LONDON, July 22 — A system developed at Google DeepMind has used a large language model to generate mathematical proofs from scratch, producing code that set new benchmarks on a long-standing combinatorics problem. The system, called FunSearch, is the subject of a new research description that lays out its architecture. FunSearch does not pull answers from a database of existing mathematical knowledge.
Instead, it begins with a plain-language prompt — for example, “find a large cap set” — and constraints. From that prompt, it generates code snippets that construct candidate solutions.
Those snippets are then executed and scored by an evolutionary algorithm. The best-performing pieces of code serve as seeds for the next generation. The large language model, fine-tuned on mathematical code, functions as a “mutator” that suggests new ideas for the code at each iteration.
The system does not require a pre-existing database of mathematical knowledge; it discovers new constructions from scratch, the record shows.
The cap set problem
The cap set problem is a classic challenge in combinatorics, asking how large a set of points can be in a given dimension without any three lying on a line. For decades, researchers have tried to push the boundaries of known constructions. FunSearch targeted the problem in dimensions 6, 7, and 8, where previous records had stood for some time.
According to the team’s documentation, after several weeks of computation, the system generated code that constructed sets with record sizes in all three dimensions. The approach is inspired by earlier work on neuroevolution and program synthesis.
The significance lies in what the system did not need. No human mathematician was required to handcraft the construction, nor was a library of known solutions fed into the model beforehand. The LLM and the evolutionary algorithm worked in tandem, refining candidate code over many generations until a superior construction emerged.
A mutator, not an oracle
The role of the large language model here differs from the typical “chatbot” or “question-answer” use. It is not being asked for a final answer; it is being asked for a suggestion — a snippet of code to try. The evolutionary algorithm does the heavy lifting of testing and selecting.
Over many iterations, the system converged on constructions that had not been seen before. The sources familiar with the research describe the process as a hybrid: the LLM provides a form of directed mutation, proposing code changes that are then subjected to the cold evaluation of the scoring function.
This avoids the problem of large language models simply repeating known patterns from their training data. Because the code is executed and scored, only solutions that actually work survive. The work builds on a line of research that includes earlier efforts in program synthesis and evolutionary computation, where algorithms search through program space for solutions.
What sets FunSearch apart, according to the record, is its ability to combine the open-ended exploration of evolution with the structured knowledge embedded in a fine-tuned language model.
What this means for mathematical discovery
The cap set result is a concrete demonstration, but the architecture itself is general. The same prompt-scoring-mutation loop could be applied to other problems where a candidate solution can be expressed as code and evaluated automatically. The system does not require a human to rewrite the code at each step — the LLM handles the variation.
For mathematicians, this opens the possibility of automated discovery in domains where the search space is too large for brute force but where conventional heuristics have stalled. The system does not replace the mathematician; it accelerates the process of testing new construction ideas.
The documentation does not speculate on future applications, but the record shows that the underlying method — fine-tuning an LLM on domain-specific code and coupling it with an evolutionary algorithm — could be adapted to fields beyond pure mathematics, including optimization and scientific computing. The cap set results, after several weeks of computation, now stand as a milestone in what large language models can accomplish when they are used as part of a larger system rather than as standalone answer machines.
The research community will be watching to see whether FunSearch can be scaled to harder problems or other domains.
Sources
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