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A Pythagorean triple consists of three 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 o ...
s , , 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 one in which , and are
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 equival ...
(that is, they have no common divisor larger than 1). For example, is a primitive Pythagorean triple whereas is not. A triangle whose sides form a Pythagorean triple is called a Pythagorean triangle, and is necessarily a
right triangle A right triangle (American English) or right-angled triangle (British), or more formally an orthogonal triangle, formerly called a rectangled triangle ( grc, ὀρθόσγωνία, lit=upright angle), is a triangle in which one angle is a right an ...
. The name is derived from 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 ...
, stating that every right triangle has side lengths satisfying the formula a^2+b^2=c^2; thus, Pythagorean triples describe the three integer side lengths of a right triangle. However, right triangles with non-integer sides do not form Pythagorean triples. For instance, the
triangle A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non-collinear ...
with sides a=b=1 and c=\sqrt2 is a right triangle, but (1,1,\sqrt2) is not a Pythagorean triple because \sqrt2 is not an integer. Moreover, 1 and \sqrt2 do not have an integer common multiple because \sqrt2 is
irrational Irrationality is cognition, thinking, talking, or acting without inclusion of rationality. It is more specifically described as an action or opinion given through inadequate use of reason, or through emotional distress or cognitive deficiency. T ...
. Pythagorean triples have been known since ancient times. The oldest known record comes from
Plimpton 322 Plimpton 322 is a Babylonian clay tablet, notable as containing an example of Babylonian mathematics. It has number 322 in the G.A. Plimpton Collection at Columbia University. This tablet, believed to have been written about 1800 BC, has a tabl ...
, a Babylonian clay tablet from about 1800 BC, written in a
sexagesimal Sexagesimal, also known as base 60 or sexagenary, is a numeral system with sixty as its base. It originated with the ancient Sumerians in the 3rd millennium BC, was passed down to the ancient Babylonians, and is still used—in a modified form� ...
number system. It was discovered by Edgar James Banks shortly after 1900, and sold to
George Arthur Plimpton George Arthur Plimpton (July 13, 1855 – July 1, 1936) was an American publisher and philanthropist. Life and career Plimpton was born in Walpole, Massachusetts, the son of Priscilla Guild (Lewis) and Calvin Gay Plimpton. He was the son and grand ...
in 1922, for $10. When searching for integer solutions, the
equation In mathematics, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign . The word ''equation'' and its cognates in other languages may have subtly different meanings; for example, in ...
is a
Diophantine equation In mathematics, a Diophantine equation is an equation, typically a polynomial equation in two or more unknowns with integer coefficients, such that the only solutions of interest are the integer ones. A linear Diophantine equation equates to a ...
. Thus Pythagorean triples are among the oldest known solutions of a
nonlinear In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other ...
Diophantine equation.


Examples

There are 16 primitive Pythagorean triples of numbers up to 100: Other small Pythagorean triples such as (6, 8, 10) are not listed because they are not primitive; for instance (6, 8, 10) is a multiple of (3, 4, 5). Each of these points (with their multiples) forms a radiating line in the scatter plot to the right. Additionally, these are the remaining primitive Pythagorean triples of numbers up to 300:


