There is no single best AI model for every mathematics task. Choose by the assurance you need: use a model plus independent computation for numerical work, a model that preserves notation and assumptions for symbolic manipulation, a reasoning model for exploring informal proofs, and a proof assistant such as Lean when the result must be formally checked.
Benchmark scores can help you shortlist systems, but they are not interchangeable rankings or guarantees on your problem. The useful comparison is task fit, verification method, tool access, evaluation quality, availability, and cost—not a single leaderboard number.
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Start with the kind of answer you need
Before comparing model names, classify the job. “Solve this equation,” “simplify this expression,” “find a proof,” and “produce a machine-checked proof” may look similar in a chat window, but they require different kinds of reliability.
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| Need | What to compare | How to verify |
|---|---|---|
| Numerical calculations | Arithmetic accuracy, consistency across several steps, ability to use code or another computation tool, and explicit assumptions | Recompute independently or execute the calculation; check units, rounding, signs, and intermediate values |
| Symbolic manipulation | Correct algebraic or calculus transformations, notation preservation, and handling of domains and conditions | Substitute the result back, test boundary cases, and use a computer algebra system when appropriate |
| Informal proof exploration | Argument structure, coverage of cases, useful lemmas, and whether each implication follows | Inspect every inference and edge case; treat the output as a draft rather than a certificate |
| Formal proof obligations | Compatibility with the proof assistant and libraries, production of accepted proof terms, and an effective tactic-search workflow | Run the artifact through the formal system and require an accepted proof |
| Model comparison | Relevant benchmark, model/version, tools, sampling, sample size, contamination controls, and grader | Read the evaluation method and uncertainty; compare only like with like |
The last row matters because a score on a short-answer contest set measures a different capability from generating a proof-assistant term or supporting open-ended scientific work.
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Numerical problems: prioritize an auditable calculation
A language model can choose a method, set up equations, explain a result, or write code. It can also make an arithmetic mistake, lose a sign, apply a formula outside its conditions, or give a correct final number for an invalid chain of reasoning. Google Research’s Minerva report documented both calculation mistakes and reasoning errors, including cases in which the final answer was correct even though the reasoning was faulty.
What to look for in a model
- Tool use: Prefer a workflow that can execute code or pass the computation to a calculator rather than relying on generated arithmetic alone.
- Intermediate visibility: Ask for equations, units, assumptions, and intermediate values that you can inspect. A polished explanation is not proof that each step is valid.
- Multi-step consistency: Test whether the same quantity is calculated consistently when the problem is decomposed into smaller checks.
- Precision control: State whether you need an exact fraction, a decimal, a tolerance, significant figures, or an interval. Do not let the model choose silently.
- Domain awareness: Require the model to identify restrictions such as nonzero denominators, positive arguments for logarithms, or a valid range for an approximation.
A reliable numerical workflow
- Define the output. State the target quantity, units, acceptable error, rounding rule, and known assumptions.
- Request a setup before a number. Ask the model to write the governing equation and identify the data it used. This separates a modeling error from a later arithmetic error.
- Have the model show a compact derivation. Request one line per transformation, with units carried through. Avoid accepting an unexplained answer.
- Execute or recompute. Run the arithmetic independently in code, a calculator, or a spreadsheet. For a high-stakes result, use a second implementation or a different method.
- Check limiting cases. Test zero, one, very large or very small inputs, and any boundary values allowed by the problem. A result that behaves impossibly at a boundary is a warning even when the central calculation looks plausible.
- Compare intermediate values. If the independent calculation disagrees, locate the first differing step instead of asking the model to “try again” without evidence.
- Record reproducibility details. Keep the prompt, code, precision, constants, and model/version used so another person can repeat the calculation.
Prompt pattern for numerical work
State the assumptions and units first. Derive the formula in numbered steps. Give an exact form where possible, then a decimal rounded to 6 significant figures. Show intermediate values. Do not perform final arithmetic by inspection: provide executable Python that recomputes it, and identify every domain or rounding condition.This pattern improves auditability, but it does not remove the need to run the code and inspect the model’s assumptions.
