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 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".

History

Fermat left no proof of the conjecture for all n, but he did prove the special case n = 4. This reduced the problem to proving 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 proved a special case for all primes less than 100. In the mid-19th century, Ernst Kummer proved the theorem for regular primes. Building on Kummer's work and using sophisticated computer studies, other mathematicians were able to prove the conjecture for all odd primes up to four million.

The final proof of the conjecture for all n came in the late 20th century. In 1984, Gerhard Frey suggested the approach of proving the conjecture through a proof of the modularity theorem for elliptic curves. Building on work of Ken Ribet, Andrew Wiles succeeded in proving enough of the modularity theorem to prove Fermat's Last Theorem, with the assistance of Richard Taylor. Wiles's achievement was reported widely in the popular press, and has been popularized in books and television programs.

Mathematical context

Pythagorean triples

A Pythagorean triple is a set of three integers (a, b, c) that satisfy a special case of Fermat's equation (n = 2)

Examples of Pythagorean triples 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. The traditional interest in Pythagorean triples connects with the Pythagorean theorem; 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 is an example of a Diophantine equation. A Diophantine equation is a polynomial equation in which the solutions must be integers. Their name derives from the 3rd-century Alexandrian mathematician, Diophantus, who developed methods for their solution. 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. Fermat's conjecture of his Last Theorem was inspired while reading a new edition of the Arithmetica, which 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 x² + y² = z² are given by the Pythagorean triples, originally solved by the Babylonians (c. 1800 BC). Solutions to linear Diophantine equations, such as 26x + 65y = 13, 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 xⁿ + yⁿ = z^m where n and m are relatively prime natural numbers.

Fermat's conjecture

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).

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

Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et generaliter nullam in infinitum ultra quadratum potestatem in duos eiusdem nominis fas est dividere cuius rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet.

it is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second, into two like powers. I have discovered a truly marvelous proof of this, which this margin is too narrow to contain.

Although Fermat's general proof is unknown, his proof of one case (n = 4) by infinite descent has survived. 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. 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. The margin note became known as Fermat's Last Theorem, as it was the last of Fermat's asserted theorems to remain unproven.

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. 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 by Frénicle de Bessy (1676), Leonhard Euler (1738), Kausler (1802), Peter Barlow (1811), Adrien-Marie Legendre (1830), Schopis (1825), Terquem (1846), Joseph Bertrand (1851), Victor Lebesgue (1853, 1859, 1862), Theophile Pepin (1883), Tafelmacher (1893), David Hilbert (1897), Bendz (1901), Gambioli (1901), Leopold Kronecker (1901), Bang (1905), Sommer (1907), Bottari (1908), Karel Rychlík (1910), Nutzhorn (1912), Robert Carmichael (1913), Hancock (1931), and Vrǎnceanu (1966).

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), Barbara (2007), and Dolan (2011).

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. 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. 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 proven 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), 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 p = 5 was proven independently by Legendre and Peter 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), and Guy Terjanian (1987). The case p = 7 was proven 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 has also been proven for the exponents n = 6, 10, and 14. Proofs for n = 6 have been published by Kausler, Thue, Tafelmacher, Lind, Kapferer, Swift, and Breusch. Similarly, Dirichlet and Terjanian each proved the case n = 14, while Kapferer and Breusch 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.

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. 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. Although some general results on Fermat's Last Theorem were published in the early 19th century by Niels Henrik Abel and Peter Barlow, the first significant work on the general theorem was done by Sophie Germain.

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 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. 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. 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 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. This conjecture was proven 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 proven for all primes less than four million.

Connection with elliptic curves

The ultimately successful strategy for proving Fermat's Last Theorem was by proving the modularity theorem. The strategy was first described by Gerhard Frey in 1984. Frey noted that if Fermat's equation had a solution (a, b, c) for exponent p > 2, the corresponding elliptic curve

y² = x (x − a^p)(x + b^p)

would have such unusual properties that the curve would likely violate the modularity theorem. This theorem, first conjectured in the mid-1950s and gradually refined through the 1960s, states that every elliptic curve is modular, meaning that it can be associated with a unique modular form.

