Pythagoras and Diophantus Pythagorean triples In ancient times it was known that a triangle whose sides were in the ratio 3:4:5 would have a right angle as one of its angles. This was used in construction and later in early geometry. It was also known to be one example of a general rule that any triangle where the length of two sides, each squared and then added together , equals the square of the length of the third side , would also be a right angle triangle. This is now known as the
Pythagorean theorem, and a triple of numbers that meets this condition is called a Pythagorean triple; both are named after the ancient Greek
Pythagoras. Examples include (3, 4, 5) and (5, 12, 13). There are infinitely many such triples, and methods for generating such triples have been studied in many cultures, beginning with the
Babylonians and later
ancient Greek,
Chinese, and
Indian mathematicians. Mathematically, the definition of a Pythagorean triple is a set of three integers that satisfy the equation .
Diophantine equations Fermat's equation, 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 Diophantine problem is to find two integers
x and
y such that their sum, and the sum of their squares, equal two given numbers
A and
B, respectively: : A = x + y : B = x^2 + y^2. Diophantus's major work is the
Arithmetica, of which only a portion has survived. 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. Diophantine equations have been studied for thousands of years. For example, the solutions to the quadratic Diophantine equation are given by the
Pythagorean triples, originally solved by the Babylonians (). Solutions to linear Diophantine equations, such as , may be found using the
Euclidean algorithm (c. 5th century BC). Many
Diophantine equations have a form similar to the equation of Fermat's Last Theorem from the point of view of algebra, in that they have no
cross terms mixing two letters, without sharing its particular properties. For example, it is known that there are infinitely many positive integers
x,
y, and
z such that , where
n and
m are
relatively prime natural numbers.
Fermat's conjecture . On the right is the margin that was too small to contain Fermat's alleged proof of his "last theorem". Problem II.8 of the asks how a given
square number is split into two other squares; in other words, for a given
rational number k, find rational numbers
u and
v such that . Diophantus shows how to solve this sum-of-squares problem for (the solutions being and ). Around 1637, Fermat wrote his Last Theorem in the margin of his copy of the next to
Diophantus's sum-of-squares problem: After Fermat's death in 1665, his son Clément-Samuel Fermat produced a new edition of the book (1670) augmented with his father's comments. Although not actually a theorem at the time (meaning a mathematical statement for which
proof exists), the marginal note became known over time as ''Fermat's Last Theorem'', as it was the last of Fermat's asserted theorems to remain unproved. It is not known whether Fermat had actually found a valid proof for all exponents
n, but it appears unlikely. Only one related proof by him has survived, namely for the case , as described in the section . While Fermat posed the cases of and of as challenges to his mathematical correspondents, such as
Marin Mersenne,
Blaise Pascal, and
John Wallis, he never posed the general case. Moreover, in the last thirty years of his life, Fermat never again wrote of his "truly marvelous proof" of the general case, and never published it. Van der Poorten as saying Fermat must have briefly deluded himself with an irretrievable idea. The techniques Fermat might have used in such a "marvelous proof" are unknown. Wiles and Taylor's proof relies on 20th-century techniques. Fermat's proof would have had to be elementary by comparison, given the mathematical knowledge of his time. While
Harvey Friedman's
grand conjecture implies that any provable theorem (including Fermat's last theorem) can be proved using only '
elementary function arithmetic', such a proof need be 'elementary' only in a technical sense and could involve millions of steps, and thus be far too long to have been Fermat's proof.
Proofs for specific exponents for Fermat's Last Theorem case n=4 in the 1670 edition of the
Arithmetica of
Diophantus (pp. 338–339). ==== Exponent = 4 ==== 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. His proof is equivalent to demonstrating that the equation : x^4 - y^4 = z^2 has no primitive solutions in integers (no pairwise
coprime solutions). In turn, this proves Fermat's Last Theorem for the case , since the equation can be written as . Alternative proofs of the case were developed later by
Frénicle de Bessy (1676),
Leonhard Euler (1738), Kausler (1802),
Adrien-Marie Legendre (1830),
Olry Terquem (1846),
Joseph Bertrand (1851),
Victor Lebesgue (1853, 1859, 1862),
Théophile Pépin (1883), Tafelmacher (1893),
David Hilbert (1897), Bendz (1901), Gambioli (1901), Bang (1905), Sommer (1907), Bottari (1908),
Karel Rychlík (1910),
Robert Carmichael (1913), Hancock (1931),
Gheorghe Vrănceanu (1966), Grant and Perella (1999), Barbara (2007), and Dolan (2011).
