The 1670 edition of Diophantus' Arithmetica includes Fermat's commentary, particularly his "Last Theorem" (Observatio Domini Petri de Fermat).

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 can satisfy the equation aⁿ + bⁿ = cⁿ for any integer value of n greater than two.

This 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. No successful proof was published until 1995 despite the efforts of countless mathematicians during the 358 intervening years. 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 is among the most famous theorems in the history of mathematics and prior to its 1995 proof was in the Guinness Book of World Records for "most difficult mathematical problems".

Overview [

The problem [

Fermat's Last Theorem (known as by this title historically although technically a conjecture–or unproven speculation–until proven in 1995) stood as an unsolved riddle in mathematics for over 3 centuries. The Theorem itself is a deceptively simple statement within mathematics that Fermat famously stated he had solved around 1637. His claim was discovered some 30 years later, after his death, as a bare statement in the margin of a book, but Fermat died without leaving any proof of his claim.

The claim eventually became one of the most famous unsolved problems of mathematics. The attempts made to prove it during that time prompted substantial development of number theory and over time Fermat's Last Theorem itself gained legendary prominence as an unsolved problem in popular mathematics. It is based upon the well known formula ("Pythagoras' Theorem") for a right-angle triangle discovered by the ancient Greek mathematician Pythagoras: a² + b² = c².

As it stands, the equation has an infinite number of whole-number solutions, representing the sides of a right-angle triangle. Fermat claimed that he had a proof that this theorem had no solutions for any other whole-number (or "integer") exponent than 2 — in other words that although a² + b² = c² had an infinity of whole-number solutions, the similar equations

a³ + b³ = c³

a⁴ + b⁴ = c⁴

aⁿ + bⁿ = cⁿ

for any other integer exponent "n" more than 2, would have no solutions. Fermat left no proof of the conjecture for all n apart from the special case n = 4.

Subsequent developments and solution [

With the special case n = 4 proved, the problem was to prove the theorem for exponents n that are prime numbers. Over the next two centuries (1637–1839), the conjecture was proven for only the primes 3, 5, and 7, although Sophie Germain innovated and proved an approach which was relevant to an entire class of primes. In the mid-19th century, Ernst Kummer extended this and proved the theorem for all regular primes, leaving irregular primes to be analyzed individually. Building on Kummer's work and using sophisticated computer studies, other mathematicians were able to extend the proof to cover all prime exponents up to four million, but a proof for all exponents was inaccessible (meaning that mathematicians generally considered a proof to be either impossible, or at best exceedingly difficult, or not achievable with current knowledge).

The proof of Fermat's Last Theorem in full, for all n, was finally accomplished, however, after 358 years, by Andrew Wiles in 1995, an achievement for which he was honoured and received numerous awards. The solution came in a roundabout manner, from a completely different area of mathematics.

Around 1955 Japanese mathematicians Japanese mathematicians Goro Shimura and Yutaka Taniyama suspected a link might exist between elliptic curves and modular forms, two completely different areas of mathematics. Known at the time as the Taniyama–Shimura-Weil conjecture, and (eventually) as the modularity theorem, it stood on its own, with no apparent connection to Fermat's Last Theorem. It was widely seen as significant and important in its own right, but was (like Fermat's equation) widely considered to be completely inaccessible to proof.

In 1984, Gerhard Frey noticed an apparent link between the modularity theorem and Fermat's Last Theorem. This potential link was confirmed two years later by Ken Ribet (see: Ribet's Theorem and Frey curve). English mathematician Andrew Wiles, who had a childhood fascination with Fermat's Last Theorem, decided on hearing this, to try and prove the modularity theorem, as a way to prove Fermat's Last Theorem. In 1993, after six years working secretly on the problem, Wiles succeeded in proving enough of the modularity theorem to prove Fermat's Last Theorem. Wiles paper was massive in size and scope. A flaw was discovered in one part of his original paper during peer review and required a further year and collaboration with a past student, Richard Taylor to resolve; as a result the final proof in 1995 was accompanied by a second, smaller, joint paper to that effect. Wiles's achievement was reported widely in the popular press, and was popularized in books and television programs. The remaining parts of the modularity theorem were subsequently proven by other mathematicians, building on Wiles' work, between 1996 and 2001.

