Fermat's Last Theorem

Fermat’s Last TheoremLast Theorema long-standing problemFermatFermat equationFermat's equationFermat's Last Theorem and Diophantus II.VIIIFermat's legendary last theoremFermat's TheoremFermats Last Theorem
In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integerswikipedia
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Conjecture

conjecturalconjecturesconjectured
The proposition was first conjectured by Pierre de Fermat in 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.
Some conjectures, such as the Riemann hypothesis (still a conjecture) or Fermat's Last Theorem (a conjecture until proven in 1995 by Andrew Wiles), have shaped much of mathematical history as new areas of mathematics are developed in order to prove them.

Wiles's proof of Fermat's Last Theorem

Wiles' proof of Fermat's Last Theoremproofproof of the 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.
Together with Ribet's theorem, it provides a proof for Fermat's Last Theorem.

Andrew Wiles

Andrew John WilesSir Andrew WilesAndrew J. Wiles
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.
He is best known for proving Fermat's Last Theorem, for which he was awarded the 2016 Abel Prize and the 2017 Copley Medal by the Royal Society.

Pierre de Fermat

FermatPierre FermatFermat, Pierre de
The proposition was first conjectured by Pierre de Fermat in 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.
He is best known for his Fermat's principle for light propagation and his Fermat's Last Theorem in number theory, which he described in a note at the margin of a copy of Diophantus' Arithmetica.

Algebraic number theory

placefinite placealgebraic
The unsolved problem stimulated the development of algebraic number theory in the 19th century and the proof of the modularity theorem in the 20th century.
Fermat's last theorem was first conjectured by Pierre de Fermat in 1637, famously in the margin of a copy of Arithmetica where he claimed he had a proof that was too large to fit in the margin.

Modularity theorem

Taniyama–Shimura conjectureShimura–Taniyama conjectureTaniyama-Shimura conjecture
The unsolved problem stimulated the development of algebraic number theory in the 19th century and the proof of the modularity theorem in the 20th century. It also proved much of the modularity theorem and opened up entire new approaches to numerous other problems and mathematically powerful modularity lifting techniques.
Andrew Wiles proved the modularity theorem for semistable elliptic curves, which was enough to imply Fermat's last theorem.

Regular prime

irregular primegave a criterionirregular
In the mid-19th century, Ernst Kummer extended this and proved the theorem for all regular primes, leaving irregular primes to be analyzed individually.
In number theory, a regular prime is a special kind of prime number, defined by Ernst Kummer in 1850 to prove certain cases of Fermat's Last Theorem.

Sophie Germain

GermainGermain, SophieHonors in number theory
Over the next two centuries (1637–1839), the conjecture was proved for only the primes 3, 5, and 7, although Sophie Germain innovated and proved an approach that was relevant to an entire class of primes.
Her work on Fermat's Last Theorem provided a foundation for mathematicians exploring the subject for hundreds of years after.

History of mathematics

historian of mathematicsmathematicshistory
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.
The Arithmetica had a significant influence on later mathematicians, such as Pierre de Fermat, who arrived at his famous Last Theorem after trying to generalize a problem he had read in the Arithmetica (that of dividing a square into two squares).

Elliptic curve

elliptic curveselliptic equationWeierstrass equation
Separately, around 1955, Japanese mathematicians Goro Shimura and Yutaka Taniyama suspected a link might exist between elliptic curves and modular forms, two completely different areas of mathematics.
Elliptic curves are especially important in number theory, and constitute a major area of current research; for example, they were used in the proof, by Andrew Wiles, of Fermat's Last Theorem.

Gerhard Frey

Frey, Gerhard
In 1984, Gerhard Frey noticed an apparent link between these two previously unrelated and unsolved problems.
His Frey curve, a construction of an elliptic curve from a purported solution to the Fermat equation, was central to Wiles's proof of Fermat's Last Theorem.

Goro Shimura

ShimuraGorō ShimuraShimura, Goro
Separately, around 1955, Japanese mathematicians Goro Shimura and Yutaka Taniyama suspected a link might exist between elliptic curves and modular forms, two completely different areas of mathematics.
He was known for developing the theory of complex multiplication of abelian varieties and Shimura varieties, as well as posing the Taniyama–Shimura conjecture which ultimately led to the proof of Fermat's Last Theorem.

Number theory

number theoristcombinatorial number theorytheory of numbers
In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers
Fermat's claim ("Fermat's last theorem") to have shown there are no solutions to for all n\geq 3 appears only in his annotations on the margin of his copy of Diophantus.

Ernst Kummer

KummerErnst Eduard KummerEduard Kummer
In the mid-19th century, Ernst Kummer extended this and proved the theorem for all regular primes, leaving irregular primes to be analyzed individually.
Kummer also proved Fermat's last theorem for a considerable class of prime exponents (see regular prime, ideal class group).

