Fermat's Last Theorem
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number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777â ...
, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three
positive Positive is a property of positivity and may refer to: Mathematics and science * Positive formula, a logical formula not containing negation * Positive number, a number that is greater than 0 * Plus sign, the sign "+" used to indicate a posi ...
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
s , , and satisfy the equation for any integer value of greater than 2. The cases and have been known since antiquity to have infinitely many solutions.Singh, pp. 18–20. The proposition was first stated as a theorem by
Pierre de Fermat Pierre de Fermat (; between 31 October and 6 December 1607 – 12 January 1665) was a French mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality. In particular, he ...
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 In additive number theory, Fermat's theorem on sums of two squares states that an odd prime ''p'' can be expressed as: :p = x^2 + y^2, with ''x'' and ''y'' integers, if and only if :p \equiv 1 \pmod. The prime numbers for which this is true ar ...
), 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 Sir Andrew John Wiles (born 11 April 1953) is an English mathematician and a Royal Society Research Professor at the University of Oxford, specializing in number theory. He is best known for proving Fermat's Last Theorem, for which he was awa ...
and formally published in 1995. It was described as a "stunning advance" in the citation for Wiles's
Abel Prize The Abel Prize ( ; no, Abelprisen ) is awarded annually by the King of Norway to one or more outstanding mathematicians. It is named after the Norwegian mathematician Niels Henrik Abel (1802–1829) and directly modeled after the Nobel Prizes. ...
award in 2016. It also proved much of the Taniyama–Shimura conjecture, subsequently known as the
modularity theorem The modularity theorem (formerly called the Taniyama–Shimura conjecture, Taniyama-Weil conjecture or modularity conjecture for elliptic curves) states that elliptic curves over the field of rational numbers are related to modular forms. And ...
, 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 The history of mathematics deals with the origin of discoveries in mathematics and the mathematical methods and notation of the past. Before the modern age and the worldwide spread of knowledge, written examples of new mathematical developments ...
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.


Overview


Pythagorean origins

The
Pythagorean equation In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite ...
, , has an infinite number of positive
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
solutions for ''x'', ''y'', and ''z''; these solutions are known as
Pythagorean triple A Pythagorean triple consists of three positive integers , , and , such that . Such a triple is commonly written , and a well-known example is . If is a Pythagorean triple, then so is for any positive integer . A primitive Pythagorean triple is ...
s (with the simplest example 3,4,5). Around 1637, Fermat wrote in the margin of a book that the more general equation 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. The claim eventually became one of the most notable unsolved problems of mathematics. Attempts to prove it prompted substantial development in
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777â ...
, and over time Fermat's Last Theorem gained prominence as an unsolved problem in mathematics.


Subsequent developments and solution

The special case , proved by Fermat himself, is sufficient to establish that if the theorem is false for some exponent ''n'' that is not a
prime number A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
, it must also be false for some smaller ''n'', so only prime values of ''n'' need further investigation.If the exponent ''n'' were not prime or 4, then it would be possible to write ''n'' either as a product of two smaller integers (''n'' = ''PQ''), in which ''P'' is a prime number greater than 2, and then ''an'' = ''aPQ'' = (''aQ'')''P'' for each of ''a'', ''b'', and ''c''. That is, an equivalent solution would ''also'' have to exist for the prime power ''P'' that is ''smaller'' than ''n''; or else as ''n'' would be a power of 2 greater than 4, and writing ''n'' = 4''Q'', the same argument would hold. Over the next two centuries (1637–1839), the conjecture was proved for only the primes 3, 5, and 7, although
Sophie Germain Marie-Sophie Germain (; 1 April 1776 – 27 June 1831) was a French mathematician, physicist, and philosopher. Despite initial opposition from her parents and difficulties presented by society, she gained education from books in her father's lib ...
innovated and proved an approach that was relevant to an entire class of primes. In the mid-19th century,
Ernst Kummer Ernst Eduard Kummer (29 January 1810 – 14 May 1893) was a German mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned ...
extended this and proved the theorem for all
regular prime 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. Regular primes may be defined via the divisibility of either class numbers or of Bernoulli nu ...
s, 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 impossible, exceedingly difficult, or unachievable with current knowledge). Separately, around 1955, Japanese mathematicians
Goro Shimura was a Japanese mathematician and Michael Henry Strater Professor Emeritus of Mathematics at Princeton University who worked in number theory, automorphic forms, and arithmetic geometry. He was known for developing the theory of complex multipli ...
and
Yutaka Taniyama was a Japanese mathematician known for the Taniyama–Shimura conjecture. Contribution Taniyama was best known for conjecturing, in modern language, automorphic properties of L-functions of elliptic curves over any number field. A partial and r ...
suspected a link might exist between
elliptic curve In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If ...
s and modular forms, two completely different areas of mathematics. Known at the time as the Taniyama–Shimura conjecture (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 theorem) widely considered completely inaccessible to proof. In 1984,
Gerhard Frey Gerhard Frey (; born 1 June 1944) is a German mathematician, known for his work in number theory. Following an original idea of Hellegouarch, he developed the notion of Frey–Hellegouarch curves, a construction of an elliptic curve from a pur ...
noticed an apparent link between these two previously unrelated and unsolved problems. An outline suggesting this could be proved was given by Frey. The full proof that the two problems were closely linked was accomplished in 1986 by
Ken Ribet Kenneth Alan Ribet (; born June 28, 1948) is an American mathematician working in algebraic number theory and algebraic geometry. He is known for the Herbrand–Ribet theorem and Ribet's theorem, which were key ingredients in the proof of Ferma ...
, building on a partial proof by
Jean-Pierre Serre Jean-Pierre Serre (; born 15 September 1926) is a French mathematician who has made contributions to algebraic topology, algebraic geometry, and algebraic number theory. He was awarded the Fields Medal in 1954, the Wolf Prize in 2000 and the ina ...
, who proved all but one part known as the "epsilon conjecture" (see: ''
Ribet's Theorem Ribet's theorem (earlier called the epsilon conjecture or Δ-conjecture) is part of number theory. It concerns properties of Galois representations associated with modular forms. It was proposed by Jean-Pierre Serre and proven by Ken Ribet. The ...
'' and ''
Frey curve In mathematics, a Frey curve or Frey–Hellegouarch curve is the elliptic curve ::y^2 = x(x - a^\ell)(x + b^\ell) associated with a (hypothetical) solution of Fermat's equation :a^\ell + b^\ell = c^\ell. The curve is named after Gerhard Frey. Hi ...
''). These papers by Frey, Serre and Ribet showed that if the Taniyama–Shimura conjecture could be proven for at least the semi-stable class of elliptic curves, a proof of Fermat's Last Theorem would also follow automatically. The connection is described below: any solution that could contradict Fermat's Last Theorem could also be used to contradict the Taniyama–Shimura conjecture. So if the modularity theorem were found to be true, then by definition no solution contradicting Fermat's Last Theorem could exist, which would therefore have to be true as well. Although both problems were daunting and widely considered to be "completely inaccessible" to proof at the time, this was the first suggestion of a route by which Fermat's Last Theorem could be extended and proved for all numbers, not just some numbers. Unlike Fermat's Last Theorem, the Taniyama–Shimura conjecture was a major active research area and viewed as more within reach of contemporary mathematics. However, general opinion was that this simply showed the impracticality of proving the Taniyama–Shimura conjecture.Singh, p. 144. Mathematician John Coates' quoted reaction was a common one: : "I myself was very sceptical that the beautiful link between Fermat’s Last Theorem and the Taniyama–Shimura conjecture would actually lead to anything, because I must confess I did not think that the Taniyama–Shimura conjecture was accessible to proof. Beautiful though this problem was, it seemed impossible to actually prove. I must confess I thought I probably wouldn’t see it proved in my lifetime." On hearing that Ribet had proven Frey's link to be correct, English mathematician
Andrew Wiles Sir Andrew John Wiles (born 11 April 1953) is an English mathematician and a Royal Society Research Professor at the University of Oxford, specializing in number theory. He is best known for proving Fermat's Last Theorem, for which he was awa ...
, who had a childhood fascination with Fermat's Last Theorem and had a background of working with elliptic curves and related fields, decided to try to prove the Taniyama–Shimura conjecture as a way to prove Fermat's Last Theorem. In 1993, after six years of working secretly on the problem, Wiles succeeded in proving enough of the conjecture to prove Fermat's Last Theorem. Wiles's paper was massive in size and scope. A flaw was discovered in one part of his original paper during
peer review Peer review is the evaluation of work by one or more people with similar competencies as the producers of the work (peers). It functions as a form of self-regulation by qualified members of a profession within the relevant field. 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 smaller joint paper showing that the fixed steps were valid. Wiles's achievement was reported widely in the popular press, and was popularized in books and television programs. The remaining parts of the Taniyama–Shimura–Weil conjecture, now proven and known as the modularity theorem, were subsequently proved by other mathematicians, who built on Wiles's work between 1996 and 2001. For his proof, Wiles was honoured and received numerous awards, including the 2016
Abel Prize The Abel Prize ( ; no, Abelprisen ) is awarded annually by the King of Norway to one or more outstanding mathematicians. It is named after the Norwegian mathematician Niels Henrik Abel (1802–1829) and directly modeled after the Nobel Prizes. ...
.


