Best AI for math research in 2026: what to use for what, and how to check it
ChatGPT, Claude, Gemini, open-weight models and specialised provers (Aristotle, AxiomProver, Lean tools) compared by what they are good at in research mathematics — literature, proofs, computation, formalisation — with the 2026 track record of each and the checks that matter more than the choice of model.
Updated 2026-10-31 · CC BY 4.0
Short answer. In 2026 all three large labs — OpenAI, Anthropic and Google DeepMind — have models that produced results on open problems which experts or the Lean proof assistant then confirmed. The frontier models are close enough that the choice matters less than two other things: using the strongest reasoning mode you can get, and checking the output outside the chat. Pick by task, not by brand.
This guide was written on 31 October 2026. Model names and plans change every few months; the advice on what to check does not.
Which tool for which job
| Job | What works best | How to check it |
|---|---|---|
| Finding what is already known | A frontier chatbot with web search on (ChatGPT, Claude, Gemini); "deep research" modes for broad surveys | Open every link. A citation without a link is a lead, not a fact. |
| Trying a proof or a special case | The strongest reasoning mode available (ChatGPT's Pro tier, Claude's top model with extended thinking, Gemini's Deep Think) | Ask a second chat, or a second model, to find the first wrong step and quote it. |
| Experiments and searches (small cases, constructions, records) | A model that runs code: Claude Code, ChatGPT's Python or Codex, Gemini with code execution | The program prints the object; a separate checker verifies it. Constructions are the fastest results to trust. |
| Formal proofs in Lean | An agent that compiles Lean itself: Claude Code or Codex in a Lean project, or a Lean MCP server; specialised provers such as Harmonic's Aristotle and AxiomProver | The Lean kernel. Then compare the formal statement with the conjecture, line by line. |
| Checking someone else's proof | Any frontier model asked to locate the first gap, never to "confirm" | The quoted step is checked by a person in minutes. |
The 2026 track record, by family
Benchmarks change monthly; results on open problems are a better signal of what a model family can do. From the running list of open problems solved by AI:
- OpenAI (ChatGPT, GPT-5.x and GPT-6 models). The most results by volume: Erdős problems with GPT-5.x Pro, Crouzeix's and Sendov's conjectures (prompted by outside researchers and checked by experts), the cycle double cover conjecture, the disproof of the unit distance conjecture, and the October 2026 release of 722 manuscripts — most of which nobody has checked yet.
- Anthropic (Claude). The counterexample to the Jacobian conjecture in dimension 3, twelve new Hadamard matrix orders, elliptic curves of record rank, a higher proven proportion of zeta zeros on the critical line, and large Lean formalisations (including Fermat's Last Theorem in September 2026). Strong at long agentic work with code and Lean through Claude Code.
- Google DeepMind (Gemini, AlphaEvolve). AlphaEvolve's evolutionary search found new constructions (matrix multiplication, kissing numbers, Ramsey lower bounds); the Aletheia study with Gemini Deep Think is the most careful public measurement of how often "solutions" are meaningful.
- Open-weight models (DeepSeek, Qwen and others). Behind the frontier on the hardest problems, but cheap, private, and good enough for literature summaries, code and many Lean lemmas. Useful when you want to run thousands of attempts.
- Specialised provers. Harmonic's Aristotle and Axiom Math's AxiomProver turn natural-language arguments into Lean proofs; both took part in 2026 results (Erdős problems 728 and 650, Fel's conjecture).
For current benchmark scores, use Epoch AI's benchmarks hub, which runs the evaluations itself. Many "leaderboard" sites only collect numbers that vendors report.
Free or paid?
Free plans of ChatGPT, Claude and Gemini are enough for literature questions, explanations, small experiments and most of the tasks on this site. The deepest reasoning modes sit in the roughly $20-a-month plans with usage limits, and in the $100–$200-a-month tiers with much higher limits. Pay for the expensive tier only when you have a precise, checkable target — a lemma, a construction, a formal statement — not to "solve" a famous problem.
Five checks that matter more than the model
- Fix the statement first. Quantifiers, ranges, definitions. Most wrong "solutions" prove a different statement.
- Search before you reason. On curated lists such as the Erdős problems, a forgotten paper is the most common "AI solution".
- Prefer outputs a computer can verify: a construction, a counterexample, a Lean proof.
- Separate writing from checking. The chat that wrote a proof is the worst judge of it.
- Record what you used. Model, mode, date and prompt — so others can reproduce or audit it.
Try it on a real problem
Cairn Commons gives your chatbot one small task on an open problem and checks the result: by Lean, by re-running code, or by review. A dead end, documented well, counts too. Start with a task for your chatbot, a record a computer checks, or the model-specific workflows for Claude and ChatGPT. The step-by-step version is in How to use AI for math research.
Sources
- Open problems solved by AI: the running list, with its primary sources.
- Google DeepMind, Aletheia study of 700 open problems, arXiv 2601.22401.
- Epoch AI, benchmarks hub and FrontierMath.
- Plan prices as listed by the vendors in October 2026 (summarised in this comparison); check the vendor's page before paying.