# Fermat's Last Theorem

Fermat's Last Theorem For other theorems named after Pierre de Fermat, see Fermat's theorem. For the book by Simon Singh, see Fermat's Last Theorem (book). Fermat's Last Theorem The 1670 edition of Diophantus's Arithmetica includes Fermat's commentary, referred to as his "Last Theorem" (Observatio Domini Petri de Fermat), posthumously published by his son.

Field Number theory Statement For any integer n > 2, the equation an + bn = cn has no positive integer solutions. First stated by Pierre de Fermat First stated in c. 1637 First proof by Andrew Wiles First proof in Released 1994 Published 1995 Implied by Effective abc conjecture Effective modified Szpiro conjecture Modularity theorem Generalizations Beal conjecture Fermat–Catalan conjecture 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 satisfy the equation an + bn = cn for any integer value of n greater than 2. The cases n = 1 and n = 2 have been known since antiquity to have infinitely many solutions.[1] The proposition was first stated as a theorem by Pierre de Fermat 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), 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 and formally published in 1995. It was described as a "stunning advance" in the citation for Wiles's Abel Prize award in 2016.[2] It also proved much of the Taniyama-Shimura conjecture, subsequently known as the modularity theorem, 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 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.[3] Contents 1 Overview 1.1 Pythagorean origins 1.2 Subsequent developments and solution 1.3 Equivalent statements of the theorem 2 Mathematical history 2.1 Pythagoras and Diophantus 2.2 Fermat's conjecture 2.3 Proofs for specific exponents 2.4 Early modern breakthroughs 2.5 Connection with elliptic curves 2.6 Wiles's general proof 2.7 Subsequent developments 3 Relationship to other problems and generalizations 3.1 Generalized Fermat equation 3.2 Inverse Fermat equation 3.3 Rational exponents 3.4 Negative integer exponents 3.5 abc conjecture 4 Prizes and incorrect proofs 5 In popular culture 6 See also 7 Footnotes 8 References 9 Bibliography 10 Further reading 11 External links Overview Pythagorean origins The Pythagorean equation, x2 + y2 = z2, has an infinite number of positive integer solutions for x, y, and z; these solutions are known as Pythagorean triples (with the simplest example 3,4,5). Around 1637, Fermat wrote in the margin of a book that the more general equation an + bn = cn 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.[4] The claim eventually became one of the most notable unsolved problems of mathematics. Attempts to prove it prompted substantial development in number theory, and over time Fermat's Last Theorem gained prominence as an unsolved problem in mathematics.

Although both problems were daunting and widely considered to be "completely inaccessible" to proof at the time,[2] 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.[8] However, general opinion was that this simply showed the impracticality of proving the Taniyama–Shimura conjecture.[9] Mathematician John Coates' quoted reaction was a common one:[9] "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, 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 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.[10][11][12] For his proof, Wiles was honoured and received numerous awards, including the 2016 Abel Prize.[13][14][15] 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 N be the set of natural numbers 1, 2, 3, ..., let Z be the set of integers 0, ±1, ±2, ..., and let Q be the set of rational numbers a/b, where a and b are in Z with b ≠ 0. In what follows we will call a solution to xn + yn = zn where one or more of x, y, or z is zero a trivial solution. A solution where all three are non-zero will be called a non-trivial solution.

Original statement. With n, x, y, z ∈ N (meaning that n, x, y, z are all positive whole numbers) and n > 2, the equation xn + yn = zn has no solutions.

Most popular treatments of the subject state it this way. It is also commonly stated over Z:[16] Equivalent statement 1: xn + yn = zn, where integer n ≥ 3, has no non-trivial solutions x, y, z ∈ Z.

The equivalence is clear if n is even. If n is odd and all three of x, y, z are negative, then we can replace x, y, z with −x, −y, −z to obtain a solution in N. If two of them are negative, it must be x and z or y and z. If x, z are negative and y is positive, then we can rearrange to get (−z)n + yn = (−x)n resulting in a solution in N; the other case is dealt with analogously. Now if just one is negative, it must be x or y. If x is negative, and y and z are positive, then it can be rearranged to get (−x)n + zn = yn again resulting in a solution in N; if y is negative, the result follows symmetrically. Thus in all cases a nontrivial solution in Z would also mean a solution exists in N, the original formulation of the problem.

