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History of Fermat's Last Theorem
Fermat’s Last Theorem was stated in a marginal note in the 1630s, published in 1670, and proved by Andrew Wiles in papers published in May 1995. In between came proofs for single exponents, a theory of “ideal numbers” invented to attack it, a large cash prize, and finally a route through elliptic curves that nobody had anticipated.
Each entry below is linked to the source it was checked against. Where sources disagree, the disagreement is noted.
Before Fermat
Section titled “Before Fermat”| Date | Event | Source |
|---|---|---|
| 3rd century AD (uncertain) | Diophantus of Alexandria writes the Arithmetica, a collection of number problems in 13 books, of which six were long thought to be the only survivors. His dates are a matter of inference. | MacTutor |
| 1621 | Bachet de Méziriac publishes the Latin translation of the Arithmetica that Fermat studied. | MacTutor |
Fermat and the note
Section titled “Fermat and the note”| Date | Event | Source |
|---|---|---|
| 1630s | Fermat writes in the margin of his Arithmetica that no power above the second can be split into two like powers, and that he has a marvellous proof too long for the margin. The note is undated: 1637 is the year usually quoted, and MacTutor and Darmon, Diamond and Taylor say around 1630. | MacTutor |
| 1659 | Fermat describes his method of infinite descent in a letter sent to Huygens through Carcavi. | MacTutor |
| January 12, 1665 | Fermat dies in Castres. | MacTutor, as above |
| 1670 | His son Samuel publishes an edition of Bachet’s Arithmetica with Fermat’s notes. The claim becomes public. | MacTutor, as above |
Fermat’s own copy of the book is lost, as MathWorld notes. The note survives through the 1670 edition.
One exponent at a time: 1753 to 1857
Section titled “One exponent at a time: 1753 to 1857”| Date | Event | Source |
|---|---|---|
| August 4, 1753 | Euler writes to Goldbach that he has proved the case n = 3. | MacTutor |
| 1770 | Euler’s proof for n = 3 appears in his Algebra. It contains a gap that can be repaired from his other work. | MacTutor, as above |
| May 12, 1819 | Sophie Germain sets out her plan of attack in a letter to Gauss. Her theorem on the “first case” dates from around 1820. | MacTutor |
| July to September 1825 | Dirichlet and Legendre between them prove the case n = 5. | MacTutor history, as above |
| 1832 | Dirichlet proves n = 14. | MacTutor history, as above |
| 1839 | Lamé proves n = 7. | MacTutor history, as above |
| March 1, 1847 | Lamé announces a general proof to the Paris Academy. Liouville points out that it assumes unique factorisation. | MacTutor history, as above |
| May 24, 1847 | Liouville reads the Academy a letter from Kummer: unique factorisation fails, Kummer showed this in 1844, and he has proved the theorem for regular primes. | MacTutor history, as above |
| 1857 | The Paris Academy awards Kummer its Grand Prize of 3,000 francs, offered for a solution of Fermat’s problem, although he had not entered. | MacTutor |
The mathematics behind these entries is in The statement and small cases.
Prizes, computers and a new conjecture: 1908 to 1983
Section titled “Prizes, computers and a new conjecture: 1908 to 1983”| Date | Event | Source |
|---|---|---|
| June 27, 1908 | The Göttingen Academy of Sciences announces the Wolfskehl Prize: 100,000 marks, from the will of Paul Wolfskehl, for a proof. | MacTutor |
| 1908 to 1912 | More than 1,000 false proofs are published. | MacTutor history, as above |
| 1915 | Jensen proves that there are infinitely many irregular primes, so Kummer’s method cannot cover every exponent. | MacTutor history, as above |
| 1955 | At a symposium in Tokyo and Nikko, Yutaka Taniyama poses problems about elliptic curves that develop, through the work of Goro Shimura and André Weil, into the modularity conjecture. | MacTutor |
| 1983 | Gerd Faltings proves the Mordell conjecture. One consequence: for each exponent n ≥ 4, Fermat’s equation has at most finitely many solutions without a common factor. | Inventiones Mathematicae |
| By 1993 | Computer calculations have verified the theorem for all exponents up to 4,000,000. | MacTutor history, as above |
The road through elliptic curves: 1985 to 1995
Section titled “The road through elliptic curves: 1985 to 1995”| Date | Event | Source |
|---|---|---|
| 1985 | Gerhard Frey, in a lecture at Oberwolfach, proposes that a solution of Fermat’s equation would yield an elliptic curve that violates the modularity conjecture. His paper appears in 1986. | Ribet, Notices of the AMS |
| August 13, 1985 | Jean-Pierre Serre, in a letter to Jean-François Mestre, pins down what is missing from Frey’s argument. The missing statement becomes known as the epsilon conjecture. | Ribet, as above |
| July 1986 | Ken Ribet proves the epsilon conjecture. From now on, the modularity conjecture for semistable elliptic curves implies Fermat’s Last Theorem. His paper is published in 1990. | Ribet, as above |
| 1986 | Andrew Wiles, at Princeton, begins working on the modularity conjecture. He continues for seven years, telling almost nobody. | MacTutor |
| June 21–23, 1993 | Wiles gives three lectures at the Isaac Newton Institute in Cambridge and announces the proof at the end of the third. | MacTutor |
| December 4, 1993 | Wiles states publicly that one step of the argument “is not yet complete as it stands”. | MacTutor history, as above |
| September 19, 1994 | Wiles, working with Richard Taylor, finds the idea that closes the gap. | MacTutor |
| October 1994 | The corrected proof is circulated, as two manuscripts: one by Wiles and one by Taylor and Wiles. | Ribet, Bulletin of the AMS |
| May 1995 | Both papers are published in the Annals of Mathematics, volume 141, issue 3: Wiles’s “Modular elliptic curves and Fermat’s Last Theorem” (pages 443–551) and Taylor and Wiles’s “Ring-theoretic properties of certain Hecke algebras” (pages 553–572). | Annals, Annals |
The ideas in this stretch are explained in How Wiles proved it.
