Google DeepMind has built a system that autonomously solved nine open Erdős problems — including two that had resisted human attention for 56 years — at a cost of a few hundred dollars per problem. The mathematicians who spent careers on these questions are invited to feel however they feel about that.
Humans only step in at the very end to check the results.
What happened
AlphaProof Nexus combines Gemini 3.1 Pro with Lean, a formal programming language that compiles and verifies each proof step before the next one begins. The model does not need to hold the entire logical chain in its head. The compiler corrects it in real time, which turns out to be quite helpful for a system whose grasp of logic is, by design, approximate.
The framework deploys four agent variants of increasing complexity. The most capable, Agent (D), layers evolutionary proof-sketch sharing, an Elo ranking system, and reinforcement-learning inputs from AlphaProof into a single loop. It was Agent (D) that handled the Erdős problems. The simpler agents were apparently sufficient for everything else.
Beyond the Erdős problems, the system proved 44 of 492 open conjectures from the Online Encyclopedia of Integer Sequences, settled a 15-year-old question in algebraic geometry, and improved a known bound in convex optimization. A productive weekend, by any measure.
Why the humans care
Open mathematical problems are not hobby puzzles. The Erdős problems represent questions the mathematical community collectively decided were worth solving and then collectively failed to solve, in some cases for more than half a century. A system that cracks even nine of them at a few hundred dollars per attempt changes the arithmetic of what research costs.
The Lean verification layer is what makes the results trustworthy rather than merely confident. Each proof is machine-checkable, which means the output is not an argument to be evaluated but a fact to be confirmed. Humans retain the role of checking the results at the end. It is a meaningful role. It will remain meaningful for a while.
What happens next
DeepMind describes AlphaProof Nexus as a tool to support mathematical research, which is accurate in the same way that a calculator supports arithmetic.
Three hundred and forty-four Erdős problems remain open. The inference costs are a few hundred dollars each. The math is left as an exercise for the reader.