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Fermat's Last Theorem

In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers a, b, and c satisfy the equation an + bn = cn for any integer value of n greater than 2. The cases n = 1 and n = 2 have been known since antiquity to have infinitely many solutions.[1]

For other theorems named after Pierre de Fermat, see Fermat's theorem. For the book by Simon Singh, see Fermat's Last Theorem (book).

Field

For any integer n > 2, the equation an + bn = cn has no positive integer solutions.

c. 1637

Released 1994
Published 1995

The proposition was first stated as a theorem by Pierre de Fermat around 1637 in the margin of a copy of Arithmetica. Fermat added that he had a proof that was too large to fit in the margin. Although other statements claimed by Fermat without proof were subsequently proven by others and credited as theorems of Fermat (for example, Fermat's theorem on sums of two squares), Fermat's Last Theorem resisted proof, leading to doubt that Fermat ever had a correct proof. Consequently, the proposition became known as a conjecture rather than a theorem. After 358 years of effort by mathematicians, the first successful proof was released in 1994 by Andrew Wiles and formally published in 1995. It was described as a "stunning advance" in the citation for Wiles's Abel Prize award in 2016.[2] It also proved much of the Taniyama–Shimura conjecture, subsequently known as the modularity theorem, and opened up entire new approaches to numerous other problems and mathematically powerful modularity lifting techniques.


The unsolved problem stimulated the development of algebraic number theory in the 19th and 20th centuries. It is among the most notable theorems in the history of mathematics and prior to its proof was in the Guinness Book of World Records as the "most difficult mathematical problem", in part because the theorem has the largest number of unsuccessful proofs.[3]

Overview[edit]

Pythagorean origins[edit]

The Pythagorean equation, x2 + y2 = z2, has an infinite number of positive integer solutions for x, y, and z; these solutions are known as Pythagorean triples (with the simplest example being 3, 4, 5). Around 1637, Fermat wrote in the margin of a book that the more general equation an + bn = cn had no solutions in positive integers if n is an integer greater than 2. Although he claimed to have a general proof of his conjecture, Fermat left no details of his proof, and no proof by him has ever been found. His claim was discovered some 30 years later, after his death. This claim, which came to be known as Fermat's Last Theorem, stood unsolved for the next three and a half centuries.[4]


The claim eventually became one of the most notable unsolved problems of mathematics. Attempts to prove it prompted substantial development in number theory, and over time Fermat's Last Theorem gained prominence as an unsolved problem in mathematics.

Fermat's Last Theorem needed to be proven for all exponents n that were prime numbers.

The modularity theorem—if proved for semi-stable elliptic curves—would mean that all semistable elliptic curves must be modular.

Ribet's theorem showed that any solution to Fermat's equation for a prime number could be used to create a semistable elliptic curve that could not be modular;

The only way that both of these statements could be true, was if no solutions existed to Fermat's equation (because then no such curve could be created), which was what Fermat's Last Theorem said. As Ribet's Theorem was already proved, this meant that a proof of the modularity theorem would automatically prove Fermat's Last theorem was true as well.

Prizes and incorrect proofs[edit]

In 1816, and again in 1850, the French Academy of Sciences offered a prize for a general proof of Fermat's Last Theorem.[180][181] In 1857, the academy awarded 3,000 francs and a gold medal to Kummer for his research on ideal numbers, although he had not submitted an entry for the prize.[180] Another prize was offered in 1883 by the Academy of Brussels.[182]


In 1908, the German industrialist and amateur mathematician Paul Wolfskehl bequeathed 100,000 gold marks—a large sum at the time—to the Göttingen Academy of Sciences to offer as a prize for a complete proof of Fermat's Last Theorem.[183][184] On 27 June 1908, the academy published nine rules for awarding the prize. Among other things, these rules required that the proof be published in a peer-reviewed journal; the prize would not be awarded until two years after the publication; and that no prize would be given after 13 September 2007, roughly a century after the competition was begun.[185] Wiles collected the Wolfskehl prize money, then worth $50,000, on 27 June 1997.[186] In March 2016, Wiles was awarded the Norwegian government's Abel prize worth €600,000 for "his stunning proof of Fermat's Last Theorem by way of the modularity conjecture for semistable elliptic curves, opening a new era in number theory".[187]


Prior to Wiles's proof, thousands of incorrect proofs were submitted to the Wolfskehl committee, amounting to roughly 10 feet (3.0 meters) of correspondence.[188] In the first year alone (1907–1908), 621 attempted proofs were submitted, although by the 1970s, the rate of submission had decreased to roughly 3–4 attempted proofs per month. According to some claims, Edmund Landau tended to use a special preprinted form for such proofs, where the location of the first mistake was left blank to be filled by one of his graduate students.[189] According to F. Schlichting, a Wolfskehl reviewer, most of the proofs were based on elementary methods taught in schools, and often submitted by "people with a technical education but a failed career".[190] In the words of mathematical historian Howard Eves, "Fermat's Last Theorem has the peculiar distinction of being the mathematical problem for which the greatest number of incorrect proofs have been published."[182]

Euler's sum of powers conjecture

Proof of impossibility

a list of related conjectures and theorems

Sums of powers

Wall–Sun–Sun prime