Equivalent statements of the theorem

There are several alternative ways to state Fermat's Last Theorem that are mathematically equivalent to the original statement of the problem. In order to state them, we use mathematical notation: let be the set of natural numbers 1, 2, 3, ..., let be the set of integers 0, ±1, ±2, ..., and let be the set of rational numbers , where and are in with . In what follows we will call a solution to where one or more of , , or is zero a ''trivial solution''. A solution where all three are non-zero will be called a ''non-trivial'' solution. For comparison's sake we start with the original formulation. * Original statement. With , , , ∈ (meaning that ''n'', ''x'', ''y'', ''z'' are all positive whole numbers) and , the equation has no solutions. Most popular treatments of the subject state it this way. It is also commonly stated over : * Equivalent statement 1: , where integer ≄ 3, has no non-trivial solutions , , ∈ . The equivalence is clear if is even. If is odd and all three of are negative, then we can replace with to obtain a solution in . If two of them are negative, it must be and or and . If are negative and is positive, then we can rearrange to get resulting in a solution in ; the other case is dealt with analogously. Now if just one is negative, it must be or . If is negative, and and are positive, then it can be rearranged to get again resulting in a solution in ; if is negative, the result follows symmetrically. Thus in all cases a nontrivial solution in would also mean a solution exists in , the original formulation of the problem. * Equivalent statement 2: , where integer ≄ 3, has no non-trivial solutions , , ∈ . This is because the exponents of and are equal (to ), so if there is a solution in , then it can be multiplied through by an appropriate common denominator to get a solution in , and hence in . * Equivalent statement 3: , where integer ≄ 3, has no non-trivial solutions , ∈ . A non-trivial solution , , ∈ to yields the non-trivial solution , ∈ for . Conversely, a solution , ∈ to yields the non-trivial solution for . This last formulation is particularly fruitful, because it reduces the problem from a problem about surfaces in three dimensions to a problem about curves in two dimensions. Furthermore, it allows working over the field , rather than over the ring ;
fields Fields may refer to: Music * Fields (band), an indie rock band formed in 2006 * Fields (progressive rock band), a progressive rock band formed in 1971 * ''Fields'' (album), an LP by Swedish-based indie rock band Junip (2010) * "Fields", a song b ...
exhibit more structure than
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, which allows for deeper analysis of their elements. * Equivalent statement 4 – connection to elliptic curves: If , , is a non-trivial solution to , odd prime, then (
Frey curve In mathematics, a Frey curve or Frey–Hellegouarch curve is the elliptic curve ::y^2 = x(x - a^\ell)(x + b^\ell) associated with a (hypothetical) solution of Fermat's equation :a^\ell + b^\ell = c^\ell. The curve is named after Gerhard Frey. Hi ...
) will be an
elliptic curve In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If ...
. Examining this elliptic curve with
Ribet's theorem Ribet's theorem (earlier called the epsilon conjecture or Δ-conjecture) is part of number theory. It concerns properties of Galois representations associated with modular forms. It was proposed by Jean-Pierre Serre and proven by Ken Ribet. The ...
shows that it does not have a modular form. However, the proof by Andrew Wiles proves that any equation of the form does have a modular form. Any non-trivial solution to (with an odd prime) would therefore create a
contradiction In traditional logic, a contradiction occurs when a proposition conflicts either with itself or established fact. It is often used as a tool to detect disingenuous beliefs and bias. Illustrating a general tendency in applied logic, Aristotle's ...
, which in turn proves that no non-trivial solutions exist. In other words, any solution that could contradict Fermat's Last Theorem could also be used to contradict the Modularity Theorem. So if the modularity theorem were found to be true, then it would follow that no contradiction to Fermat's Last Theorem could exist either. As described above, the discovery of this equivalent statement was crucial to the eventual solution of Fermat's Last Theorem, as it provided a means by which it could be "attacked" for all numbers at once.