Symbolic mathematics: test transformations, not fluency
Symbolic tasks include factoring, solving equations, differentiating, integrating, simplifying expressions, and manipulating matrices or series. The central question is whether each transformation is valid under the stated conditions. A model may produce a plausible-looking expression while silently assuming a variable is positive, cancelling a factor that could be zero, or changing the branch of a complex function.
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- State the domain. Specify whether variables are real or complex and list sign, nonzero, integer, or interval assumptions.
- Preserve conditions. Ask the model to retain excluded values and branch conditions rather than returning only a simplified expression.
- Substitute back. For an equation solution, substitute every candidate into the original equation, not a rearranged intermediate version.
- Use equivalent forms deliberately. Compare expanded, factored, rationalized, and numerical forms to catch dropped terms or sign changes.
- Test singular and boundary cases. Check denominators approaching zero, endpoints of an interval, repeated roots, and parameter values where the expression changes behavior.
- Use a computer algebra system as a checker. A symbolic tool can verify an identity or derivative, but its result still needs interpretation when assumptions or branch cuts matter.
Google Research describes Minerva as performing numerical calculations and symbolic manipulation without external tools, while also noting that its natural-language approach has no explicit underlying mathematical structure and cannot automatically verify answers. That distinction is useful: a model can propose a transformation, while a separate symbolic system or substitution check tests it.
Example of a safe symbolic request
Assume x is real and x ≠ 0. Simplify the expression, listing every condition under which each step is reversible. Give an independently checkable substitution into the original expression and test x = 1, x = -1, and values near any excluded point.For calculus, ask whether an antiderivative is being given up to a constant, whether an integral is improper, and whether differentiation under an integral sign requires conditions that have not been stated. For matrix work, check dimensions and rank before trusting a compact formula.
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Informal proofs: use models for exploration, not certification
Reasoning models are useful for generating conjectures, proposing lemmas, finding a proof strategy, or explaining a known argument. Their prose can still hide a false implication. A contest benchmark result tells you how a system performed on that test; it does not validate a new proof written for your problem.
How to inspect a generated proof
- Write the exact claim and quantifiers. Many apparent proofs change “for every” into “for a typical case” or omit a required existence condition.
- List dependencies. Ask the model to identify each lemma, theorem, definition, and assumption used.
- Check the first nontrivial inference. If that step fails, later fluent paragraphs do not repair the proof.
- Separate cases explicitly. Look for zero, negative, degenerate, equality, and boundary cases that the model may have treated as generic.
- Try to construct a counterexample. Ask a second pass—or a different tool—to search for a small counterexample to every broad intermediate claim.
- Verify independently. Re-derive the key step yourself or encode it in a formal system when the result matters.
OpenAI’s process-supervision research reported better results from evaluating intermediate steps than from evaluating only final answers in its MATH experiments. It also left generalization beyond those experiments unresolved. Step-level checking is therefore a useful evaluation idea, not a guarantee that a visible explanation is trustworthy.
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Give two possible proof strategies, then develop the shorter one. State all definitions and quantify every variable. Mark each lemma used, explain why its hypotheses hold here, and include a separate search for counterexamples to the intermediate claims. Do not call a statement proved until every case is covered.Requesting a concise, checkable derivation is safer than treating a long chain of generated prose as evidence of correctness. The model’s explanation is an object to audit, not an authority.
Formal proofs: require an accepted proof artifact
A formal proof has a different assurance level from a persuasive natural-language argument. In a proof assistant, the model may suggest tactics, definitions, or mathematical terms, but the formal system checks whether the resulting term proves the exact statement in the selected logic and library.
OpenAI’s formal-mathematics work describes this interaction between a language model and a formal system. It also cautions that the reported system closed challenging olympiad problems only occasionally, rather than consistently reaching the level of the best students. That is encouraging evidence that models can assist formalization, not evidence that they can replace a proof assistant or a human reviewer.