Generating a triple

Euclid's formula is a fundamental formula for generating Pythagorean triples given an arbitrary pair of integers and with . The formula states that the integers : a = m^2 - n^2 ,\ \, b = 2mn ,\ \, c = m^2 + n^2 form a Pythagorean triple. The triple generated by
Euclid Euclid (; grc-gre, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the ''Elements'' treatise, which established the foundations of ge ...
's formula is primitive if and only if and are
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 equival ...
and one of them is even. When both and are odd, then , , and will be even, and the triple will not be primitive; however, dividing , , and by 2 will yield a primitive triple when and are coprime. ''Every'' primitive triple arises (after the exchange of and , if is even) from a ''unique pair'' of coprime numbers , , one of which is even. It follows that there are infinitely many primitive Pythagorean triples. This relationship of , and to and from Euclid's formula is referenced throughout the rest of this article. Despite generating all primitive triples, Euclid's formula does not produce all triples—for example, (9, 12, 15) cannot be generated using integer and . This can be remedied by inserting an additional parameter to the formula. The following will generate all Pythagorean triples uniquely: : a = k\cdot(m^2 - n^2) ,\ \, b = k\cdot(2mn) ,\ \, c = k\cdot(m^2 + n^2) where , , and are positive integers with , and with and coprime and not both odd. That these formulas generate Pythagorean triples can be verified by expanding using
elementary algebra Elementary algebra encompasses the basic concepts of algebra. It is often contrasted with arithmetic: arithmetic deals with specified numbers, whilst algebra introduces variables (quantities without fixed values). This use of variables entail ...
and verifying that the result equals . Since every Pythagorean triple can be divided through by some integer to obtain a primitive triple, every triple can be generated uniquely by using the formula with and to generate its primitive counterpart and then multiplying through by as in the last equation. Choosing and from certain integer sequences gives interesting results. For example, if and are consecutive
Pell number In mathematics, the Pell numbers are an infinite sequence of integers, known since ancient times, that comprise the denominators of the closest rational approximations to the square root of 2. This sequence of approximations begins , , , , an ...
s, and will differ by 1. Many formulas for generating triples with particular properties have been developed since the time of Euclid.


Proof of Euclid's formula

That satisfaction of Euclid's formula by ''a, b, c'' is sufficient for the triangle to be Pythagorean is apparent from the fact that for positive integers and , , the , , and given by the formula are all positive integers, and from the fact that : a^2+b^2 = (m^2 - n^2)^2 + (2mn)^2 = (m^2 + n^2)^2 = c^2. A proof of the ''necessity'' that ''a, b, c'' be expressed by Euclid's formula for any primitive Pythagorean triple is as follows. All such primitive triples can be written as where and , , are
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 equival ...
. Thus , , are
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 equival ...
(if a prime number divided two of them, it would be forced also to divide the third one). As and are coprime, at least one of them is odd, so we may suppose that is odd, by exchanging, if needed, and . This implies that is even and is odd (if were odd, would be even, and would be a multiple of 4, while would be congruent to 2 modulo 4, as an odd square is congruent to 1 modulo 4). From a^2+b^2=c^2 we obtain c^2-a^2=b^2 and hence (c-a)(c+a)=b^2. Then \tfrac=\tfrac. Since \tfrac is rational, we set it equal to \tfrac in lowest terms. Thus \tfrac=\tfrac, being the reciprocal of \tfrac. Then solving :\frac+\frac=\frac, \quad \quad \frac-\frac=\frac for \tfrac and \tfrac gives :\frac=\frac\left(\frac+\frac\right)=\frac, \quad \quad \frac=\frac\left(\frac-\frac\right)=\frac. As \tfrac is fully reduced, and are coprime, and they cannot both be even. If they were both odd, the numerator of \tfrac would be a multiple of 4 (because an odd square is congruent to 1 modulo 4), and the denominator 2''mn'' would not be a multiple of 4. Since 4 would be the minimum possible even factor in the numerator and 2 would be the maximum possible even factor in the denominator, this would imply to be even despite defining it as odd. Thus one of and is odd and the other is even, and the numerators of the two fractions with denominator 2''mn'' are odd. Thus these fractions are fully reduced (an odd prime dividing this denominator divides one of and but not the other; thus it does not divide ). One may thus equate numerators with numerators and denominators with denominators, giving Euclid's formula : a = m^2 - n^2 ,\ \, b = 2mn ,\ \, c = m^2 + n^2 with and coprime and of opposite parities. A longer but more commonplace proof is given in Maor (2007) and Sierpiński (2003). Another proof is given in , as an instance of a general method that applies to every
homogeneous Homogeneity and heterogeneity are concepts often used in the sciences and statistics relating to the uniformity of a substance or organism. A material or image that is homogeneous is uniform in composition or character (i.e. color, shape, size, ...
Diophantine equation of degree two.