Mathematical history


Pythagoras and Diophantus


Pythagorean triples

In ancient times it was known that a triangle whose sides were in the
ratio In mathematics, a ratio shows how many times one number contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio of oranges to lemons is eight to six (that is, 8:6, which is equivalent to the ...
3:4:5 would have a right angle as one of its angles. This was used in
construction Construction is a general term meaning the art and science to form objects, systems, or organizations,"Construction" def. 1.a. 1.b. and 1.c. ''Oxford English Dictionary'' Second Edition on CD-ROM (v. 4.0) Oxford University Press 2009 and com ...
and later in early
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
. 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 In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite t ...
, and a triple of numbers that meets this condition is called a Pythagorean triple – both are named after the ancient Greek
Pythagoras Pythagoras of Samos ( grc, Î Ï…ÎžÎ±ÎłÏŒÏÎ±Ï‚ ᜁ ÎŁÎŹÎŒÎčÎżÏ‚, PythagĂłras ho SĂĄmios, Pythagoras the Samos, Samian, or simply ; in Ionian Greek; ) was an ancient Ionians, Ionian Ancient Greek philosophy, Greek philosopher and the eponymou ...
. 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 Babylonia (; Akkadian: , ''māt Akkadī'') was an ancient Akkadian-speaking state and cultural area based in the city of Babylon in central-southern Mesopotamia (present-day Iraq and parts of Syria). It emerged as an Amorite-ruled state c. 1 ...
and later
ancient Greek Ancient Greek includes the forms of the Greek language used in ancient Greece and the ancient world from around 1500 BC to 300 BC. It is often roughly divided into the following periods: Mycenaean Greek (), Dark Ages (), the Archaic peri ...
,
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, and
Indian Indian or Indians may refer to: Peoples South Asia * Indian people, people of Indian nationality, or people who have an Indian ancestor ** Non-resident Indian, a citizen of India who has temporarily emigrated to another country * South Asia ...
mathematicians. Mathematically, the definition of a Pythagorean triple is a set of three integers (''a'', ''b'', ''c'') that satisfy the equation a^2 + b^2 = c^2.


Diophantine equations

Fermat's equation, ''x''''n'' + ''y''''n'' = ''z''''n'' with positive
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
solutions, is an example of a Diophantine equation, named for the 3rd-century
Alexandria Alexandria ( or ; ar, Ù±Ù„Ù’Ű„ÙŰłÙ’ÙƒÙŽÙ†Ù’ŰŻÙŽŰ±ÙÙŠÙŽÙ‘Ű©Ù ; grc-gre, Î‘Î»Î”ÎŸÎŹÎœÎŽÏÎ”Îčα, AlexĂĄndria) is the second largest city in Egypt, and the largest city on the Mediterranean coast. Founded in by Alexander the Great, Alexandria ...
n 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 ''x''2 + ''y''2 = ''z''2 are given by the
Pythagorean triple A Pythagorean triple consists of three positive integers , , and , such that . Such a triple is commonly written , and a well-known example is . If is a Pythagorean triple, then so is for any positive integer . A primitive Pythagorean triple is ...
s, originally solved by the Babylonians (c. 1800 BC). Solutions to linear Diophantine equations, such as 26''x'' + 65''y'' = 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 ''x''''n'' + ''y''''n'' = ''z''''m'' where ''n'' and ''m'' are
relatively prime In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivale ...
natural numbers.For example, \left((j^r+1)^s\right)^r + \left(j(j^r+1)^s\right)^r = (j^r+1)^.


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 In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ration ...
''k'', find rational numbers ''u'' and ''v'' such that ''k''2 = ''u''2 + ''v''2. 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's sum-of-squares problem: After Fermat's death in 1665, his son ClĂ©ment-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'',Dickson, p. 731. 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 ''n'' = 4, as described in the section '' Proofs for specific exponents''. While Fermat posed the cases of ''n'' = 4 and of ''n'' = 3 as challenges to his mathematical correspondents, such as Marin Mersenne,
Blaise Pascal Blaise Pascal ( , , ; ; 19 June 1623 – 19 August 1662) was a French mathematician, physicist, inventor, philosopher, and Catholic Church, Catholic writer. He was a child prodigy who was educated by his father, a tax collector in Rouen. Pa ...
, 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 suggests that while the absence of a proof is insignificant, the lack of challenges means Fermat realised he did not have a proof; he quotes Weil 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 __NOTOC__ Harvey Friedman (born 23 September 1948)Handbook of Philosophical Logic, , p. 38 is an American mathematical logician at Ohio State University in Columbus, Ohio. He has worked on reverse mathematics, a project intended to derive the axi ...
's
grand conjecture In proof theory, a branch of mathematical logic, elementary function arithmetic (EFA), also called elementary arithmetic and exponential function arithmetic,C. SmoryƄski, "Nonstandard Models and Related Developments" (p. 217). From ''Harvey Fri ...
implies that any provable theorem (including Fermat's last theorem) can be proved using only '
elementary function arithmetic In proof theory, a branch of mathematical logic, elementary function arithmetic (EFA), also called elementary arithmetic and exponential function arithmetic,C. SmoryƄski, "Nonstandard Models and Related Developments" (p. 217). From ''Harvey Frie ...
', 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