Mathematical History [

Pythagoras and Diophantus [

Pythagorean triples [

A Pythagorean triple - named for the ancient Greek Pythagoras - is a set of three integers (a, b, c) that satisfy a special case of Fermat's equation (n = 2)^[1]

Examples of Pythagorean triples include (3, 4, 5) and (5, 12, 13). There are infinitely many such triples,^[2] and methods for generating such triples have been studied in many cultures, beginning with the Babylonians^[3] and later ancient Greek, Chinese, and Indian mathematicians.^[4] The traditional interest in Pythagorean triples connects with the Pythagorean theorem;^[5] in its converse form, it states that a triangle with sides of lengths a, b, and c has a right angle between the a and b legs when the numbers are a Pythagorean triple. Right angles have various practical applications, such as surveying, carpentry, masonry, and construction. Fermat's Last Theorem is an extension of this problem to higher powers, stating that no solution exists when the exponent 2 is replaced by any larger integer.

Diophantine equations [

Fermat's equation, xⁿ + yⁿ = zⁿ with positive integer solutions, is an example of a Diophantine equation,^[6] named for the 3rd-century Alexandrian mathematician, Diophantus, who studied them and developed methods for the solution of some kinds of Diophantine equation. 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:

Diophantus's major work is the Arithmetica, of which only a portion has survived.^[7] Fermat's conjecture of his Last Theorem was inspired while reading a new edition of the Arithmetica,^[8] which was translated into Latin and published in 1621 by Claude Bachet.^[9]

Diophantine equations have been studied for thousands of years. For example, the solutions to the quadratic Diophantine equation x² + y² = z² are given by the Pythagorean triples, originally solved by the Babylonians (c. 1800 BC).^[10] Solutions to linear Diophantine equations, such as 26x + 65y = 13, may be found using the Euclidean algorithm (c. 5th century BC).^[11] 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 xⁿ + yⁿ = z^m where n and m are relatively prime natural numbers.^{[note 1]}

Fermat's conjecture [

Problem II.8 in the 1621 edition of the Arithmetica of Diophantus. On the right is the famous margin which was too small to contain Fermat's alleged proof of his "last theorem".

Problem II.8 of the Arithmetica 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 k² = u² + v². Diophantus shows how to solve this sum-of-squares problem for k = 4 (the solutions being u = 16/5 and v = 12/5).^[12]

Around 1637, Fermat wrote his Last Theorem in the margin of his copy of the Arithmetica next to Diophantus' sum-of-squares problem:^[13]

It is not known whether Fermat had actually found a valid proof. His proof of one case (n = 4) by infinite descent has survived.^[15] Fermat posed the cases of n = 4 and of n = 3 as challenges to his mathematical correspondents, such as Marin Mersenne, Blaise Pascal, and John Wallis.^[16] However, in the last thirty years of his life, Fermat never again wrote of his "truly marvellous proof" of the general case.

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.^[17] The margin note became known as Fermat's Last Theorem,^[18] as it was the last of Fermat's asserted theorems to remain unproven.^[19]

Proofs for specific exponents [

Only one mathematical proof by Fermat has survived, in which Fermat 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.^[20] His proof is equivalent to demonstrating that the equation

has no primitive solutions in integers (no pairwise coprime solutions). In turn, this proves Fermat's Last Theorem for the case n=4, since the equation a⁴ + b⁴ = c⁴ can be written as c⁴ − b⁴ = (a²)².