Interpretation of parameters in Euclid's formula

Suppose the sides of a Pythagorean triangle have lengths , , and , and suppose the angle between the leg of length and the
hypotenuse In geometry, a hypotenuse is the longest side of a right-angled triangle, the side opposite the right angle. The length of the hypotenuse can be found using the Pythagorean theorem, which states that the square of the length of the hypotenuse equ ...
of length is denoted as . Then \tan=\tfrac and the full-angle trigonometric values are \sin=\tfrac, \cos=\tfrac, and \tan=\tfrac.


A variant

The following variant of Euclid's formula is sometimes more convenient, as being more symmetric in and (same parity condition on and ). If and are two odd integers such that , then : a = mn ,\ \, b =\frac ,\ \, c = \frac are three integers that form a Pythagorean triple, which is primitive if and only if and are coprime. Conversely, every primitive Pythagorean triple arises (after the exchange of and , if is even) from a unique pair of coprime odd integers.


Elementary properties of primitive Pythagorean triples


General properties

The properties of a primitive Pythagorean triple with (without specifying which of or is even and which is odd) include: * \tfrac is always a perfect square. As it is only a necessary condition but not a sufficient one, it can be used in checking if a given triple of numbers is ''not'' a Pythagorean triple when they fail the test. For example, the triples and each pass the test that is a perfect square, but neither is a Pythagorean triple. *When a triple of numbers , and forms a primitive Pythagorean triple, then and one-half of are both perfect squares; however this is not a sufficient condition, as the numbers pass the perfect squares test but are not a Pythagorean triple since . *At most one of , , is a square. *The area of a Pythagorean triangle cannot be the square or twice the square of a natural number. *Exactly one of , is
divisible In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...
by 2 (is even), but never . *Exactly one of , is divisible by 3, but never . *Exactly one of , is divisible by 4, but never (because is never even). *Exactly one of , , is divisible by 5. *The largest number that always divides ''abc'' is 60. *Any odd number of the form , where is an integer and , can be the odd leg of a primitive Pythagorean triple PT See almost-isosceles PPT section below. However, only even numbers divisible by 4 can be the even leg of a PPT. This is because Euclid's formula for the even leg given above is and one of or must be even. *The hypotenuse is the sum of two squares. This requires all of its prime factors to be primes of the form . Therefore, c is of the form . A sequence of possible hypotenuse numbers for a PPT can be found at . *The area is a
congruent number In number theory, a congruent number is a positive integer that is the area of a right triangle with three rational number sides. A more general definition includes all positive rational numbers with this property. The sequence of (integer) cong ...
divisible by 6. *In every Pythagorean triangle, the radius of the
incircle In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter. ...
and the radii of the three excircles are natural numbers. Specifically, for a primitive triple the radius of the incircle is , and the radii of the excircles opposite the sides , ''2mn'', and the hypotenuse are respectively , , and . *As for any right triangle, the converse of
Thales' theorem In geometry, Thales's theorem states that if A, B, and C are distinct points on a circle where the line is a diameter, the angle ABC is a right angle. Thales's theorem is a special case of the inscribed angle theorem and is mentioned and proved ...
says that the diameter of the circumcircle equals the hypotenuse; hence for primitive triples the circumdiameter is , and the circumradius is half of this and thus is rational but non-integer (since and have opposite parity). *When the area of a Pythagorean triangle is multiplied by the
curvature In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane. For curves, the canon ...
s of its incircle and 3 excircles, the result is four positive integers , respectively. Integers satisfy Descartes's Circle Equation. Equivalently, the radius of the outer Soddy circle of any right triangle is equal to its semiperimeter. The outer Soddy center is located at , where is a rectangle, the right triangle and its hypotenuse. *Only two sides of a primitive Pythagorean triple can be simultaneously prime because by Euclid's formula for generating a primitive Pythagorean triple, one of the legs must be composite and even. However, only one side can be an integer of perfect power p \ge 2 because if two sides were integers of perfect powers with equal exponent p it would contradict the fact that there are no integer solutions to the
Diophantine equation In mathematics, a Diophantine equation is an equation, typically a polynomial equation in two or more unknowns with integer coefficients, such that the only solutions of interest are the integer ones. A linear Diophantine equation equates to a ...
x^ \pm y^=z^2, with x, y and z being pairwise coprime.H. Darmon and L. Merel. Winding quotients and some variants of Fermat’s Last Theorem, J. Reine Angew. Math. 490 (1997), 81–100. *There are no Pythagorean triangles in which the hypotenuse and one leg are the legs of another Pythagorean triangle; this is one of the equivalent forms of
Fermat's right triangle theorem Fermat's right triangle theorem is a non-existence proof in number theory, published in 1670 among the works of Pierre de Fermat, soon after his death. It is the only complete proof given by Fermat. It has several equivalent formulations, one o ...
. *Each primitive Pythagorean triangle has a ratio of area, , to squared
semiperimeter In geometry, the semiperimeter of a polygon is half its perimeter. Although it has such a simple derivation from the perimeter, the semiperimeter appears frequently enough in formulas for triangles and other figures that it is given a separate name ...
, , that is unique to itself and is given by :: \frac = \frac = 1-\frac. *No primitive Pythagorean triangle has an integer altitude from the hypotenuse; that is, every primitive Pythagorean triangle is indecomposable. *The set of all primitive Pythagorean triples forms a rooted
ternary tree : In computer science, a ternary tree is a tree data structure in which each node has at most three child nodes, usually distinguished as "left", “mid” and "right". Nodes with children are parent nodes, and child nodes may contain references ...
in a natural way; see
Tree of primitive Pythagorean triples 500px, Berggrens's tree of primitive Pythagorean triples. In mathematics, a tree of primitive Pythagorean triples is a data tree in which each node branches to three subsequent nodes with the infinite set of all nodes giving all (and only) primi ...
. *Neither of the
acute angle In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the '' vertex'' of the angle. Angles formed by two rays lie in the plane that contains the rays. Angles ...
s of a Pythagorean triangle can be a
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 ratio ...
of degrees. (This follows from
Niven's theorem In mathematics, Niven's theorem, named after Ivan Niven, states that the only rational values of ''θ'' in the interval 0° ≤ ''θ'' ≤ 90° for which the sine of ''θ'' degrees is also a rational number ...
.)