Following this strategy, the proof of Fermat's Last Theorem required two steps. First, it was necessary to show that Frey's intuition was correct: that the above elliptic curve, if it exists, is always non-modular. Frey did not succeed in proving this rigorously; the missing piece was identified by Jean-Pierre Serre. This missing piece, the so-called " epsilon conjecture", was proven by Ken Ribet in 1986. Second, it was necessary to prove a special case of the modularity theorem. This special case (for semistable elliptic curves) was proven by Andrew Wiles in 1995.

Thus, the epsilon conjecture showed that any solution to Fermat's equation could be used to generate a non-modular semistable elliptic curve, whereas Wiles' proof showed that all such elliptic curves must be modular. This contradiction implies that there can be no solutions to Fermat's equation, thus proving Fermat's Last Theorem.

Wiles's general proof

Ribet's proof of the epsilon conjecture in 1986 accomplished the first half of Frey's strategy for proving Fermat's Last Theorem. Upon hearing of Ribet's proof, Andrew Wiles 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 almost complete secrecy. He based his initial approach on his area of expertise, Horizontal Iwasawa theory, but by the summer of 1991, this approach seemed inadequate to the task. In response, he exploited an Euler system recently developed by Victor Kolyvagin and Matthias Flach. Since Wiles was unfamiliar with such methods, he asked his Princeton colleague, Nick Katz, to check his reasoning over the spring semester of 1993.

By mid-1993, Wiles was sufficiently confident of his results that he presented them in three lectures delivered on June 21–23, 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 soon became apparent that Wiles's initial proof was incorrect. A critical portion of the proof 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, who alerted Wiles on 23 August 1993.

Wiles and his former student Richard Taylor spent almost a year trying to repair the proof, without success. On 19 September 1994, Wiles had a flash of insight that the proof could be saved by returning to his original Horizontal Iwasawa theory approach, which he had abandoned in favour of the Kolyvagin–Flach approach. 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. The two papers were vetted and published as the entirety of the May 1995 issue of the Annals of Mathematics. 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.

Exponents other than positive integers

Rational exponents

All solutions of the Diophantine equation when n=1 were computed by Lenstra in 1992. In the case in which the m^th roots are required to be real and positive, all solutions are given by

for positive integers r, s, t with s and t coprime.

In 2004, for n>2, Bennett, Glass, and Szekely proved that if gcd(n,m)=1, then there are integer solutions if and only if 6 divides m, and , and are different complex 6th roots of the same real number.

Negative exponents

n = –1

All primitive (pairwise coprime) integer solutions to can be written as

for positive, coprime integers m, n.

n = –2

The case n = –2 also has an infinitude of solutions, and these have a geometric interpretation in terms of right triangles with integer sides and an integer altitude to the hypotenuse. All primitive solutions to are given by

for coprime integers u, v with v > u. The geometric interpretation is that a and b are the integer legs of a right triangle and d is the integer altitude to the hypotenuse. Then the hypotenuse itself is the integer

so (a, b, c) is a Pythagorean triple.

Integer n < –2

There are no solutions in integers for for integer n < –2. If there were, the equation could be multiplied through by to obtain , which is impossible by Fermat's Last Theorem.

Did Fermat possess a general proof?

The mathematical techniques used in Fermat's "marvelous" proof are unknown. Only one detailed proof of Fermat has survived, the above proof that no three coprime integers (x, y, z) satisfy the equation x⁴ − y⁴ = z².

Taylor and Wiles's proof relies on mathematical techniques developed in the twentieth century, which would be unknown to mathematicians who had worked on Fermat's Last Theorem even a century earlier. Fermat's alleged "marvellous proof", by comparison, would have had to be elementary, given mathematical knowledge of the time, and so could not have been the same as Wiles' proof. Most mathematicians and science historians doubt that Fermat had a valid proof of his theorem for all exponents n.