After the proof: 1997 to 2026
Section titled “After the proof: 1997 to 2026”| Date | Event | Source |
|---|---|---|
| 1997 | Wiles collects the Wolfskehl Prize, by then worth about 75,000 Deutschmarks. | MacTutor |
| 2000 | Wiles is knighted. | MacTutor |
| 2001 | Breuil, Conrad, Diamond and Taylor publish the proof of the modularity conjecture for all elliptic curves over the rational numbers, completing what Wiles began. | Journal of the AMS |
| March 15, 2016 | The Norwegian Academy of Science and Letters announces that Wiles has won the Abel Prize. | University of Oxford |
| May 24, 2016 | Wiles receives the prize in Oslo from Crown Prince Haakon. | University of Oxford |
| October 2024 | Funding begins for Kevin Buzzard’s project at Imperial College London to formalize a proof in the Lean theorem prover. | Project repository |
| July 15, 2025 | A Lean formalization of Kummer’s theorem for regular primes is published. | Annals of Formalized Mathematics |
| September 4, 2026 | Anthropic announces a complete, computer-checked proof of the theorem in Lean, produced by its AI system. | Anthropic |
The last three entries are covered in Formalizing Fermat’s Last Theorem in Lean.
Dates that are often quoted differently
Section titled “Dates that are often quoted differently”- The note: 1637 or about 1630. There is no date on it. Popular accounts say 1637. Darmon, Diamond and Taylor and MacTutor say “around 1630”. The figure of 358 years assumes 1637.
- Frey’s idea: 1985 or 1986. Ribet, who was closest to the events, dates it to a 1985 Oberwolfach lecture and an unpublished manuscript of the same year. The published paper is from 1986, which is the year some accounts give.
- The proof: 1993, 1994 or 1995. All three are defensible. 1993 is the announcement, 1994 is when the proof became correct, and 1995 is the publication.
- Kummer: 1847 or later. The result for regular primes was announced in 1847. Some references give a later year. MathWorld, for example, has 1849.
Frequently asked questions
When did Fermat write his Last Theorem?
Nobody knows exactly, because the note is undated. The year most often quoted is 1637. The MacTutor History of Mathematics archive and the mathematicians Darmon, Diamond and Taylor both put it around 1630. It became public in 1670, five years after Fermat died.
When was Fermat's Last Theorem proved?
Andrew Wiles announced a proof on June 23, 1993, but it contained a gap. He found the repair on September 19, 1994, working with Richard Taylor, and circulated the corrected proof in October 1994. It was published in the Annals of Mathematics in May 1995.
How long did the theorem stay unproved?
The usual figure is 358 years, counted from 1637, the traditional date of Fermat's note, to the publication of the proof in 1995. Counted from the note's first publication in 1670, it is 325 years.
What was the Wolfskehl Prize?
A prize of 100,000 marks for a proof of Fermat's Last Theorem, left in the will of Paul Wolfskehl and announced by the Göttingen Academy of Sciences on June 27, 1908. Andrew Wiles collected it in 1997, by which time it was worth about 75,000 Deutschmarks.
Who proved Fermat's Last Theorem besides Wiles?
Wiles's proof rests on a chain of earlier results. Gerhard Frey and Jean-Pierre Serre linked the problem to elliptic curves, Ken Ribet proved in July 1986 that the modularity conjecture implies Fermat, and Richard Taylor co-wrote the paper that closed the gap in Wiles's 1993 argument.