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Fermat's Last Theorem

A note in a book margin in the 1630s, more than 350 years of failed attempts, and a proof in 1995. Here is what the theorem says, why it was so hard, and where it stands now.

The statement and small cases How Wiles proved it

The statement

Fermat’s Last Theorem states that no three positive whole numbers x, y and z satisfy the equation xⁿ + yⁿ = zⁿ when n is a whole number greater than 2. A cube is never the sum of two cubes, a fourth power is never the sum of two fourth powers, and so on for every higher power.

Pierre de Fermat wrote the claim in the margin of a book in the 1630s and said he had a proof. He never wrote it down. The first accepted proof was published by Andrew Wiles in 1995.

Why n = 2 is different

With squares, the equation has plenty of solutions. They are the whole-number right triangles of Pythagoras’s theorem:

3² + 4² = 9 + 16 = 25 = 5²
5² + 12² = 25 + 144 = 169 = 13²
8² + 15² = 64 + 225 = 289 = 17²

There are infinitely many. Pick any two whole numbers m and k with m larger than k, and the three numbers m² − k², 2mk and m² + k² always work. Choosing m = 2 and k = 1 gives 3, 4, 5.

Now try cubes. You can come painfully close:

6³ + 8³ = 216 + 512 = 728, and 9³ = 729.

One short. Fermat’s claim is that you will never hit it exactly, for cubes or for any higher power, however far you search. Changing the exponent from 2 to 3 turns a problem with infinitely many answers into one with none.

The story in brief

Pierre de Fermat was a lawyer and councillor at the parliament of Toulouse who did mathematics in his spare time. He owned a copy of the Arithmetica, a book of number problems by the ancient Greek mathematician Diophantus, in the Latin edition published by Bachet in 1621. In its margin he wrote a note in Latin. In English it says, roughly:

It is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general any power higher than the second into two powers of the same kind. I have discovered a truly marvellous proof of this, which this margin is too narrow to contain.

The note is undated. The year usually quoted is 1637. Some historians and mathematicians put it earlier, around 1630. Fermat died in 1665, and the world learned of the note only in 1670, when his son Samuel published an edition of the Arithmetica that included his father’s annotations.

Despite its name, the statement was a conjecture, not a theorem, for the next three centuries. Fermat never mentioned the general claim again in any letter that survives.

Progress came one exponent at a time. Fermat himself left a proof for n = 4. Leonhard Euler handled n = 3 in the 18th century. Sophie Germain, Dirichlet, Legendre and Lamé extended the list in the 19th, and Ernst Kummer proved the theorem for a large class of prime exponents in 1847. The statement and small cases walks through these.

The breakthrough came from an unexpected direction. In the mid-1980s Gerhard Frey, Jean-Pierre Serre and Ken Ribet showed that a counterexample to Fermat would produce an elliptic curve so strange that it would contradict a famous unproved conjecture about such curves. Andrew Wiles, who had been fascinated by Fermat’s problem since he was ten, spent seven years working in near secrecy to prove enough of that conjecture. He announced his proof in Cambridge on June 23, 1993. A gap was found during checking. Wiles and Richard Taylor closed it in September 1994, and the proof was published in the Annals of Mathematics in May 1995.

Counting from the traditional date of 1637 to the publication in 1995 gives the figure of 358 years.

A short timeline

Year	Event
1630s	Fermat writes his marginal note.
1670	The note is published by his son, five years after Fermat’s death.
1753	Euler tells Goldbach he has a proof for n = 3.
1825	Dirichlet and Legendre settle n = 5.
1847	Kummer proves the theorem for all “regular” prime exponents.
1955	Yutaka Taniyama poses the problems that grow into the modularity conjecture.
1985–1986	Frey, Serre and Ribet link Fermat’s equation to that conjecture.
June 23, 1993	Wiles announces a proof at the Isaac Newton Institute in Cambridge.
September 19, 1994	Wiles finds the repair for the gap, with Richard Taylor.
May 1995	The proof is published in the Annals of Mathematics.
2016	Wiles receives the Abel Prize.
September 4, 2026	A complete computer-checked proof in Lean is announced.