Other exponents After Fermat proved the special case , the general proof for all
n required only that the theorem be established for all odd prime exponents. In other words, it was necessary to prove only that the equation has no positive integer solutions when
n is an odd
prime number. This follows because a solution for a given
n is equivalent to a solution for all the factors of
n. For illustration, let
n be factored into
d and
e,
n =
de. The general equation :
an +
bn =
cn implies that is a solution for the exponent
e : (
ad)
e + (
bd)
e = (
cd)
e. Thus, to prove that Fermat's equation has no solutions for , it would suffice to prove that it has no solutions for at least one prime factor of every
n. Each integer is divisible by 4 or by an odd prime number (or both). Therefore, Fermat's Last Theorem could be proved for all
n if it could be proved for and for all odd primes
p. In the two centuries following its conjecture (1637–1839), Fermat's Last Theorem was proved for three odd prime exponents
p = 3, 5 and 7. The case was first stated by
Abu-Mahmud Khojandi (10th century), but his attempted proof of the theorem was incorrect. In 1770,
Leonhard Euler gave a proof of
p = 3, but his proof by infinite descent contained a major gap. However, since Euler himself had proved the lemma necessary to complete the proof in other work, he is generally credited with the first proof. Independent proofs were published by Kausler (1802), Legendre (1823, 1830), Calzolari (1855),
Gabriel Lamé (1865),
Peter Guthrie Tait (1872),
Siegmund Günther (1878), Gambioli (1901), Krey (1909), Rychlík (1910), Stockhaus (1910), Carmichael (1915),
Johannes van der Corput (1915),
Axel Thue (1917), and Duarte (1944). The case was proved independently by Legendre and
Peter Gustav Lejeune Dirichlet around 1825. Alternative proofs were developed by
Carl Friedrich Gauss (1875, posthumous), Lebesgue (1843), Lamé (1847), Gambioli (1901), Werebrusow (1905), Rychlík (1910), van der Corput (1915), The case was proved by Lamé in 1839. His rather complicated proof was simplified in 1840 by Lebesgue, and still simpler proofs were published by
Angelo Genocchi in 1864, 1874 and 1876. Alternative proofs were developed by Théophile Pépin (1876) and Edmond Maillet (1897). Fermat's Last Theorem was also proved for the exponents
n = 6, 10, and 14. Proofs for were published by Kausler, Tafelmacher, Lind, Kapferer, Swift, and Breusch. Similarly, Dirichlet and Terjanian each proved the case
n = 14, while Kapferer the first significant work on the general theorem was done by
Sophie Germain.
Early modern breakthroughs Sophie Germain In the early 19th century,
Sophie Germain developed several novel approaches to prove Fermat's Last Theorem for all exponents. First, she defined a set of auxiliary primes
θ constructed from the prime exponent
p by the equation , where
h is any integer not divisible by three. She showed that, if no integers raised to the
pth power were adjacent modulo
θ (the
non-consecutivity condition), then
θ must divide the product
xyz. Her goal was to use
mathematical induction to prove that, for any given
p, infinitely many auxiliary primes
θ satisfied the non-consecutivity condition and thus divided
xyz; since the product
xyz can have at most a finite number of prime factors, such a proof would have established Fermat's Last Theorem. Although she developed many techniques for establishing the non-consecutivity condition, she did not succeed in her strategic goal. She also worked to set lower limits on the size of solutions to Fermat's equation for a given exponent
p, a modified version of which was published by
Adrien-Marie Legendre. 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, In 1985,
Leonard Adleman,
Roger Heath-Brown and
Étienne Fouvry proved that the first case of Fermat's Last Theorem holds for infinitely many odd primes
p.
Ernst Kummer and the theory of ideals In 1847,
Gabriel Lamé outlined a proof of Fermat's Last Theorem based on factoring the equation in
complex numbers, specifically the
cyclotomic field based on the
roots of the number 1. His proof failed, however, because it assumed incorrectly that such complex numbers can be
factored uniquely into primes, similar to integers. This gap was pointed out immediately by
Joseph Liouville, who later read a paper that demonstrated this failure of unique factorisation, written by
Ernst Kummer. Kummer set himself the task of determining whether the
cyclotomic field could be generalized to include new prime numbers such that unique factorisation was restored. He succeeded in that task by developing the
ideal numbers. (It is often stated that Kummer was led to his "ideal complex numbers" by his interest in Fermat's Last Theorem; there is even a story often told that Kummer, like
Lamé, believed he had proven Fermat's Last Theorem until
Lejeune Dirichlet told him his argument relied on unique factorization; but the story was first told by
Kurt Hensel in 1910 and the evidence indicates it likely derives from a confusion by one of Hensel's sources.
Harold Edwards said the belief that Kummer was mainly interested in Fermat's Last Theorem "is surely mistaken". See
the history of ideal numbers.) Using the general approach outlined by Lamé, Kummer proved both cases of Fermat's Last Theorem for all
regular prime numbers. However, he could not prove the theorem for the exceptional primes (irregular primes) that
conjecturally occur approximately 39% of the time; the only irregular primes below 270 are 37, 59, 67, 101, 103, 131, 149, 157, 233, 257 and 263.
