Fermat's Last Theorem

There are number triples like 3, 4 and 5. For them: 3 times 3 plus 4 times 4 equals exactly 5 times 5, that is 9 plus 16 equals 25. Such triples are known from math class, and there are infinitely many of them. The French lawyer and amateur mathematician Pierre de Fermat raised the question around 1637 of whether this also works with third, fourth, or higher powers. In other words: Are there positive integers where a times a times a plus b times b times b equals exactly c times c times c? Fermat claimed that from the third power onward there is not a single such solution, and noted in the margin of a book that he had a proof for which there was no room there. This proof was never found.

A Marginal Note Puzzle Spanning 350 Years

The claim is easy to understand but extremely hard to prove. Exactly this combination made it famous. One can explain it to a student in two minutes, yet generations of professionals failed at it. For centuries it was regarded as the most famous open problem in mathematics.

A common misconception is that something like this could be checked by computer. In fact, very many cases have been calculated without finding a counterexample. But this proves nothing. The numbers are infinite, and a computer never reaches their end. A proof must handle all cases at once, not billions of individual ones.

The side effect was more important than the result itself. In the search for a proof, entire subfields of mathematics emerged. Many tools that are today embedded in the encryption of messages stem from this research tradition. An apparently useless puzzle thus produced practical technology.

The Detour via Elliptic Curves

The proof came in 1994 from the British mathematician Andrew Wiles. He did not attack the problem directly. Instead, he used a translation into a completely different field. Experts had previously shown: if a solution to the Fermat equation existed, it would have to give rise to a very peculiar geometric object.

These objects are called elliptic curves. These are curves described by certain equations, whose points can be added together in a special way. A conjecture unproven at the time stated that all such curves belong to a particular ordered family. But the curve arising from a Fermat solution could not possibly belong to it. Wiles proved enough of this conjecture to derive a contradiction from it. Therefore, no solution can exist.

The path there was bumpy. Wiles worked largely in secret for seven years. During the review of his first manuscript, a referee found a gap in 1993. Only after a further year of work, partly with his former student Richard Taylor, was the proof complete. It spans over a hundred pages and is not readable without years of study.

Why AI Researchers Cite the Theorem

In tech news today, Fermat’s Last Theorem usually appears as a benchmark. When a company claims its software can prove things mathematically, people like to ask how far it is from a problem of this magnitude. Wiles’s proof is regarded as an example of creativity that goes far beyond fast computation.

Specifically, this concerns proof assistants. These are programs that check every step of a mathematical argument for completeness. An international group has been working since 2024 to fully encode the Fermat proof into such a system. The project is planned to span several years. Language models, of the kind behind chatbots, are being tested as aids in this, not as a replacement for the experts.

The theorem also appears in pop culture. It shows up in novels, in episodes of The Simpsons, and in popular science books. Anyone who hears the name Fermat should not confuse this with Fermat’s little theorem: that is a different, much simpler result and is indeed used directly in cryptography.