Special cases

In addition, special Pythagorean triples with certain additional properties can be guaranteed to exist: *Every integer greater than 2 that is not congruent to 2 mod 4 (in other words, every integer greater than 2 which is ''not'' of the form ) is part of a primitive Pythagorean triple. (If the integer has the form , one may take and in Euclid's formula; if the integer is , one may take and .) *Every integer greater than 2 is part of a primitive or non-primitive Pythagorean triple. For example, the integers 6, 10, 14, and 18 are not part of primitive triples, but are part of the non-primitive triples , and . *There exist infinitely many Pythagorean triples in which the hypotenuse and the longest leg differ by exactly one. Such triples are necessarily primitive and have the form . This results from Euclid's formula by remarking that the condition implies that the triple is primitive and must verify . This implies , and thus . The above form of the triples results thus of substituting for in Euclid's formula. *There exist infinitely many primitive Pythagorean triples in which the hypotenuse and the longest leg differ by exactly two. They are all primitive, and are obtained by putting in Euclid's formula. More generally, for every integer , there exist infinitely many primitive Pythagorean triples in which the hypotenuse and the odd leg differ by . They are obtained by putting in Euclid's formula. *There exist infinitely many Pythagorean triples in which the two legs differ by exactly one. For example, 20 + 21 = 29; these are generated by Euclid's formula when \tfrac is a convergent to \sqrt2. *For each natural number , there exist Pythagorean triples with different hypotenuses and the same area. *For each natural number , there exist at least different primitive Pythagorean triples with the same leg , where is some natural number (the length of the even leg is 2''mn'', and it suffices to choose with many factorizations, for example , where is a product of different odd primes; this produces at least different primitive triples). *For each natural number , there exist at least different Pythagorean triples with the same hypotenuse. *If is a
prime power In mathematics, a prime power is a positive integer which is a positive integer power of a single prime number. For example: , and are prime powers, while , and are not. The sequence of prime powers begins: 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17 ...
, there exists a primitive Pythagorean triple if and only if the prime has the form ; this triple is unique
up to Two mathematical objects ''a'' and ''b'' are called equal up to an equivalence relation ''R'' * if ''a'' and ''b'' are related by ''R'', that is, * if ''aRb'' holds, that is, * if the equivalence classes of ''a'' and ''b'' with respect to ''R'' a ...
the exchange of ''a'' and ''b''. *More generally, a positive integer is the hypotenuse of a primitive Pythagorean triple if and only if each
prime factor 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 ...
of is congruent to modulo ; that is, each prime factor has the form . In this case, the number of primitive Pythagorean triples with is , where is the number of distinct prime factors of . *There exist infinitely many Pythagorean triples with square numbers for both the hypotenuse and the sum of the legs . According to Fermat, the smallest such triple has sides ; ; and . Here and . This is generated by Euclid's formula with parameter values and . *There exist non-primitive Pythagorean triangles with integer altitude from the hypotenuse. Such Pythagorean triangles are known as decomposable since they can be split along this altitude into two separate and smaller Pythagorean triangles.