Mordell conjecture In the 1920s,
Louis Mordell posed a conjecture that implied that Fermat's equation has at most a finite number of nontrivial primitive integer solutions, if the exponent
n is greater than two. This conjecture was proved in 1983 by
Gerd Faltings, and is now known as
Faltings' theorem.
Computational studies In the latter half of the 20th century, computational methods were used to extend Kummer's approach to the irregular primes. In 1954,
Harry Vandiver used a
SWAC computer to prove Fermat's Last Theorem for all primes up to 2521. By 1978,
Samuel Wagstaff had extended this to all primes less than 125,000. By 1993, Fermat's Last Theorem had been proved for all primes less than four million. However, despite these efforts and their results, no proof existed of Fermat's Last Theorem. Proofs of individual exponents by their nature could never prove the
general case: even if all exponents were verified up to an extremely large number X, a higher exponent beyond X might still exist for which the claim was not true. (This had been the case with some other past conjectures, such as with
Skewes' number, and it could not be ruled out in this conjecture.)
Connection with elliptic curves The strategy that ultimately led to a successful proof of Fermat's Last Theorem arose from the "astounding"
Taniyama–Shimura–Weil conjecture, proposed around 1955—which many mathematicians believed would be near to impossible to prove, and was linked in the 1980s by
Gerhard Frey,
Jean-Pierre Serre and
Ken Ribet to Fermat's equation. By accomplishing a partial proof of this conjecture in 1994,
Andrew Wiles ultimately succeeded in proving Fermat's Last Theorem, as well as leading the way to a full proof by others of what is now known as the
modularity theorem.
Taniyama–Shimura–Weil conjecture Around 1955, Japanese mathematicians
Goro Shimura and
Yutaka Taniyama observed a possible link between two apparently completely distinct branches of mathematics,
elliptic curves and
modular forms. The resulting
modularity theorem (at the time known as the Taniyama–Shimura conjecture) states that every elliptic curve is
modular, meaning that it can be associated with a unique
modular form. The link was initially dismissed as unlikely or highly speculative, but was taken more seriously when number theorist
André Weil found evidence supporting it, though not proving it; as a result the conjecture was often known as the Taniyama–Shimura–Weil conjecture. Even after gaining serious attention, the conjecture was seen by contemporary mathematicians as extraordinarily difficult or perhaps inaccessible to proof. For example, Wiles's doctoral supervisor
John Coates states that it seemed "impossible to actually prove", and Ken Ribet considered himself "one of the vast majority of people who believed [it] was completely inaccessible", adding that "Andrew Wiles was probably one of the few people on earth who had the audacity to dream that you can actually go and prove [it]."
Ribet's theorem for Frey curves In 1984,
Gerhard Frey noted a link between Fermat's equation and the modularity theorem, then still a conjecture. If Fermat's equation had any solution for exponent , then it could be shown that the semi-stable
elliptic curve (now known as a
Frey-Hellegouarch) :
y2 = would have such unusual properties that it was unlikely to be modular. This would conflict with the modularity theorem, which asserted that all elliptic curves are modular. As such, Frey observed that a proof of the Taniyama–Shimura–Weil conjecture might also simultaneously prove Fermat's Last Theorem. By
contraposition, a
disproof or refutation of Fermat's Last Theorem would disprove the Taniyama–Shimura–Weil conjecture. In plain English, Frey had shown that, if this intuition about his equation was correct, then any set of four numbers (
a,
b,
c,
n) capable of disproving Fermat's Last Theorem, could also be used to disprove the Taniyama–Shimura–Weil conjecture. Therefore, if the latter were true, the former could not be disproven, and would also have to be true. Following this strategy, a proof of Fermat's Last Theorem required two steps. First, it was necessary to prove the modularity theorem, or at least to prove it for the types of elliptical curves that included Frey's equation (known as
semistable elliptic curves). This was widely believed inaccessible to proof by contemporary mathematicians. Second, it was necessary to show that Frey's intuition was correct: that if an elliptic curve were constructed in this way, using a set of numbers that were a solution of Fermat's equation, the resulting elliptic curve could not be modular. Frey showed that this was
plausible but did not go as far as giving a full proof. The missing piece (the so-called "epsilon conjecture", now known as
Ribet's theorem) was identified by
Jean-Pierre Serre who also gave an almost-complete proof and the link suggested by Frey was finally proved in 1986 by
Ken Ribet. Following Frey, Serre and Ribet's work, this was where matters stood: • 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.
