolved for 56 years. It also proved 44 conjectures from the OEIS, resolved a 15‑year‑old algebraic‑geometry question, and improved a theoretical bound in convex optimization. Each theorem required only a few hundred dollars of compute and the proof code is released on GitHub.

Selected problems and AI‑generated solutions

Erdős #12 (1970)

The problem asks for an infinite set of integers such that for any three distinct elements a, b, c, a never divides b + c and the set has positive natural density. The system decomposes the construction using the Chinese Remainder Theorem into independent blocks, each built from a three‑term arithmetic‑progression avoidance set, and then stitches the blocks together to obtain a single infinite set satisfying both constraints.

Erdős #125 (1996)

Given the set of ternary numbers using only digits 0 and 1 and the set of quaternary numbers using only digits 0 and 1, the problem asks whether the pairwise sum set has positive lower density. The proof exploits the irrationality of log₄ / log₃, which implies that powers of 3 and 4 can be made arbitrarily close. By constructing a sequence of scales that approximate each other, the density is shown to decay by a factor of 0.99 at each step, converging to zero.

Erdős #846 (1992)

The problem seeks an infinite planar point set in which every finite subset contains many non‑collinear points, yet the whole set cannot be partitioned into finitely many subsets with no three points collinear. The system encodes each edge of a complete graph as a planar point using quadratic polynomial coordinates, then applies the infinite Ramsey theorem to translate the geometric statement into a graph‑theoretic one, completing the proof.

The remaining six problems involve divisor‑set constructions, Van der Waerden gaps, Sidon‑set isolation, and set‑splitting density.

Architecture and agent design

The core loop is:

Gemini 3.1 Pro generates Lean proof steps → Lean compiler checks each line → error messages are fed back to the model → the model revises the proof → repeat until the proof passes.

Four agents were implemented:

Agent A : launches multiple independent sub‑agents that use Gemini 3.1 Pro to draft proof code and rely solely on the Lean‑compiler feedback loop.

Agent B : invokes the pre‑existing AlphaProof reinforcement‑learning prover when Agent A stalls on a difficult sub‑step.

Agent C : maintains a population of proof sketches; each sub‑module produces a draft that is scored on plausibility, clarity, and novelty with an Elo‑like system, and high‑scoring drafts are combined while low‑scoring ones are discarded.

Agent D : combines the evolutionary selection of Agent C with the specialized tools of Agent B and the simple LLM‑plus‑compiler loop of Agent A.

Empirically, the simplest Agent A alone solved all nine target problems; the more complex agents did not provide additional advantage beyond higher computational cost.

Why the simple loop succeeded

Two factors were identified:

Gemini 3.1 Pro’s intrinsic capability is sufficient for these proofs.

The line‑by‑line feedback from the Lean compiler supplies a strong guidance signal.

Cost and impact

Each theorem required only a few hundred dollars of compute, demonstrating that massive monetary rewards are not necessary for solving long‑standing Erdős problems.

Paper: https://arxiv.org/abs/2605.22763v1

GitHub repository: https://github.com/google-deepmind/alphaproof-nexus-results