Equivalent statement 2: xn + yn = zn, where integer n ≥ 3, has no non-trivial solutions x, y, z ∈ Q.

This is because the exponents of x, y, and z are equal (to n), so if there is a solution in Q, then it can be multiplied through by an appropriate common denominator to get a solution in Z, and hence in N.

Equivalent statement 3: xn + yn = 1, where integer n ≥ 3, has no non-trivial solutions x, y ∈ Q.

A non-trivial solution a, b, c ∈ Z to xn + yn = zn yields the non-trivial solution a/c, b/c ∈ Q for vn + wn = 1. Conversely, a solution a/b, c/d ∈ Q to vn + wn = 1 yields the non-trivial solution ad, cb, bd for xn + yn = zn.

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 Q, rather than over the ring Z; fields exhibit more structure than rings, which allows for deeper analysis of their elements.

Equivalent statement 4 – connection to elliptic curves: If a, b, c is a non-trivial solution to ap + bp = cp, p odd prime, then y2 = x(x − ap)(x + bp) (Frey curve) will be an elliptic curve.[17] Examining this elliptic curve with Ribet's theorem shows that it does not have a modular form. However, the proof by Andrew Wiles proves that any equation of the form y2 = x(x − an)(x + bn) does have a modular form. Any non-trivial solution to xp + yp = zp (with p an odd prime) would therefore create a contradiction, which in turn proves that no non-trivial solutions exist.[18] 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 Main article: Pythagorean triple In ancient times it was known that a triangle whose sides were in the ratio 3:4:5 would have a right angle as one of its angles. This was used in construction and later in early geometry. It was also known to be one example of a general rule that any triangle where the length of two sides, each squared and then added together (32 + 42 = 9 + 16 = 25), equals the square of the length of the third side (52 = 25), would also be a right angle triangle. This is now known as the Pythagorean theorem, and a triple of numbers that meets this condition is called a Pythagorean triple – both are named after the ancient Greek Pythagoras. Examples include (3, 4, 5) and (5, 12, 13). There are infinitely many such triples,[19] and methods for generating such triples have been studied in many cultures, beginning with the Babylonians[20] and later ancient Greek, Chinese, and Indian mathematicians.[1] Mathematically, the definition of a Pythagorean triple is a set of three integers (a, b, c) that satisfy the equation[21] {displaystyle a^{2}+b^{2}=c^{2}.} Diophantine equations Main article: Diophantine equation Fermat's equation, xn + yn = zn with positive integer solutions, is an example of a Diophantine equation,[22] named for the 3rd-century Alexandrian mathematician, Diophantus, who studied them and developed methods for the solution of some kinds of Diophantine equations. A typical Diophantine problem is to find two integers x and y such that their sum, and the sum of their squares, equal two given numbers A and B, respectively: {displaystyle A=x+y} {displaystyle B=x^{2}+y^{2}.} Diophantus's major work is the Arithmetica, of which only a portion has survived.[23] Fermat's conjecture of his Last Theorem was inspired while reading a new edition of the Arithmetica,[24] that was translated into Latin and published in 1621 by Claude Bachet.[25] 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).[26] Solutions to linear Diophantine equations, such as 26x + 65y = 13, may be found using the Euclidean algorithm (c. 5th century BC).[27] 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 = zm where n and m are relatively prime natural numbers.[note 2] Fermat's conjecture Problem II.8 in the 1621 edition of the Arithmetica of Diophantus. On the right is the margin that was too small to contain Fermat's alleged proof of his "last theorem".

Problem II.8 of the 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).[28] Around 1637, Fermat wrote his Last Theorem in the margin of his copy of the Arithmetica next to Diophantus's sum-of-squares problem:[29] Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos & 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.[30][31] 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.[32] 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,[33] as it was the last of Fermat's asserted theorems to remain unproved.[34] 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, and John Wallis,[35] he never posed the general case.[36] 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[37] 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[38] 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.[39] Fermat's proof would have had to be elementary by comparison, given the mathematical knowledge of his time.

