AI solves mathematics problems. What counts as a proof?

Take a nonnegative whole number n. Calculate n² + n + 41. For n = 0 you get 41; for 1, you get 43; for 2, you get 47. All primes: numbers greater than 1 divisible only by 1 and themselves. It looks like a handy prime-number machine. Would you bet that it always works?

At n = 40, the result is 1,600 + 40 + 41 = 1,681 = 41 × 41. One number destroys the claim “always prime.” Our first three tests were correct. The conclusion went too far. This fully worked teaching example unlocks the current mathematics story: A convincing run of successes and a proof answer different questions.

On 8 September 2026, OpenAI announced a solution to the Navier–Stokes problem and published an analytical paper and a formalized version. According to the company, the internal AI system used was more capable than publicly available GPT-6 Astra. Announcement and setup.

The Navier–Stokes equations describe fluid motion. A difficult part is the interaction: motion transports itself, while viscosity smooths differences. Can initially smooth conditions become so concentrated that the mathematical model breaks down in finite time? A few computed flows would not settle that question for every permitted case. Clay Mathematics Institute problem statement.

The published paper constructs a case with a smooth external force: the fluid starts at rest, velocity becomes unbounded in finite time in the model, and kinetic energy remains bounded. This concerns alternatives C and D in the official problem. The claim is more precise than “AI understands turbulence.” It also does not say that real water becomes infinitely fast. Paper, introduction and Theorem 1.1, formalized results.

Remove the external force from the explanation and you are describing a different task. A simulation can suggest an interesting candidate; a proof must also satisfy the required conditions. For a major result, those conditions are part of what the achievement means.

Return to something we can calculate ourselves. This time the claim is: The difference between consecutive squares is always odd. Between 2² and 3² it is 5. Between 3² and 4² it is 7. Again, we could collect examples for a long time.

Now there is a short general reason: (n + 1)² − n² = n² + 2n + 1 − n² = 2n + 1. For every nonnegative whole number n, 2n is even; adding 1 makes it odd. The calculation covers all n in the stated set. In a drawing, an n-by-n square grows into the next square by adding an L-shaped layer of n + n + 1 cells.

Our prime formula supplied tested examples and a counterexample. The square calculation supplies a general proof. Both are much smaller than Navier–Stokes. That is why we can see the difference here without relying on a model’s authority.

Lean is a proof assistant. Its purpose is to check a mathematical statement and its justification in a formal language. It receives definitions, assumptions, the exact claim and a constructed proof. Its checking kernel verifies whether that proof establishes the claim under the system’s rules. The output is an accepted formal proof — or a point where checking fails. Lean: propositions and proofs.

This also helps with AI: a model can propose approaches and develop proof steps; the decisive check does not depend on the most fluent explanation. People still need to examine whether the formal statement expresses the intended problem, which assumptions and axioms are used, and what the proof contributes. A correctly proved statement about the wrong problem does not resolve your original question.

On 6 October 2026, OpenAI publicly shared further mathematics results. Publication. Its linked mathematics repository explicitly distinguishes verification stages: not every result has a formalization, and unformalized work may still contain problems. A manuscript collection is therefore not a list of equally established, independently confirmed solutions.

For a particular result, separate three questions: What exact statement is claimed? What evidence is available? Who actually performed which checks? A readable research paper, a formally checkable artifact and expert review provide different information. This article explains those layers; it does not claim that we independently checked the entire large fluid proof.

In an exam, questions and correct answers are fixed in advance. In research, you often first need to find a productive question, approach or counterexample. A checkable new result would therefore represent a different kind of progress from a higher test score. Its value also includes what happens next: others can inspect the proof, build on it or locate an error.

When the next headline says AI has cracked mathematics, follow the same small path as with our formula: Write down the claim, retain the conditions, open the evidence. For a test, ask which cases were tried; for a counterexample, ask for the concrete contradiction; for a proof, ask for the general reason. That makes the result easier to understand and its potential value clearer.