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Choosing a formal-proof workflow
- Fix the environment first. Select the proof assistant version, imported libraries, and project configuration before asking for code. A tactic that works in one environment may fail in another.
- State the theorem exactly. Include types, hypotheses, and desired conclusions. Ambiguous prose invites the model to prove a nearby but different proposition.
- Ask for small increments. Have the model produce a definition or lemma, run it, report the error, and then make one targeted correction.
- Compile every step. Never infer validity from syntax that looks plausible. Acceptance by the checker is the decisive test.
- Review imports and automation. Confirm that automation did not use an unintended axiom, an overly strong hypothesis, or a theorem that changes the intended statement.
- Preserve the artifact. Save the source, toolchain version, library revision, and successful build output.
For a publication, safety-critical argument, or result that others must audit, the meaningful claim is “the formal system accepted this proof artifact,” not “the model explained why the theorem is true.”
How to interpret current benchmark evidence
Benchmark numbers are useful only with their owner, year, model/version, settings, tools, sample count, and grading method. Scores from different suites should not be merged into an overall ranking.
| Evaluation | Reported result | Conditions and interpretation |
|---|---|---|
| International Math Olympiad 2025 evaluation | Gemini 3.1 Deep Think: 81.5%; GPT-5.2 Thinking (xhigh): 71.4%; Gemini 3 Pro Preview Thinking (High): 14.3% | Google DeepMind page, table headed Gemini 3.1 Deep Think Feb 2026. Publisher-reported figures under its linked methodology; not an independently reproduced head-to-head test. |
| SMT 2025 | GPT-5: 91.8% ± 1.5 | NIST CAISI report, 2025; 58 problems; percentages are tasks solved and the uncertainty is the reported standard error. |
| OTIS-AIME 2025 | GPT-5: 91.9% ± 2.0 | NIST CAISI report, 2025; 30 problems; grading used o4-mini to judge equivalence of submitted mathematical expressions to ground-truth answers. |
| PUMaC 2024 | GPT-5: 85.9% ± 3.5 | NIST CAISI report, 2025; 55 problems; the same report describes its task-set and grading methodology. |
| MATH | Minerva 540B: 50.3% | Google Research, 2022; historical research result, not a current product recommendation. |
| MMLU-STEM | Minerva 540B: 75% | Google Research, 2022; historical result from the reported evaluation. |
| OCWCourses | Minerva 540B: 30.8% | Google Research, 2022; historical result from the reported evaluation. |
| GSM8k | Minerva 540B: 78.5% | Google Research, 2022; historical result from the reported evaluation. |
| GPQA | GPT-4: 39%; GPT-5.2: 92% | OpenAI, 2025; GPQA is a science benchmark, not a math-specific proof or symbolic-math evaluation. |
The NIST CAISI math table covers three task sets with 58, 30, and 55 problems and reports standard errors. Its grader checks expression equivalence, which is appropriate for those answer formats but does not establish proof correctness or broad symbolic reliability.
The Minerva results also came with an important limitation: Google reported that its natural-language method could not automatically verify answers and that a correct final answer could still contain faulty reasoning. Historical scores help explain progress and failure modes; they should not determine which system you use today.
Why benchmark comparisons can mislead
- Different tasks test different abilities. An olympiad evaluation, an answer-equivalence grader, a proof-assistant benchmark, and open-ended research questions are not interchangeable.
- Tools change the task. A model with code execution, retrieval, or a formal checker is being evaluated under different conditions from a model answering unaided.
- Sampling changes results. Temperature, number of attempts, selection of the best attempt, and hidden retry policies can materially affect a percentage.
- Small samples create wide uncertainty. A few problems can move a score substantially, so retain the reported standard error and task count.
- Publisher comparisons have selection effects. OpenAI notes that some FrontierScience tasks were selected against its internal models, which may bias results against those models relative to others. The same article discusses model-based grading and the limitations of rubric-based evaluation.