Alternative proofs of the case n = 4 were developed later^[21] by Frénicle de Bessy (1676),^[22] Leonhard Euler (1738),^[23] Kausler (1802),^[24] Peter Barlow (1811),^[25] Adrien-Marie Legendre (1830),^[26] Schopis (1825),^[27] Terquem (1846),^[28] Joseph Bertrand (1851),^[29] Victor Lebesgue (1853, 1859, 1862),^[30] Theophile Pepin (1883),^[31] Tafelmacher (1893),^[32] David Hilbert (1897),^[33] Bendz (1901),^[34] Gambioli (1901),^[35] Leopold Kronecker (1901),^[36] Bang (1905),^[37] Sommer (1907),^[38] Bottari (1908),^[39] Karel Rychlík (1910),^[40] Nutzhorn (1912),^[41] Robert Carmichael (1913),^[42] Hancock (1931),^[43] and Vrǎnceanu (1966).^[44]

For another proof for n=4 by infinite descent, see Infinite descent: Non-solvability of r² + s⁴ = t⁴. For various proofs for n=4 by infinite descent, see Grant and Perella (1999),^[45] Barbara (2007),^[46] and Dolan (2011).^[47]

After Fermat proved the special case n = 4, the general proof for all n required only that the theorem be established for all odd prime exponents.^[48] In other words, it was necessary to prove only that the equation aⁿ + bⁿ = cⁿ has no integer solutions (a, b, c) when n is an odd prime number. This follows because a solution (a, b, c) 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 (a^d, b^d, c^d) is a solution for the exponent e

(a^d)^e + (b^d)^e = (c^d)^e.

Thus, to prove that Fermat's equation has no solutions for n > 2, it suffices to prove that it has no solutions for at least one prime factor of every n. All integers n > 2 contain a factor of 4, or an odd prime number, or both. Therefore, Fermat's Last Theorem can be proven for all n if it can be proven for n = 4 and for all odd primes p (the only even prime number is the number 2).

In the two centuries following its conjecture (1637–1839), Fermat's Last Theorem was proven for three odd prime exponents p = 3, 5 and 7. The case p = 3 was first stated by Abu-Mahmud Khojandi (10th century), but his attempted proof of the theorem was incorrect.^[49] In 1770, Leonhard Euler gave a proof of p = 3,^[50] but his proof by infinite descent^[51] contained a major gap.^[52] However, since Euler himself had proven the lemma necessary to complete the proof in other work, he is generally credited with the first proof.^[53] Independent proofs were published^[54] by Kausler (1802),^[24] Legendre (1823, 1830),^[26][55] Calzolari (1855),^[56] Gabriel Lamé (1865),^[57] Peter Guthrie Tait (1872),^[58] Günther (1878),^[59] Gambioli (1901),^[35] Krey (1909),^[60] Rychlík (1910),^[40] Stockhaus (1910),^[61] Carmichael (1915),^[62] Johannes van der Corput (1915),^[63] Axel Thue (1917),^[64] and Duarte (1944).^[65] The case p = 5 was proven^[66] independently by Legendre and Peter Gustav Lejeune Dirichlet around 1825.^[67] Alternative proofs were developed^[68] by Carl Friedrich Gauss (1875, posthumous),^[69] Lebesgue (1843),^[70] Lamé (1847),^[71] Gambioli (1901),^[35][72] Werebrusow (1905),^[73] Rychlík (1910),^[74] van der Corput (1915),^[63] and Guy Terjanian (1987).^[75] The case p = 7 was proven^[76] by Lamé in 1839.^[77] His rather complicated proof was simplified in 1840 by Lebesgue,^[78] and still simpler proofs^[79] were published by Angelo Genocchi in 1864, 1874 and 1876.^[80] Alternative proofs were developed by Théophile Pépin (1876)^[81] and Edmond Maillet (1897).^[82]

Fermat's Last Theorem has also been proven for the exponents n = 6, 10, and 14. Proofs for n = 6 have been published by Kausler,^[24] Thue,^[83] Tafelmacher,^[84] Lind,^[85] Kapferer,^[86] Swift,^[87] and Breusch.^[88] Similarly, Dirichlet^[89] and Terjanian^[90] each proved the case n = 14, while Kapferer^[86] and Breusch^[88] each proved the case n = 10. Strictly speaking, these proofs are unnecessary, since these cases follow from the proofs for n = 3, 5, and 7, respectively. Nevertheless, the reasoning of these even-exponent proofs differs from their odd-exponent counterparts. Dirichlet's proof for n = 14 was published in 1832, before Lamé's 1839 proof for n = 7.^[91]