Every entry is sourced on the dated history page.

Why it was so hard

You cannot check your way to a proof. By 1993 computers had confirmed the theorem for every exponent up to 4,000,000. That is evidence. It says nothing about exponent 4,000,001.

Each exponent seemed to need its own trick. The proofs for 3, 4, 5 and 7 are all different, and each is harder than the last. Nothing in them suggests a method for every exponent at once.

The obvious general method breaks. In 1847 Gabriel Lamé announced a proof that factorised xⁿ + yⁿ using complex numbers. It assumed that these new numbers factor into primes in only one way, as ordinary whole numbers do. They do not, in general. Kummer rescued part of the idea with his “ideal numbers”, which is how he reached the regular primes. The concept of an ideal grew into a central tool of modern algebra.

The real proof lives somewhere else. Wiles did not study the equation xⁿ + yⁿ = zⁿ directly. He proved a statement about elliptic curves and modular forms, two kinds of object that Fermat never saw. In the words of the Imperial College London project that is formalizing the theorem, all known proofs “involve much deeper mathematics” than the counting numbers in the statement. How Wiles proved it follows the route.

This is also why most mathematicians doubt that Fermat had a valid proof. The MacTutor archive’s verdict is that “it is now believed that Fermat’s ‘proof’ was wrong although it is impossible to be completely certain”.

Where it stands today

The theorem is proved, and nobody seriously disputes it. Wiles’s argument has been studied, simplified and extended for three decades. In 2001 Christophe Breuil, Brian Conrad, Fred Diamond and Richard Taylor completed the proof of the full conjecture that Wiles had proved in part. In 2016 Wiles received the Abel Prize, one of the highest honours in mathematics, “for his stunning proof of Fermat’s Last Theorem by way of the modularity conjecture for semistable elliptic curves”.

No elementary proof has been found. Every known proof uses the 20th-century machinery.

The proof has now been checked by machine. On September 4, 2026, Anthropic announced a complete proof written in the Lean theorem prover by its AI system, in which every logical step is verified by software. Kevin Buzzard of Imperial College London, who leads a separate project to formalize a more modern proof, compiled and checked it. His project continues. Formalizing Fermat’s Last Theorem in Lean explains what was done and what it does and does not show.

Frequently asked questions

What does Fermat's Last Theorem say?

Fermat's Last Theorem says that no three positive whole numbers x, y and z satisfy xⁿ + yⁿ = zⁿ when n is a whole number greater than 2. For n = 2 there are infinitely many solutions, such as 3² + 4² = 5². For cubes, fourth powers and every higher power there are none.

Who proved Fermat's Last Theorem?

Andrew Wiles, a British mathematician then at Princeton University. He announced a proof in June 1993, a gap was found, and he repaired it in 1994 with his former student Richard Taylor. The two papers containing the proof were published in the Annals of Mathematics in May 1995.

Did Fermat really have a proof?

Almost certainly not a correct one. Fermat left a proof only for the exponent 4, and every known proof of the full theorem uses mathematics developed centuries after his death. It is generally believed that his argument was flawed, and he may have realised that himself, although nobody can be sure.

Why did it take 358 years to prove?

The elementary methods that work for single exponents such as 3, 4, 5 and 7 do not extend to all exponents. The eventual proof went through a deep link between elliptic curves and modular forms, two subjects that did not exist in Fermat's time, and that link was only conjectured in the 1950s.

Is there a simple proof of Fermat's Last Theorem?

No short or elementary proof is known. Wiles's proof and its companion paper fill 129 journal pages and rest on a large body of earlier work. All known proofs use elliptic curves, modular forms and Galois representations.

Has the proof been checked by a computer?

Yes. On September 4, 2026, Anthropic announced a complete proof of Fermat's Last Theorem written in the Lean theorem prover by its AI system, and the mathematician Kevin Buzzard confirmed that the code compiles and proves the correct statement. A separate, human-led project at Imperial College London is formalizing a more modern proof.