Exponent = 4

Only one relevant proof by Fermat has survived, in which he uses the technique of
infinite descent In mathematics, a proof by infinite descent, also known as Fermat's method of descent, is a particular kind of proof by contradiction used to show that a statement cannot possibly hold for any number, by showing that if the statement were to hold f ...
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 In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivale ...
solutions). In turn, this proves Fermat's Last Theorem for the case ''n'' = 4, since the equation ''a''4 + ''b''4 = ''c''4 can be written as ''c''4 − ''b''4 = (''a''2)2. Alternative proofs of the case ''n'' = 4 were developed later by FrĂ©nicle de Bessy (1676),
Leonhard Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
(1738),. Reprinted ''Opera omnia'', ser. I, "Commentationes Arithmeticae", vol. I, pp. 38–58, Leipzig:Teubner (1915). Kausler (1802), Peter Barlow (1811),
Adrien-Marie Legendre Adrien-Marie Legendre (; ; 18 September 1752 – 9 January 1833) was a French mathematician who made numerous contributions to mathematics. Well-known and important concepts such as the Legendre polynomials and Legendre transformation are name ...
(1830), Schopis (1825),
Olry Terquem Olry Terquem (16 June 1782 – 6 May 1862) was a French mathematician. He is known for his works in geometry and for founding two scientific journals, one of which was the first journal about the history of mathematics. He was also the pseudon ...
(1846),
Joseph Bertrand Joseph Louis François Bertrand (; 11 March 1822 – 5 April 1900) was a French mathematician who worked in the fields of number theory, differential geometry, probability theory, economics and thermodynamics. Biography Joseph Bertrand was ...
(1851), Victor Lebesgue (1853, 1859, 1862), Théophile Pépin (1883), Tafelmacher (1893), David Hilbert (1897), Bendz (1901), Gambioli (1901),
Leopold Kronecker Leopold Kronecker (; 7 December 1823 – 29 December 1891) was a German mathematician who worked on number theory, algebra and logic. He criticized Georg Cantor's work on set theory, and was quoted by as having said, "'" ("God made the integers, ...
(1901), Bang (1905), Sommer (1907), Bottari (1908),
Karel Rychlík Karel Rychlík (; 1885–1968) was a Czechoslovak mathematician who contributed significantly to the fields of algebra, number theory, mathematical analysis, and the history of mathematics The history of mathematics deals with the origin of dis ...
(1910), Nutzhorn (1912),
Robert Carmichael Robert Daniel Carmichael (March 1, 1879 – May 2, 1967) was an American mathematician. Biography Carmichael was born in Goodwater, Alabama. He attended Lineville College, briefly, and he earned his bachelor's degree in 1898, while he was s ...
(1913), Hancock (1931),
Gheorghe Vrănceanu Gheorghe Vrănceanu (June 30, 1900 – April 27, 1979) was a Romanian mathematician, best known for his work in differential geometry and topology. He was titular member of the Romanian Academy and vice-president of the International Mathematical ...
(1966), Grant and Perella (1999), Barbara (2007), and Dolan (2011).


Other exponents

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 ''a''''n'' + ''b''''n'' = ''c''''n'' has no positive integer solutions (''a'', ''b'', ''c'') when ''n'' is an odd
prime number A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
. 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 : ''a''''n'' + ''b''''n'' = ''c''''n'' 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 would suffice to prove that it has no solutions for at least one prime factor of every ''n''. Each integer ''n'' > 2 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 ''n'' = 4 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 ''p'' = 3 was first stated by
Abu-Mahmud Khojandi Abu Mahmud Hamid ibn al-Khidr al-Khojandi (known as Abu Mahmood Khojandi, Alkhujandi or al-Khujandi, Persian: Ű§ŰšÙˆÙ…Ű­Ù…ÙˆŰŻ ŰźŰŹÙ†ŰŻÛŒ, c. 940 - 1000) was a Muslim Transoxanian astronomer and mathematician born in Khujand (now part of Tajikista ...
(10th century), but his attempted proof of the theorem was incorrect. In 1770,
Leonhard Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
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), Reprinted in 1955 by A. Blanchard (Paris). Reprinted in 1825 as the "Second SupplĂ©ment" for a printing of the 2nd edition of ''Essai sur la ThĂ©orie des Nombres'', Courcier (Paris). Also reprinted in 1909 in ''Sphinx-Oedipe'', 4, 97–128. Calzolari (1855),
Gabriel LamĂ© Gabriel LamĂ© (22 July 1795 – 1 May 1870) was a French mathematician who contributed to the theory of partial differential equations by the use of curvilinear coordinates, and the mathematical theory of elasticity (for which linear elasticity ...
(1865),
Peter Guthrie Tait Peter Guthrie Tait FRSE (28 April 1831 – 4 July 1901) was a Scottish mathematical physicist and early pioneer in thermodynamics. He is best known for the mathematical physics textbook ''Treatise on Natural Philosophy'', which he co-wrote wi ...
(1872), GĂŒnther (1878), Gambioli (1901), Krey (1909), RychlĂ­k (1910), Stockhaus (1910), Carmichael (1915),
Johannes van der Corput Johannes Gaultherus van der Corput (4 September 1890 – 16 September 1975) was a Dutch mathematician, working in the field of analytic number theory. He was appointed professor at the University of Fribourg (Switzerland) in 1922, at the Univers ...
(1915),
Axel Thue Axel Thue (; 19 February 1863 – 7 March 1922) was a Norwegian mathematician, known for his original work in diophantine approximation and combinatorics. Work Thue published his first important paper in 1909. He stated in 1914 the so-calle ...
(1917), and Duarte (1944). The case ''p'' = 5 was proved independently by Legendre and
Peter Gustav Lejeune Dirichlet Johann Peter Gustav Lejeune Dirichlet (; 13 February 1805 – 5 May 1859) was a German mathematician who made deep contributions to number theory (including creating the field of analytic number theory), and to the theory of Fourier series and ...
around 1825. Alternative proofs were developed by
Carl Friedrich Gauss Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
(1875, posthumous), Lebesgue (1843), Lamé (1847), Gambioli (1901), Werebrusow (1905), Rychlík (1910), van der Corput (1915), and
Guy Terjanian Guy Terjanian is a French mathematician who has worked on algebraic number theory. He achieved his Ph.D. under Claude Chevalley in 1966, and at that time published a counterexample to the original form of a conjecture of Emil Artin, which suitabl ...
(1987). The case ''p'' = 7 was proved by LamĂ© in 1839.
His rather complicated proof was simplified in 1840 by Lebesgue, and still simpler proofs were published by
Angelo Genocchi Angelo Genocchi (5 March 1817 – 7 March 1889) was an Italian mathematician who specialized in number theory. He worked with Giuseppe Peano. The Genocchi number In mathematics, the Genocchi numbers G''n'', named after Angelo Genocchi, are ...
in 1864, 1874 and 1876. Alternative proofs were developed by ThĂ©ophile PĂ©pin (1876) and Edmond Maillet (1897). Fermat's Last Theorem was also proved for the exponents ''n'' = 6, 10, and 14. Proofs for ''n'' = 6 were 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 LamĂ©'s 1839 proof for ''n'' = 7. All proofs for specific exponents used Fermat's technique of
infinite descent In mathematics, a proof by infinite descent, also known as Fermat's method of descent, is a particular kind of proof by contradiction used to show that a statement cannot possibly hold for any number, by showing that if the statement were to hold f ...
, either in its original form, or in the form of descent on elliptic curves or abelian varieties. The details and auxiliary arguments, however, were often ''ad hoc'' and tied to the individual exponent under consideration.Edwards, p. 74. Since they became ever more complicated as ''p'' increased, it seemed unlikely that the general case of Fermat's Last Theorem could be proved 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 Niels Henrik Abel ( , ; 5 August 1802 – 6 April 1829) was a Norwegian mathematician who made pioneering contributions in a variety of fields. His most famous single result is the first complete proof demonstrating the impossibility of solvin ...
and Peter Barlow, the first significant work on the general theorem was done by
Sophie Germain Marie-Sophie Germain (; 1 April 1776 – 27 June 1831) was a French mathematician, physicist, and philosopher. Despite initial opposition from her parents and difficulties presented by society, she gained education from books in her father's lib ...
.