Geometry of Euclid's formula


Rational points on a unit circle

Euclid's formula for a Pythagorean triple :a = m^2-n^2,\quad b=2mn,\quad c=m^2+n^2 can be understood in terms of the geometry of
rational point In number theory and algebraic geometry, a rational point of an algebraic variety is a point whose coordinates belong to a given field. If the field is not mentioned, the field of rational numbers is generally understood. If the field is the field ...
s on the
unit circle In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucl ...
. In fact, a point in the
Cartesian plane A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...
with coordinates belongs to the unit circle if . The point is ''rational'' if and are
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 ratio ...
s, that is, if there are coprime integers such that :\biggl(\frac\biggr)^2 + \biggl(\frac\biggr)^2=1. By multiplying both members by , one can see that the rational points on the circle are in one-to-one correspondence with the primitive Pythagorean triples. The unit circle may also be defined by a
parametric equation In mathematics, a parametric equation defines a group of quantities as functions of one or more independent variables called parameters. Parametric equations are commonly used to express the coordinates of the points that make up a geometric ob ...
:x=\frac\quad y=\frac. Euclid's formula for Pythagorean triples and the inverse relationship mean that, except for , a point on the circle is rational if and only if the corresponding value of is a rational number. Note that is also the tangent of half of the angle that is opposite the triangle side of length .


Stereographic approach

There is a correspondence between points on the unit circle with rational coordinates and primitive Pythagorean triples. At this point, Euclid's formulae can be derived either by methods of
trigonometry Trigonometry () is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. ...
or equivalently by using the
stereographic projection In mathematics, a stereographic projection is a perspective projection of the sphere, through a specific point on the sphere (the ''pole'' or ''center of projection''), onto a plane (the ''projection plane'') perpendicular to the diameter thro ...
. For the stereographic approach, suppose that ′ is a point on the -axis with rational coordinates :P' = \left(\frac,0\right). Then, it can be shown by basic algebra that the point has coordinates : P = \left( \frac, \frac \right) = \left( \frac, \frac \right). This establishes that each
rational point In number theory and algebraic geometry, a rational point of an algebraic variety is a point whose coordinates belong to a given field. If the field is not mentioned, the field of rational numbers is generally understood. If the field is the field ...
of the -axis goes over to a rational point of the unit circle. The converse, that every rational point of the unit circle comes from such a point of the -axis, follows by applying the inverse stereographic projection. Suppose that is a point of the unit circle with and rational numbers. Then the point ′ obtained by stereographic projection onto the -axis has coordinates :\left(\frac,0\right) which is rational. In terms of
algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrica ...
, the
algebraic variety Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. ...
of rational points on the unit circle is
birational In mathematics, birational geometry is a field of algebraic geometry in which the goal is to determine when two algebraic varieties are isomorphic outside lower-dimensional subsets. This amounts to studying mappings that are given by rational ...
to the
affine line In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties related ...
over the rational numbers. The unit circle is thus called a
rational curve In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane ...
, and it is this fact which enables an explicit parameterization of the (rational number) points on it by means of rational functions.