While Harvey Friedman's grand conjecture implies that any provable theorem (including Fermat's last theorem) can be proved using only 'elementary function arithmetic', such a proof need be 'elementary' only in a technical sense and could involve millions of steps, and thus be far too long to have been Fermat's proof.

Proofs for specific exponents Main article: Proof of Fermat's Last Theorem for specific exponents Fermat's infinite descent for Fermat's Last Theorem case n=4 in the 1670 edition of the Arithmetica of Diophantus (pp. 338–339). Exponent = 4 Only one relevant proof by Fermat has survived, in which he uses the technique of infinite descent to show that the area of a right triangle with integer sides can never equal the square of an integer.[40][41] His proof is equivalent to demonstrating that the equation {displaystyle x^{4}-y^{4}=z^{4}} 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[42] by Frénicle de Bessy (1676),[43] Leonhard Euler (1738),[44] Kausler (1802),[45] Peter Barlow (1811),[46] Adrien-Marie Legendre (1830),[47] Schopis (1825),[48] Olry Terquem (1846),[49] Joseph Bertrand (1851),[50] Victor Lebesgue (1853, 1859, 1862),[51] Théophile Pépin (1883),[52] Tafelmacher (1893),[53] David Hilbert (1897),[54] Bendz (1901),[55] Gambioli (1901),[56] Leopold Kronecker (1901),[57] Bang (1905),[58] Sommer (1907),[59] Bottari (1908),[60] Karel Rychlík (1910),[61] Nutzhorn (1912),[62] Robert Carmichael (1913),[63] Hancock (1931),[64] Gheorghe Vrănceanu (1966),[65] Grant and Perella (1999),[66] Barbara (2007),[67] and Dolan (2011).[68] 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.[69] In other words, it was necessary to prove only that the equation an + bn = cn has no positive 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 (ad, bd, cd) is a solution for the exponent e (ad)e + (bd)e = (cd)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.

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.

(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 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".[121] See the history of ideal numbers.) Using the general approach outlined by Lamé, Kummer proved both cases of Fermat's Last Theorem for all regular prime numbers. However, he could not prove the theorem for the exceptional primes (irregular primes) that conjecturally occur approximately 39% of the time; the only irregular primes below 270 are 37, 59, 67, 101, 103, 131, 149, 157, 233, 257 and 263.

Mordell conjecture In the 1920s, Louis Mordell posed a conjecture that implied that Fermat's equation has at most a finite number of nontrivial primitive integer solutions, if the exponent n is greater than two.[122] This conjecture was proved in 1983 by Gerd Faltings,[123] and is now known as Faltings's 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.[124] By 1978, Samuel Wagstaff had extended this to all primes less than 125,000.[125] By 1993, Fermat's Last Theorem had been proved for all primes less than four million.[5] 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.)[126] Connection with elliptic curves The strategy that ultimately led to a successful proof of Fermat's Last Theorem arose from the "astounding"[127]: 211  Taniyama–Shimura–Weil conjecture, proposed around 1955—which many mathematicians believed would be near to impossible to prove,[127]: 223  and was linked in the 1980s by Gerhard Frey, Jean-Pierre Serre and Ken Ribet to Fermat's equation. By accomplishing a partial proof of this conjecture in 1994, Andrew Wiles ultimately succeeded in proving Fermat's Last Theorem, as well as leading the way to a full proof by others of what is now known as the modularity theorem.

Taniyama–Shimura–Weil conjecture Main article: Modularity theorem Around 1955, Japanese mathematicians Goro Shimura and Yutaka Taniyama observed a possible link between two apparently completely distinct branches of mathematics, elliptic curves and modular forms. The resulting modularity theorem (at the time known as the Taniyama–Shimura conjecture) states that every elliptic curve is modular, meaning that it can be associated with a unique modular form.