Early modern breakthroughs


Sophie Germain

In the early 19th century,
Sophie Germain Marie-Sophie Germain (; 1 April 1776 – 27 June 1831) was a French mathematician, physicist, and philosopher. Despite initial opposition from her parents and difficulties presented by society, she gained education from books in her father's lib ...
developed several novel approaches to prove Fermat's Last Theorem for all exponents. First, she defined a set of auxiliary primes \theta constructed from the prime exponent p by the equation \theta = 2hp + 1, where h is any integer not divisible by three. She showed that, if no integers raised to the p^ power were adjacent modulo \theta (the ''non-consecutivity condition''), then \theta must divide the product xyz. Her goal was to use
mathematical induction Mathematical induction is a method for proving that a statement ''P''(''n'') is true for every natural number ''n'', that is, that the infinitely many cases ''P''(0), ''P''(1), ''P''(2), ''P''(3), ...  all hold. Informal metaphors help ...
to prove that, for any given p, infinitely many auxiliary primes \theta 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 Adrien-Marie Legendre (; ; 18 September 1752 – 9 January 1833) was a French mathematician who made numerous contributions to mathematics. Well-known and important concepts such as the Legendre polynomials and Legendre transformation are name ...
. As a byproduct of this latter work, she proved
Sophie Germain's theorem In number theory, Sophie Germain's theorem is a statement about the divisibility of solutions to the equation x^p + y^p = z^p of Fermat's Last Theorem for odd prime p. Formal statement Specifically, Sophie Germain proved that at least one of the n ...
, 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 prime In number theory, a prime number ''p'' is a if 2''p'' + 1 is also prime. The number 2''p'' + 1 associated with a Sophie Germain prime is called a . For example, 11 is a Sophie Germain prime and 2 × 11 +  ...
s). Germain tried unsuccessfully to prove the first case of Fermat's Last Theorem for all even exponents, specifically for n = 2p, which was proved by
Guy Terjanian Guy Terjanian is a French mathematician who has worked on algebraic number theory. He achieved his Ph.D. under Claude Chevalley in 1966, and at that time published a counterexample to the original form of a conjecture of Emil Artin, which suitabl ...
in 1977. In 1985,
Leonard Adleman Leonard Adleman (born December 31, 1945) is an American computer scientist. He is one of the creators of the RSA encryption algorithm, for which he received the 2002 Turing Award, often called the Nobel prize of Computer science. He is also kno ...
,
Roger Heath-Brown David Rodney "Roger" Heath-Brown FRS (born 12 October 1952), is a British mathematician working in the field of analytic number theory. Education He was an undergraduate and graduate student of Trinity College, Cambridge; his research supervi ...
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 LamĂ© Gabriel LamĂ© (22 July 1795 – 1 May 1870) was a French mathematician who contributed to the theory of partial differential equations by the use of curvilinear coordinates, and the mathematical theory of elasticity (for which linear elasticity ...
outlined a proof of Fermat's Last Theorem based on factoring the equation in complex numbers, specifically the
cyclotomic field In number theory, a cyclotomic field is a number field obtained by adjoining a complex root of unity to , the field of rational numbers. Cyclotomic fields played a crucial role in the development of modern algebra and number theory because of ...
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 Joseph Liouville (; ; 24 March 1809 – 8 September 1882) was a French mathematician and engineer. Life and work He was born in Saint-Omer in France on 24 March 1809. His parents were Claude-Joseph Liouville (an army officer) and ThĂ©rĂšse ...
, who later read a paper that demonstrated this failure of unique factorisation, written by
Ernst Kummer Ernst Eduard Kummer (29 January 1810 – 14 May 1893) was a German mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned ...
. Kummer set himself the task of determining whether the
cyclotomic field In number theory, a cyclotomic field is a number field obtained by adjoining a complex root of unity to , the field of rational numbers. Cyclotomic fields played a crucial role in the development of modern algebra and number theory because of ...
could be generalized to include new prime numbers such that unique factorisation was restored. He succeeded in that task by developing the
ideal number In number theory an ideal number is an algebraic integer which represents an ideal in the ring of integers of a number field; the idea was developed by Ernst Kummer, and led to Richard Dedekind's definition of ideals for rings. An ideal in the rin ...
s. (Note: 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 Kurt Wilhelm Sebastian Hensel (29 December 1861 – 1 June 1941) was a German mathematician born in Königsberg. Life and career Hensel was born in Königsberg, East Prussia (today Kaliningrad, Russia), the son of Julia (nĂ©e von Adelson) and lan ...
in 1910 and the evidence indicates it likely derives from a confusion by one of Hensel's sources. Harold Edwards says 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 Louis Joel Mordell (28 January 1888 – 12 March 1972) was an American-born British mathematician, known for pioneering research in number theory. He was born in Philadelphia, United States, in a Jewish family of Lithuanian extraction. Educati ...
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 Gerd Faltings (; born 28 July 1954) is a German mathematician known for his work in arithmetic geometry. Education From 1972 to 1978, Faltings studied mathematics and physics at the University of MĂŒnster. In 1978 he received his PhD in mathema ...
, and is now known as
Faltings's theorem In arithmetic geometry, the Mordell conjecture is the conjecture made by Louis Mordell that a curve of Genus (mathematics), genus greater than 1 over the field Q of rational numbers has only finitely many rational points. In 1983 it was proved by ...
.