Wiles's general proof Ribet's proof of the
epsilon conjecture in 1986 accomplished the first of the two goals proposed by Frey. Upon hearing of Ribet's success,
Andrew Wiles, an English mathematician with a childhood fascination with Fermat's Last Theorem, and who had worked on elliptic curves, decided to commit himself to accomplishing the second half: proving a special case of the
modularity theorem (then known as the Taniyama–Shimura conjecture) for semistable elliptic curves. Wiles worked on that task for six years in near-total secrecy, covering up his efforts by releasing prior work in small segments as separate papers and confiding only in his wife. His initial study suggested
proof by
induction, and he based his initial work and first significant breakthrough on
Galois theory before switching to an attempt to extend
horizontal Iwasawa theory for the inductive argument around 1990–91 when it seemed that there was no existing approach adequate to the problem. However, by mid-1991, Iwasawa theory also seemed to not be reaching the central issues in the problem. In response, he approached colleagues to seek out any hints of cutting-edge research and new techniques, and discovered an
Euler system recently developed by
Victor Kolyvagin and
Matthias Flach that seemed "tailor made" for the inductive part of his proof.259–260 Wiles studied and extended this approach, which worked. Since his work relied extensively on this approach, which was new to mathematics and to Wiles, in January 1993 he asked his Princeton colleague,
Nick Katz, to help him check his reasoning for subtle errors. Their conclusion at the time was that the techniques Wiles used seemed to work correctly. By mid-May 1993, Wiles was ready to tell his wife he thought he had solved the proof of Fermat's Last Theorem, and by June he felt sufficiently confident to present his results in three lectures delivered on 21–23 June 1993 at the
Isaac Newton Institute for Mathematical Sciences. Specifically, Wiles presented his proof of the Taniyama–Shimura conjecture for semistable elliptic curves; together with Ribet's proof of the epsilon conjecture, this implied Fermat's Last Theorem. However, it became apparent during
peer review that a critical point in the proof was incorrect. It contained an error in a bound on the order of a particular
group. The error was caught by several mathematicians refereeing Wiles's manuscript including Katz (in his role as reviewer), who alerted Wiles on 23 August 1993. The error would not have rendered his work worthless: each part of Wiles's work was highly significant and innovative by itself, as were the many developments and techniques he had created in the course of his work, and only one part was affected. However, without this part proved, there was no actual proof of Fermat's Last Theorem. Wiles spent almost a year trying to repair his proof, initially by himself and then in collaboration with his former student
Richard Taylor, without success. By the end of 1993, rumours had spread that under scrutiny, Wiles's proof had failed, but how seriously was not known. Mathematicians were beginning to pressure Wiles to disclose his work whether it was complete or not, so that the wider community could explore and use whatever he had managed to accomplish. But instead of being fixed, the problem, which had originally seemed minor, now seemed very significant, far more serious, and less easy to resolve. Wiles states that on the morning of 19 September 1994, he was on the verge of giving up and was almost resigned to accepting that he had failed, and to publishing his work so that others could build on it and fix the error. He adds that he was having a final look to try and understand the fundamental reasons why his approach could not be made to work, when he had a sudden insight: that the specific reason why the Kolyvagin–Flach approach would not work directly meant that his original attempts using
Iwasawa theory could be made to work, if he strengthened it using his experience gained from the Kolyvagin–Flach approach. Fixing one approach with tools from the other approach would resolve the issue for all the cases that were not already proven by his refereed paper. He described later that Iwasawa theory and the Kolyvagin–Flach approach were each inadequate on their own, but together they could be made powerful enough to overcome this final hurdle. On 24 October 1994, Wiles submitted two manuscripts, "Modular elliptic curves and Fermat's Last Theorem" and "Ring theoretic properties of certain Hecke algebras", the second of which was co-authored with Taylor and proved that certain conditions were met that were needed to justify the corrected step in the main paper. The two papers were vetted and published as the entirety of the May 1995 issue of the
Annals of Mathematics. The proof's method of identification of a deformation ring with a Hecke algebra (now referred to as an
R=T theorem) to prove modularity lifting theorems has been an influential development in
algebraic number theory. These papers established the modularity theorem for semistable elliptic curves, the last step in proving Fermat's Last Theorem, 358 years after it was conjectured.
Subsequent developments The full Taniyama–Shimura–Weil conjecture was finally proved by Diamond (1996), Conrad et al. (1999), and Breuil et al. (2001) who, building on Wiles's work, incrementally chipped away at the remaining cases until the full result was proved. The now fully proved conjecture became known as the
modularity theorem. Several other theorems in number theory similar to Fermat's Last Theorem also follow from the same reasoning, using the modularity theorem. For example: no cube can be written as a sum of two coprime
nth powers, . (The case was already known by
Euler.) == Relationship to other problems and generalizations ==