Many proofs for specific exponents use Fermat's technique of infinite descent, which Fermat used to prove the case n = 4, but many do not. However, the details and auxiliary arguments are often ad hoc and tied to the individual exponent under consideration.^[92] Since they became ever more complicated as p increased, it seemed unlikely that the general case of Fermat's Last Theorem could be proven by building upon the proofs for individual exponents.^[92] Although some general results on Fermat's Last Theorem were published in the early 19th century by Niels Henrik Abel and Peter Barlow,^[93][94] the first significant work on the general theorem was done by Sophie Germain.^[95]

Sophie Germain [

In the early 19th century, Sophie Germain developed several novel approaches to prove Fermat's Last Theorem for all exponents.^[96] First, she defined a set of auxiliary primes θ constructed from the prime exponent p by the equation θ = 2hp+1, where h is any integer not divisible by three. She showed that if no integers raised to the p^th 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 100.^[96][97] Germain tried unsuccessfully to prove the first case of Fermat's Last Theorem for all even exponents, specifically for n = 2p, which was proven by Guy Terjanian in 1977.^[98] 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.^[99]

Ernst Kummer and the theory of ideals [

In 1847, Gabriel Lamé outlined a proof of Fermat's Last Theorem based on factoring the equation x^p + y^p = z^p 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. 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) which conjecturally occur approximately 39% of the time; the only irregular primes below 100 are 37, 59 and 67.

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.^[100] This conjecture was proven in 1983 by Gerd Faltings,^[101] 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.^[102] By 1978, Samuel Wagstaff had extended this to all primes less than 125,000.^[103] By 1993, Fermat's Last Theorem had been proven for all primes less than four million.^[104]

Connection with elliptic curves [

The strategy that ultimately led to a successful proof of Fermat's Last Theorem arose from the "astounding"[105]^:211 Taniyama–Shimura-Weil conjecture, proposed around 1955, which many mathematicians believed would be near to impossible to prove,^[105]:223 and which was linked in the 1980s by Gerhard Frey and Ken Ribet to Fermat's equation. By accomplishing a partial proof of this conjecture in 1995, 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 the modularity theorem.

The 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. It was initially dismissed as unlikely or highly speculative, and was taken more seriously when number theorist André Weil found evidence supporting it, but no proof; as a result the "astounding"^[105]:211 conjecture was often known as the Taniyama–Shimura-Weil conjecture. It became a part of the Langlands programme, a list of important conjectures needing proof or disproof.^{[105]:211-215}

Even after gaining serious attention, the conjecture was seen by contemporary mathematicians as extraordinarily difficult or perhaps inaccessible to proof.^{[105]:203-205, 223, 226} For example, Wiles' ex-supervisor John Coates states that it seemed "impossible to actually prove",^[105]:226 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]." ^[105]:223

Frey's equation / Ribet's theorem [

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 (a, b, c) for exponent p > 2, then it could be shown that the elliptic curve (now known as a Frey curve ^{[note 2]})

y² = x (x − a^p)(x + b^p)

would have such unusual properties that it was unlikely to be able to be modular.^[106] 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 would simultaneously prove Fermat's Last Theorem.^[107] (Equally, a disproof or refutation of Fermat's Last Theorem would disprove the conjecture)

Following this strategy, a proof of Fermat's Last Theorem required two steps. First, 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 did not quite succeed in proving this rigorously; the missing piece (the so-called "epsilon conjecture", now known as Ribet's theorem) was noticed by Jean-Pierre Serre^{[citation needed]} and proven in 1986 by Ken Ribet. Second, it was necessary to prove the modularity theorem - or at least to prove it for thee sub-class of cases (known as semistable elliptic curves) which included Frey's equation - and this was widely believed inaccessible to proof by contemporary mathematicians.^{[105]:203-205, 223, 226}

• Ribet's theorem - proven in 1986 - showed that if a solution to Fermat's equation existed, it could be used to create a semistable elliptic curve that was not modular;
• The modularity theorem - if proven for Frey's equation - would mean all elliptic curves (and hence all semistable elliptic curves) are of necessity modular.
• The contradiction implies that no solutions can exist to Fermat's equation, thus proving Fermat's Last Theorem.

Wiles's general proof [

British mathematician Andrew Wiles