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 Harry Schultz Vandiver (21 October 1882 – 9 January 1973) was an American mathematician, known for work in number theory. He was born in Philadelphia, Pennsylvania to John Lyon and Ida Frances (Everett) Vandiver. He did not complete a form ...
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, 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"Fermat's Last Theorem, Simon Singh, 1997,
Taniyama–Shimura–Weil conjecture The modularity theorem (formerly called the Taniyama–Shimura conjecture, Taniyama-Weil conjecture or modularity conjecture for elliptic curves) states that elliptic curves over the field of rational numbers are related to modular forms. And ...
, proposed around 1955—which many mathematicians believed would be near to impossible to prove, and was linked in the 1980s by
Gerhard Frey Gerhard Frey (; born 1 June 1944) is a German mathematician, known for his work in number theory. Following an original idea of Hellegouarch, he developed the notion of Frey–Hellegouarch curves, a construction of an elliptic curve from a pur ...
,
Jean-Pierre Serre Jean-Pierre Serre (; born 15 September 1926) is a French mathematician who has made contributions to algebraic topology, algebraic geometry, and algebraic number theory. He was awarded the Fields Medal in 1954, the Wolf Prize in 2000 and the ina ...
and
Ken Ribet Kenneth Alan Ribet (; born June 28, 1948) is an American mathematician working in algebraic number theory and algebraic geometry. He is known for the Herbrand–Ribet theorem and Ribet's theorem, which were key ingredients in the proof of Ferma ...
to Fermat's equation. By accomplishing a partial proof of this conjecture in 1994,
Andrew Wiles Sir Andrew John Wiles (born 11 April 1953) is an English mathematician and a Royal Society Research Professor at the University of Oxford, specializing in number theory. He is best known for proving Fermat's Last Theorem, for which he was awa ...
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 The modularity theorem (formerly called the Taniyama–Shimura conjecture, Taniyama-Weil conjecture or modularity conjecture for elliptic curves) states that elliptic curves over the field of rational numbers are related to modular forms. And ...
.


Taniyama–Shimura–Weil conjecture

Around 1955, Japanese mathematicians
Goro Shimura was a Japanese mathematician and Michael Henry Strater Professor Emeritus of Mathematics at Princeton University who worked in number theory, automorphic forms, and arithmetic geometry. He was known for developing the theory of complex multipli ...
and
Yutaka Taniyama was a Japanese mathematician known for the Taniyama–Shimura conjecture. Contribution Taniyama was best known for conjecturing, in modern language, automorphic properties of L-functions of elliptic curves over any number field. A partial and r ...
observed a possible link between two apparently completely distinct branches of mathematics,
elliptic curve In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If ...
s and modular forms. The resulting
modularity theorem The modularity theorem (formerly called the Taniyama–Shimura conjecture, Taniyama-Weil conjecture or modularity conjecture for elliptic curves) states that elliptic curves over the field of rational numbers are related to modular forms. And ...
(at the time known as the Taniyama–Shimura conjecture) states that every elliptic curve is
modular Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. The concept of modularity is used primarily to reduce complexity by breaking a s ...
, 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 AndrĂ© 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 twas 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 t"


Ribet's theorem for Frey curves

In 1984,
Gerhard Frey Gerhard Frey (; born 1 June 1944) is a German mathematician, known for his work in number theory. Following an original idea of Hellegouarch, he developed the notion of Frey–Hellegouarch curves, a construction of an elliptic curve from a pur ...
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 semi-stable
elliptic curve In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If ...
(now known as a Frey-HellegouarchThis elliptic curve was first suggested in the 1960s by , but he did not call attention to its non-modularity. For more details, see ) :''y''2 = ''x'' (''x'' âˆ’ ''a''''p'')(''x'' + ''b''''p'') 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 In logic and mathematics, contraposition refers to the inference of going from a conditional statement into its logically equivalent contrapositive, and an associated proof method known as proof by contraposition. The contrapositive of a stateme ...
, 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 4 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 curve In algebraic geometry, a semistable abelian variety is an abelian variety defined over a global or local field, which is characterized by how it reduces at the primes of the field. For an abelian variety A defined over a field F with ring of intege ...
s). 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 Epsilon (, ; uppercase , lowercase or lunate ; el, έψÎčÎ»ÎżÎœ) is the fifth letter of the Greek alphabet, corresponding phonetically to a mid front unrounded vowel or . In the system of Greek numerals it also has the value five. It was der ...
", now known as
Ribet's theorem Ribet's theorem (earlier called the epsilon conjecture or Δ-conjecture) is part of number theory. It concerns properties of Galois representations associated with modular forms. It was proposed by Jean-Pierre Serre and proven by Ken Ribet. The ...
) was identified by
Jean-Pierre Serre Jean-Pierre Serre (; born 15 September 1926) is a French mathematician who has made contributions to algebraic topology, algebraic geometry, and algebraic number theory. He was awarded the Fields Medal in 1954, the Wolf Prize in 2000 and the ina ...
who also gave an almost-complete proof and the link suggested by Frey was finally proved in 1986 by
Ken Ribet Kenneth Alan Ribet (; born June 28, 1948) is an American mathematician working in algebraic number theory and algebraic geometry. He is known for the Herbrand–Ribet theorem and Ribet's theorem, which were key ingredients in the proof of Ferma ...
. 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 Epsilon (, ; uppercase , lowercase or lunate ; el, έψÎčÎ»ÎżÎœ) is the fifth letter of the Greek alphabet, corresponding phonetically to a mid front unrounded vowel or . In the system of Greek numerals it also has the value five. It was der ...
in 1986 accomplished the first of the two goals proposed by Frey. Upon hearing of Ribet's success,
Andrew Wiles Sir Andrew John Wiles (born 11 April 1953) is an English mathematician and a Royal Society Research Professor at the University of Oxford, specializing in number theory. He is best known for proving Fermat's Last Theorem, for which he was awa ...
, 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 The modularity theorem (formerly called the Taniyama–Shimura conjecture, Taniyama-Weil conjecture or modularity conjecture for elliptic curves) states that elliptic curves over the field of rational numbers are related to modular forms. And ...
(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 Induction, Inducible or Inductive may refer to: Biology and medicine * Labor induction (birth/pregnancy) * Induction chemotherapy, in medicine * Induced stem cells, stem cells derived from somatic, reproductive, pluripotent or other cell t ...
, and he based his initial work and first significant breakthrough on
Galois theory In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems in field theory to ...
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 In mathematics, an Euler system is a collection of compatible elements of Galois cohomology groups indexed by fields. They were introduced by in his work on Heegner points on modular elliptic curves, which was motivated by his earlier paper and ...
recently developed by
Victor Kolyvagin Victor Alexandrovich Kolyvagin (russian: ВоĐșŃ‚ĐŸŃ€ АлДĐșŃĐ°ĐœĐŽŃ€ĐŸĐČоч ĐšĐŸĐ»Ń‹ĐČĐ°ĐłĐžĐœ, born 11 March, 1955) is a Russian mathematician who wrote a series of papers on Euler systems, leading to breakthroughs on the Birch and Swinnerto ...
and Matthias Flach that seemed "tailor made" for the inductive part of his proof. 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 Nicholas Michael Katz (born December 7, 1943) is an American mathematician, working in arithmetic geometry, particularly on ''p''-adic methods, monodromy and moduli problems, and number theory. He is currently a professor of Mathematics at ...
, 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 The Isaac Newton Institute for Mathematical Sciences is an international research institute for mathematics and its many applications at the University of Cambridge. It is named after one of the university's most illustrious figures, the mathemat ...
. 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 Peer review is the evaluation of work by one or more people with similar competencies as the producers of the work (peers). It functions as a form of self-regulation by qualified members of a profession within the relevant field. 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 A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
. 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.Singh, pp. 269–277. 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 for why his approach could not be made to work, when he had a sudden
insight Insight is the understanding of a specific cause and effect within a particular context. The term insight can have several related meanings: *a piece of information *the act or result of understanding the inner nature of things or of seeing intu ...
– that the specific reason why the Kolyvagin–Flach approach would not work directly ''also'' meant that his original attempts using
Iwasawa theory In number theory, Iwasawa theory is the study of objects of arithmetic interest over infinite towers of number fields. It began as a Galois module theory of ideal class groups, initiated by (), as part of the theory of cyclotomic fields. In th ...
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. : "I was sitting at my desk examining the Kolyvagin–Flach method. It wasn't that I believed I could make it work, but I thought that at least I could explain why it didn’t work. Suddenly I had this incredible revelation. I realised that, the Kolyvagin–Flach method wasn't working, but it was all I needed to make my original Iwasawa theory work from three years earlier. So out of the ashes of Kolyvagin–Flach seemed to rise the true answer to the problem. It was so indescribably beautiful; it was so simple and so elegant. I couldn't understand how I'd missed it and I just stared at it in disbelief for twenty minutes. Then during the day I walked around the department, and I'd keep coming back to my desk looking to see if it was still there. It was still there. I couldn't contain myself, I was so excited. It was the most important moment of my working life. Nothing I ever do again will mean as much." ::— Andrew Wiles, as quoted by Simon Singh 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 ''Annals of Mathematics'' is a mathematical journal published every two months by Princeton University and the Institute for Advanced Study. History The journal was established as ''The Analyst'' in 1874 and with Joel E. Hendricks as the ...
''. 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 The modularity theorem (formerly called the Taniyama–Shimura conjecture, Taniyama-Weil conjecture or modularity conjecture for elliptic curves) states that elliptic curves over the field of rational numbers are related to modular forms. And ...
. 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 ''n''-th powers, ''n'' â‰„ 3. (The case ''n'' = 3 was already known by Euler.)


