Fermat's Last Theorem

Fermat’s Last Theorem is a statement about integers, that is, numbers without a decimal point such as 3, 17, or 480. From school, one knows the equation a² + b² = c². For it, there are neat solutions in integers, such as 3² + 4² = 5², i.e. 9 + 16 = 25. The French lawyer and amateur mathematician Pierre de Fermat claimed in 1637: as soon as you replace the exponent 2 with a larger exponent, that’s the end of it. There are then no longer any three positive integers that satisfy the equation. So for third, fourth, hundredth powers and all further ones, you will never find a matching trio of numbers.

A theorem that remained open for 358 years

Fermat noted his claim in the margin of a book. Alongside it, he wrote that he had a proof, but the margin was too narrow for it. This proof was never found. Today, experts assume that Fermat was mistaken, because the actual proof requires methods that did not yet exist at the time.

After that, generations of mathematicians tried their luck. Individual exponents were checked off one by one: Euler managed the third, others the fourth, fifth, seventh. The problem: there are infinitely many exponents. You cannot go through them one by one. That is precisely the appeal of such statements. They can be explained in a single sentence, but not proven in a single sentence.

For mathematics, the value did not lie in the statement itself. Nobody needs the information in everyday life that a³ + b³ = c³ is unsolvable. What was valuable were the tools invented in the search for the proof. Entire subfields of number theory emerged as a byproduct of this pursuit.

The detour via elliptic curves

The proof came in 1994 from the British mathematician Andrew Wiles, supported by his former student Richard Taylor. Wiles did not attack the problem directly. Instead, he proved a completely different statement, from which Fermat’s theorem automatically follows.

The trick works like this: it had previously been shown that a solution to Fermat’s equation would inevitably produce a very peculiar mathematical object. Such objects are called elliptic curves. You can picture them as curves whose points can be combined with one another according to fixed rules. The curve arising from a Fermat solution would have properties that no regular object of this kind is allowed to have.

Wiles then proved that all elliptic curves of a certain type are well-behaved, meaning no such outliers exist. This means there can be no Fermat solution either, because otherwise the outlier would exist. This is a proof by contradiction: you assume a solution exists and show that something impossible follows from it. The proof fills over 100 pages. An initial attempt from 1993 contained a gap, which Wiles took more than a year to close.

From marginal-note puzzle to AI test case

Today, the theorem is a standard example of how difficult mathematical proofs can be. It appears in documentaries, novels, and TV series, for instance in an episode of Star Trek. Anyone who comes across the name in the news often finds it in discussions about the limits of computing power, since no computer could solve the problem by trial and error.

In the AI world, the theorem is present for two reasons. First, it serves as a benchmark: if a language model claims to have found a short proof, that is a sure sign of a fabricated answer. Such fabrications are called hallucinations. Second, research groups are working on translating large proofs into software that verifies every step by machine. For Wiles' proof, such a project has been running for years.

It is important to distinguish this from open problems. Fermat’s Last Theorem is settled and considered proven. Other famous questions, such as the Riemann Hypothesis, are not. So if a report suggests that an AI has solved Fermat’s theorem, it is already wrong from the outset.