Frey curve

The full proof that the two problems were closely linked was accomplished in 1986 by Ken Ribet, building on a partial proof by Jean-Pierre Serre, who proved all but one part known as the "epsilon conjecture" (see: Ribet's Theorem and Frey curve).
::associated with a (hypothetical) solution of Fermat's equation

Yutaka Taniyama

TaniyamaTaniyama, Yutaka
Separately, around 1955, Japanese mathematicians Goro Shimura and Yutaka Taniyama suspected a link might exist between elliptic curves and modular forms, two completely different areas of mathematics.
This conjecture proved to be a major factor in the proof of Fermat's Last Theorem by Andrew Wiles.”

Diophantus

Diophantus of AlexandriaDiophantosDiophantus the Arab
Fermat's equation, x n + y n = z n with positive integer solutions, is an example of a Diophantine equation, named for the 3rd-century Alexandrian mathematician, Diophantus, who studied them and developed methods for the solution of some kinds of Diophantine equations.
While reading Claude Gaspard Bachet de Méziriac's edition of Diophantus' Arithmetica, Pierre de Fermat concluded that a certain equation considered by Diophantus had no solutions, and noted in the margin without elaboration that he had found "a truly marvelous proof of this proposition," now referred to as Fermat's Last Theorem.

Diophantine equation

Diophantine equationsDiophantine analysisDiophantine
Fermat's equation, x n + y n = z n with positive integer solutions, is an example of a Diophantine equation, named for the 3rd-century Alexandrian mathematician, Diophantus, who studied them and developed methods for the solution of some kinds of Diophantine equations.
A typical such equation is the equation of Fermat's Last Theorem

Ken Ribet

Kenneth Alan RibetKenneth RibetKenneth A. Ribet
The full proof that the two problems were closely linked was accomplished in 1986 by Ken Ribet, building on a partial proof by Jean-Pierre Serre, who proved all but one part known as the "epsilon conjecture" (see: Ribet's Theorem and Frey curve).
Ribet is credited with paving the way towards Andrew Wiles's proof of Fermat's last theorem.

Fermat's right triangle theorem

hereproof by Fermat
Only one relevant proof by Fermat has survived, in which he uses the technique of infinite descent to show that the area of a right triangle with integer sides can never equal the square of an integer.
But, if Fermat's last theorem were false for the exponent n=4, then squaring one of the three numbers in any counterexample would also give three numbers that solve this equation.

Prime number

primeprime factorprime numbers
, proved by Fermat himself, is sufficient to establish that if the theorem is false for some exponent n that is not a prime number, it must also be false for some smaller n, so only prime values of n need further investigation.
Early attempts to prove Fermat's Last Theorem led to Kummer's introduction of regular primes, integer prime numbers connected with the failure of unique factorization in the cyclotomic integers.

Jean-Pierre Serre

SerreSerre, Jean-PierreJ.-P. Serre
The full proof that the two problems were closely linked was accomplished in 1986 by Ken Ribet, building on a partial proof by Jean-Pierre Serre, who proved all but one part known as the "epsilon conjecture" (see: Ribet's Theorem and Frey curve).
Amongst his most original contributions were: his "Conjecture II" (still open) on Galois cohomology; his use of group actions on trees (with Hyman Bass); the Borel–Serre compactification; results on the number of points of curves over finite fields; Galois representations in ℓ-adic cohomology and the proof that these representations have often a "large" image; the concept of p-adic modular form; and the Serre conjecture (now a theorem) on mod-p representations that made Fermat's last theorem a connected part of mainstream arithmetic geometry.

Pythagorean triple

Pythagorean triplesPythagorean triangleEuclid's formula
The Pythagorean equation, x 2 + y 2 = z 2, has an infinite number of positive integer solutions for x, y, and z; these solutions are known as Pythagorean triples (with the simplest example 3,4,5).
Pierre de Fermat in 1637 claimed that no such triple exists, a claim that came to be known as Fermat's Last Theorem because it took longer than any other conjecture by Fermat to be proven or disproven.

Claude Gaspard Bachet de Méziriac

BachetClaude BachetBachet (Claude Gaspard Bachet de Méziriac)
Fermat's conjecture of his Last Theorem was inspired while reading a new edition of the Arithmetica, that was translated into Latin and published in 1621 by Claude Bachet.
It was this very translation in which Fermat wrote his famous margin note claiming that he had a proof of Fermat's last theorem.

Sophie Germain prime

Maximally periodic reciprocalsSophie Germain conjecture
As a byproduct of this latter work, she proved Sophie Germain's theorem, which verified the first case of Fermat's Last Theorem (namely, the case in which p does not divide xyz) for every odd prime exponent less than 270, and for all primes p such that at least one of 2p+1, 4p+1, 8p+1, 10p+1, 14p+1 and 16p+1 is prime (specially, the primes p such that 2p+1 is prime are called Sophie Germain primes).
Sophie Germain primes are named after French mathematician Sophie Germain, who used them in her investigations of Fermat's Last Theorem.