Relationship to other problems and generalizations

Fermat's Last Theorem considers solutions to the Fermat equation: with positive integers , , and and an integer greater than 2. There are several generalizations of the Fermat equation to more general equations that allow the exponent to be a negative integer or rational, or to consider three different exponents.


Generalized Fermat equation

The generalized Fermat equation generalizes the statement of Fermat's last theorem by considering positive integer solutions ''a, b, c, m, n, k'' satisfying In particular, the exponents ''m'', ''n'', ''k'' need not be equal, whereas Fermat's last theorem considers the case The Beal conjecture, also known as the Mauldin conjecture and the Tijdeman-Zagier conjecture, states that there are no solutions to the generalized Fermat equation in positive integers ''a'', ''b'', ''c'', ''m'', ''n'', ''k'' with ''a'', ''b'', and ''c'' being pairwise coprime and all of ''m'', ''n'', ''k'' being greater than 2. The
Fermat–Catalan conjecture In number theory, the Fermat–Catalan conjecture is a generalization of Fermat's Last Theorem and of Catalan's conjecture, hence the name. The conjecture states that the equation has only finitely many solutions (''a'',''b'',''c'',''m'',''n'','' ...
generalizes Fermat's last theorem with the ideas of the
Catalan conjecture Catalan's conjecture (or Mihăilescu's theorem) is a theorem in number theory that was conjectured by the mathematician EugÚne Charles Catalan in 1844 and proven in 2002 by Preda Mihăilescu at Paderborn University. The integers 23 and 32 are ...
. The conjecture states that the generalized Fermat equation has only ''finitely many'' solutions (''a'', ''b'', ''c'', ''m'', ''n'', ''k'') with distinct triplets of values (''a''''m'', ''b''''n'', ''c''''k''), where ''a'', ''b'', ''c'' are positive coprime integers and ''m'', ''n'', ''k'' are positive integers satisfying The statement is about the finiteness of the set of solutions because there are 10 known solutions.


Inverse Fermat equation

When we allow the exponent to be the reciprocal of an integer, i.e. for some integer , we have the inverse Fermat equation a^ + b^ = c^. All solutions of this equation were computed by
Hendrik Lenstra Hendrik Willem Lenstra Jr. (born 16 April 1949, Zaandam) is a Dutch mathematician. Biography Lenstra received his doctorate from the University of Amsterdam in 1977 and became a professor there in 1978. In 1987 he was appointed to the faculty o ...
in 1992. In the case in which the ''m''th roots are required to be real and positive, all solutions are given by :a=rs^m :b=rt^m :c=r(s+t)^m for positive integers ''r, s, t'' with ''s'' and ''t'' coprime.


Rational exponents

For the Diophantine equation a^ + b^ = c^ with ''n'' not equal to 1, Bennett, Glass, and Székely proved in 2004 for ''n'' > 2, that if ''n'' and ''m'' are coprime, then there are integer solutions if and only if 6 divides ''m'', and a^, b^, and c^ are different complex 6th roots of the same real number.


Negative integer exponents


''n'' = âˆ’1

All primitive integer solutions (i.e., those with no prime factor common to all of ''a'', ''b'', and ''c'') to the
optic equation In number theory, the optic equation is an equation that requires the sum of the reciprocals of two positive integers ''a'' and ''b'' to equal the reciprocal of a third positive integer ''c'':Dickson, L. E., ''History of the Theory of Numbers, V ...
a^ + b^ = c^ can be written as : a = mk + m^2, : b = mk + k^2, : c = mk for positive, coprime integers ''m'', ''k''.