Harvey Friedman's grand conjecture implies that Fermat's last theorem can be proved in elementary arithmetic, a rather weak form of arithmetic with addition, multiplication, exponentiation, and a limited form of induction for formulas with bounded quantifiers. Any such proof would be elementary but possibly too long to write down.

Monetary prizes

In 1816 and again in 1850, the French Academy of Sciences offered a prize for a general proof of Fermat's Last Theorem. In 1857, the Academy awarded 3000 francs and a gold medal to Kummer for his research on ideal numbers, although he had not submitted an entry for the prize. Another prize was offered in 1883 by the Academy of Brussels.

In 1908, the German industrialist and amateur mathematician Paul Wolfskehl bequeathed 100,000 marks to the Göttingen Academy of Sciences to be offered as a prize for a complete proof of Fermat's Last Theorem. 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. Wiles collected the Wolfskehl prize money, then worth $50,000, on 27 June 1997.

Prior to Wiles' proof, thousands of incorrect proofs were submitted to the Wolfskehl committee, amounting to roughly 10 feet (3 meters) of correspondence. 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 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". 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."

In popular culture

• An episode in the television series Star Trek: The Next Generation, titled " The Royale", refers to the theorem in the first act. Riker visits Captain Jean-Luc Picard in his ready room to report only to find Picard puzzling over Fermat's last theorem. Picard's interest in this theorem goes beyond the difficulty of the puzzle; he also feels humbled that despite their advanced technology, they are still unable to solve a problem set forth by a man who had no computer. An episode in Star Trek: Deep Space 9, titled " Facets", refers to the theorem as well. In a scene involving O'Brien, Tobin Dax mentions continuing work on his own attempt to solve Fermat's last theorem.
• "The Proof" – Nova ( PBS) documentary about Andrew Wiles's proof of Fermat's Last Theorem.
• On August 17, 2011, a Google doodle was shown on the Google homepage, showing a blackboard with the theorem on it. When hovered over, it displays the text "I have discovered a truly marvelous proof of this theorem, which this doodle is too small to contain." This is a reference to the note made by Fermat in the margins of Arithmetica. It commemorated the 410th birth anniversary of de Fermat.
• In the book The Girl Who Played with Fire, main character Lisbeth Salander becomes obsessed with the theorem in the opening chapters of the book. Her continuing effort to come up with a proof on her own is a running sub-plot throughout the story, and is used as a way to demonstrate her exceptional intelligence. In the end she comes up with a proof (the actual proof is not featured in the book). But after being shot in the head and surviving, she has lost the proof.
• In the Harold Ramis re-make of the movie Bedazzled, starring Brendan Fraser and Elizabeth Hurley, Fermat's Last Theorem appears written on the chalkboard in the classroom that the protagonist Elliot finds himself teleported to after he aborts his failed fourth wish. In the director's commentary for the DVD release, director Ramis comments that nobody has seemed to notice that the equation on the board is Fermat's Last Theorem.
• In Doctor Who, Season 5 Episode 1 " The Eleventh Hour", the Doctor transmits a proof of Fermat's Last Theorem by typing it in just a few seconds on Jeff's laptop to prove his genius to a collection of world leaders discussing the latest threat to the human race. This implies that the Doctor knew a proof which was quite short and easy for others to comprehend.
• In The IT Crowd, Series 3 Episode 6 " Calendar Geeks" Fermat's Last Theorem is referenced during a photo shoot for a calendar about geeks and achievements in Science and Mathematics.
• The song "Bizarro Genius Baby" by MC Frontalot contains the lyrics "And no dust had settled when she’d disproved Fermat by finding A³ + B³ that = C³".
• In the manga and anime series of Zatch Bell! one of the questions from the gatekeeper Unko Tintin was to prove Fermat's Last Theorem. The lead protagonist managed to avoid it by asking whether Unko Tintin could answer it himself, which he could not.