The link was initially dismissed as unlikely or highly speculative, but was taken more seriously when number theorist André Weil found evidence supporting it, though not proving it; as a result the conjecture was often known as the Taniyama–Shimura–Weil conjecture.[127]: 211–215  Even after gaining serious attention, the conjecture was seen by contemporary mathematicians as extraordinarily difficult or perhaps inaccessible to proof.[127]: 203–205, 223, 226  For example, Wiles's doctoral supervisor John Coates states that it seemed "impossible to actually prove",[127]: 226  and Ken Ribet considered himself "one of the vast majority of people who believed [it] was completely inaccessible", adding that "Andrew Wiles was probably one of the few people on earth who had the audacity to dream that you can actually go and prove [it]."[127]: 223  Ribet's theorem for Frey curves Main articles: Frey curve and Ribet's theorem In 1984, Gerhard Frey noted a link between Fermat's equation and the modularity theorem, then still a conjecture. If Fermat's equation had any solution (a, b, c) for exponent p > 2, then it could be shown that the semi-stable elliptic curve (now known as a Frey-Hellegouarch[note 3]) y2 = x (x − ap)(x + bp) would have such unusual properties that it was unlikely to be modular.[128] 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.[129] By contraposition, a disproof or refutation of Fermat's Last Theorem would disprove the Taniyama–Shimura–Weil conjecture.

In plain English, Frey had shown that, if this intuition about his equation was correct, then any set of 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.

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),[10] Conrad et al. (1999),[11] and Breuil et al. (2001)[12] who, building on Wiles's work, incrementally chipped away at the remaining cases until the full result was proved. The now fully proved conjecture became known as the modularity theorem.

Several other theorems in number theory similar to Fermat's Last Theorem also follow from the same reasoning, using the modularity theorem. For example: no cube can be written as a sum of two coprime n-th powers, n ≥ 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: an + bn = cn with positive integers a, b, and c and an integer n greater than 2. There are several generalizations of the Fermat equation to more general equations that allow the exponent n 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[146] {displaystyle a^{m}+b^{n}=c^{k}.}         (1) In particular, the exponents m, n, k need not be equal, whereas Fermat's last theorem considers the case m = n = k.

The Beal conjecture, also known as the Mauldin conjecture[147] and the Tijdeman-Zagier conjecture,[148][149][150] 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.[151] The Fermat–Catalan conjecture generalizes Fermat's last theorem with the ideas of the Catalan conjecture.[152][153] The conjecture states that the generalized Fermat equation has only finitely many solutions (a, b, c, m, n, k) with distinct triplets of values (am, bn, ck), where a, b, c are positive coprime integers and m, n, k are positive integers satisfying {displaystyle {frac {1}{m}}+{frac {1}{n}}+{frac {1}{k}}<1.}         (2) The statement is about the finiteness of the set of solutions because there are 10 known solutions.[146] Inverse Fermat equation When we allow the exponent n to be the reciprocal of an integer, i.e. n = 1/m for some integer m, we have the inverse Fermat equation {displaystyle a^{1/m}+b^{1/m}=c^{1/m}.} All solutions of this equation were computed by Hendrik Lenstra in 1992.[154] In the case in which the mth roots are required to be real and positive, all solutions are given by[155] {displaystyle a=rs^{m}} {displaystyle b=rt^{m}} {displaystyle c=r(s+t)^{m}} for positive integers r, s, t with s and t coprime. Rational exponents For the Diophantine equation {displaystyle a^{n/m}+b^{n/m}=c^{n/m}} 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 {displaystyle a^{1/m}} , {displaystyle b^{1/m},} and {displaystyle c^{1/m}} are different complex 6th roots of the same real number.[156] Negative integer exponents n = −1 All primitive integer solutions (i.e., those with no prime factor common to all of a, b, and c) to the optic equation {displaystyle a^{-1}+b^{-1}=c^{-1}} can be written as[157] {displaystyle a=mk+m^{2},} {displaystyle b=mk+k^{2},} {displaystyle c=mk} for positive, coprime integers m, k.

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.[158][159] All primitive solutions to {displaystyle a^{-2}+b^{-2}=d^{-2}} are given by {displaystyle a=(v^{2}-u^{2})(v^{2}+u^{2}),} {displaystyle b=2uv(v^{2}+u^{2}),} {displaystyle 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 {displaystyle c=(v^{2}+u^{2})^{2},} so (a, b, c) is a Pythagorean triple.