''n'' = âˆ’2

The case ''n'' = âˆ’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 a^ + b^ = d^ are given by : a = (v^2 - u^2)(v^2 + u^2), : b = 2uv(v^2 + u^2), : d = 2uv(v^2 - u^2), 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 : c = (v^2 + u^2)^2, so (''a, b, c'') is a
Pythagorean triple A Pythagorean triple consists of three positive integers , , and , such that . Such a triple is commonly written , and a well-known example is . If is a Pythagorean triple, then so is for any positive integer . A primitive Pythagorean triple is ...
.


''n'' < âˆ’2

There are no solutions in integers for a^n + b^n = c^n for integers ''n'' < âˆ’2. If there were, the equation could be multiplied through by a^ b^ c^ to obtain (bc)^ + (ac)^ = (ab)^, which is impossible by Fermat's Last Theorem.


abc conjecture

The
abc conjecture The ''abc'' conjecture (also known as the Oesterlé–Masser conjecture) is a conjecture in number theory that arose out of a discussion of Joseph OesterlĂ© and David Masser in 1985. It is stated in terms of three positive integers ''a'', ''b'' ...
roughly states that if three positive integers ''a'', ''b'' and ''c'' (hence the name) are coprime and satisfy ''a'' + ''b'' = ''c'', then the radical ''d'' of ''abc'' is usually not much smaller than ''c''. In particular, the abc conjecture in its most standard formulation implies Fermat's last theorem for ''n'' that are sufficiently large. The modified Szpiro conjecture is equivalent to the abc conjecture and therefore has the same implication. An effective version of the abc conjecture, or an effective version of the modified Szpiro conjecture, implies Fermat's Last Theorem outright.


Prizes and incorrect proofs

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 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. Another prize was offered in 1883 by the Academy of Brussels. In 1908, the German industrialist and amateur mathematician
Paul Wolfskehl Paul Friedrich Wolfskehl (30 June 1856 in Darmstadt – 13 September 1906 in Darmstadt), was a physician with an interest in mathematics. He bequeathed 100,000 marks (equivalent to 1,000,000 pounds in 1997 money) to the first person to prove Ferm ...
bequeathed 100,000
gold marks The German mark (german: Goldmark ; sign: ℳ) was the currency of the German Empire, which spanned from 1871 to 1918. The mark was paired with the minor unit of the pfennig (₰); 100 pfennigs were equivalent to 1 mark. The mark was on the ...
—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. 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. In March 2016, Wiles was awarded the Norwegian government's
Abel prize The Abel Prize ( ; no, Abelprisen ) is awarded annually by the King of Norway to one or more outstanding mathematicians. It is named after the Norwegian mathematician Niels Henrik Abel (1802–1829) and directly modeled after the Nobel Prizes. ...
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." Prior to Wiles's 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 some claims,
Edmund Landau Edmund Georg Hermann Landau (14 February 1877 – 19 February 1938) was a German mathematician who worked in the fields of number theory and complex analysis. Biography Edmund Landau was born to a Jewish family in Berlin. His father was Leopol ...
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. 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 Howard Whitley Eves (10 January 1911, New Jersey – 6 June 2004) was an American mathematician, known for his work in geometry and the history of mathematics. Eves received his B.S. from the University of Virginia, an M.A. from Harvard Universi ...
, "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

The popularity of the theorem outside science has led to it being described as achieving "that rarest of mathematical accolades: A niche role in pop culture."
Arthur Porges Arthur Porges (; 20 August 1915 – 12 May 2006) was an American writer of numerous short stories, most notably during the 1950s and 1960s, though he continued to write and publish stories until his death. Life Arthur Porges was born in Chica ...
' 1954 short story "
The Devil and Simon Flagg Arthur Porges (; 20 August 1915 – 12 May 2006) was an American writer of numerous short stories, most notably during the 1950s and 1960s, though he continued to write and publish stories until his death. Life Arthur Porges was born in Chica ...
" features a
mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...
who bargains with the
Devil A devil is the personification of evil as it is conceived in various cultures and religious traditions. It is seen as the objectification of a hostile and destructive force. Jeffrey Burton Russell states that the different conceptions of t ...
that the latter cannot produce a proof of Fermat's Last Theorem within twenty-four hours. In ''
The Simpsons ''The Simpsons'' is an American animated sitcom created by Matt Groening for the Fox Broadcasting Company. The series is a satirical depiction of American life, epitomized by the Simpson family, which consists of Homer, Marge, Bart, Lisa, ...
'' episode " The Wizard of Evergreen Terrace," Homer Simpson writes the equation 3987^ + 4365^ = 4472^ on a blackboard, which appears to be a
counterexample A counterexample is any exception to a generalization. In logic a counterexample disproves the generalization, and does so rigorously in the fields of mathematics and philosophy. For example, the fact that "John Smith is not a lazy student" is a ...
to Fermat's Last Theorem. The equation is wrong, but it appears to be correct if entered in a calculator with 10
significant figures Significant figures (also known as the significant digits, ''precision'' or ''resolution'') of a number in positional notation are digits in the number that are reliable and necessary to indicate the quantity of something. If a number expres ...
.


See also

*
Euler's sum of powers conjecture Euler's conjecture is a disproved conjecture in mathematics related to Fermat's Last Theorem. It was proposed by Leonhard Euler in 1769. It states that for all integers and greater than 1, if the sum of many th powers of positive integers i ...
*
Proof of impossibility In mathematics, a proof of impossibility is a proof that demonstrates that a particular problem cannot be solved as described in the claim, or that a particular set of problems cannot be solved in general. Such a case is also known as a negative ...
* Sums of powers, a list of related conjectures and theorems *
Wall–Sun–Sun prime In number theory, a Wall–Sun–Sun prime or Fibonacci–Wieferich prime is a certain kind of prime number which is conjectured to exist, although none are known. Definition Let p be a prime number. When each term in the sequence of Fibonac ...


Footnotes


References


Bibliography

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Further reading

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External links

* * * * Blog that covers the history of Fermat's Last Theorem from Fermat to Wiles. * * Discusses various material that is related to the proof of Fermat's Last Theorem: elliptic curves, modular forms, Galois representations and their deformations, Frey's construction, and the conjectures of Serre and of Taniyama–Shimura. * The story, the history and the mystery. * * * The title of one edition of the PBS television series NOVA, discusses Andrew Wiles's effort to prove Fermat's Last Theorem. * Simon Singh and John Lynch's film tells the story of Andrew Wiles. {{Authority control 1637 in science 1637 introductions Pythagorean theorem Theorems in number theory Conjectures that have been proved 1995 in mathematics