Run a small task-specific evaluation before committing
If your work is specialized, your own representative set is more informative than a general leaderboard. Build a set that includes routine, difficult, and adversarial examples from the notation and domains you actually use.
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- Sample real tasks. Include numerical calculations, symbolic transformations, proof sketches, and any formalization obligations that occur in your work.
- Define grading before testing. For numbers, specify tolerance and units. For symbolic answers, require equivalence under stated assumptions. For proofs, grade each lemma and case, not only the conclusion. For formal work, require a successful build.
- Lock the conditions. Record model/version, context length, tools, temperature or other sampling settings, number of attempts, and whether a human selected the answer.
- Measure failure types. Separate arithmetic mistakes, invalid assumptions, omitted cases, notation drift, hallucinated theorems, and tool or compilation failures.
- Test repeatability. Re-run a subset to see whether success is stable or depends on one lucky sample.
- Measure verification cost. A model that is slightly less accurate but produces easy-to-check intermediate work may be safer and faster than a model whose answers require a full re-derivation.
- Keep an error log. Save prompts, outputs, corrections, and the smallest counterexample that exposed each failure. Use it to refine prompts and decide where automation is mandatory.
Do not report the resulting percentage as a universal model ranking. It is evidence about your workload under your stated conditions.
Common failure modes and practical fixes
A correct number with an invalid derivation
This is especially dangerous when a result is copied into a report. Recompute independently, compare intermediate values, and check the assumptions used to derive the formula. Treat a matching final number as necessary but not sufficient.
Silent domain changes
Expressions involving square roots, logarithms, division, inverse trigonometric functions, absolute values, or complex powers can change meaning when a variable’s domain changes. Put the domain in the prompt, require conditions beside every reversible step, and substitute candidates into the original statement.
Missing or duplicated proof cases
Ask for an explicit case partition and verify that the cases are exhaustive and mutually compatible. Search for degenerate examples such as zero vectors, repeated roots, equal parameters, empty sets, or boundary values.
Confidently cited but nonexistent mathematics
Require the exact theorem statement and hypotheses, then check it in a trusted reference or formal library. A plausible theorem name or citation is not evidence that the result exists.
Benchmark overconfidence
Do not select a model solely because it leads one published table. Preserve the benchmark’s date, task definition, tool conditions, and uncertainty, then test a sample of your own work.
Tool and environment mismatch
A code snippet may depend on a package, precision mode, or library version that is absent from your environment. Run it in a clean, controlled setup and record the versions. For formal proofs, compile against the exact project configuration.
Access and cost are separate checks
The evidence summarized here does not establish current regional availability, subscription prices, usage caps, or enterprise terms. Those conditions change independently of benchmark performance. Before adopting a model, verify the current access tier in your region, whether the required reasoning mode and tools are included, data-retention and privacy terms, rate limits, and the cost of the verification tools you will need alongside it.
A lower-cost model may be adequate for drafting algebra or suggesting lemmas when every result is checked. A more capable reasoning model may save time on difficult exploration but still require a proof assistant or independent computation. Compare the full workflow cost, including human checking and failed attempts, rather than the chat subscription alone.
Decision rules you can apply immediately
- If the answer is a number: choose a model that can explain the setup and use computation, then independently recompute every important result.
- If the answer is an exact expression: choose for notation and assumption handling, substitute back, test edge cases, and use symbolic software where suitable.
- If you need ideas for a proof: use a strong reasoning model to propose strategies and lemmas, but audit every implication and search for counterexamples.
- If the proof must be correct: make the proof assistant the authority. Use the model as a tactic and formalization assistant, and accept only a checked artifact.
- If you are comparing products: compare the same task, same tool conditions, same sampling policy, same grader, and same uncertainty. Keep benchmark names attached to their scores.
The practical answer to “Which AI is best at math?” is therefore conditional: the best choice is the system and verification workflow that fit your problem and provide the assurance your decision requires. For consequential mathematics, model output is a proposal; independent computation or formal checking supplies the evidence.
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