Pythagorean triangles in a 2D lattice

A 2D lattice is a regular array of isolated points where if any one point is chosen as the Cartesian origin (0, 0), then all the other points are at where and range over all positive and negative integers. Any Pythagorean triangle with triple can be drawn within a 2D lattice with vertices at coordinates , and . The count of lattice points lying strictly within the bounds of the triangle is given by   \tfrac; for primitive Pythagorean triples this interior lattice count is  \tfrac. The area (by
Pick's theorem In geometry, Pick's theorem provides a formula for the area of a simple polygon with integer vertex coordinates, in terms of the number of integer points within it and on its boundary. The result was first described by Georg Alexander Pick in ...
equal to one less than the interior lattice count plus half the boundary lattice count) equals  \tfrac . The first occurrence of two primitive Pythagorean triples sharing the same area occurs with triangles with sides and common area 210 . The first occurrence of two primitive Pythagorean triples sharing the same interior lattice count occurs with and interior lattice count 2287674594 . Three primitive Pythagorean triples have been found sharing the same area: , , with area 13123110. As yet, no set of three primitive Pythagorean triples have been found sharing the same interior lattice count.


Enumeration of primitive Pythagorean triples

By Euclid's formula all primitive Pythagorean triples can be generated from integers m and n with m>n>0, m+n odd and \gcd(m, n)=1. Hence there is a 1 to 1 mapping of rationals (in lowest terms) to primitive Pythagorean triples where \tfrac is in the interval (0,1) and m+n odd. The reverse mapping from a primitive triple (a , b , c) where c>b>a>0 to a rational \tfrac is achieved by studying the two sums a+c and b+c. One of these sums will be a square that can be equated to (m+n)^2 and the other will be twice a square that can be equated to 2m^2. It is then possible to determine the rational \tfrac. In order to enumerate primitive Pythagorean triples the rational can be expressed as an ordered pair (n,m) and mapped to an integer using a pairing function such as Cantor's pairing function. An example can be seen at . It begins ::8,18,19,32,33,34,\dots and gives rationals ::\tfrac,\tfrac,\tfrac,\tfrac,\tfrac,\tfrac,\dots these, in turn, generate primitive triples ::(3,4,5),(5,12,13),(8,15,17),(7,24,25),(20,21,29),(12,35,37),\dots


Spinors and the modular group

Pythagorean triples can likewise be encoded into a
square matrix In mathematics, a square matrix is a matrix with the same number of rows and columns. An ''n''-by-''n'' matrix is known as a square matrix of order Any two square matrices of the same order can be added and multiplied. Square matrices are ofte ...
of the form :X = \begin c+b & a\\ a & c-b \end. A matrix of this form is
symmetric Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definit ...
. Furthermore, the
determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if an ...
of is :\det X = c^2 - a^2 - b^2\, which is zero precisely when is a Pythagorean triple. If corresponds to a Pythagorean triple, then as a matrix it must have
rank Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as: Level or position in a hierarchical organization * Academic rank * Diplomatic rank * Hierarchy * H ...
1. Since is symmetric, it follows from a result in
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrices. ...
that there is a
column vector In linear algebra, a column vector with m elements is an m \times 1 matrix consisting of a single column of m entries, for example, \boldsymbol = \begin x_1 \\ x_2 \\ \vdots \\ x_m \end. Similarly, a row vector is a 1 \times n matrix for some n, c ...
such that the
outer product In linear algebra, the outer product of two coordinate vectors is a matrix. If the two vectors have dimensions ''n'' and ''m'', then their outer product is an ''n'' × ''m'' matrix. More generally, given two tensors (multidimensional arrays of nu ...
holds, where the denotes the
matrix transpose In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal; that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations). The tr ...
. The vector ξ is called a
spinor In geometry and physics, spinors are elements of a complex vector space that can be associated with Euclidean space. Like geometric vectors and more general tensors, spinors transform linearly when the Euclidean space is subjected to a sligh ...
(for the
Lorentz group In physics and mathematics, the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical and quantum setting for all (non-gravitational) physical phenomena. The Lorentz group is named for the Dutch physic ...
SO(1, 2)). In abstract terms, the Euclid formula means that each primitive Pythagorean triple can be written as the outer product with itself of a spinor with integer entries, as in (). The
modular group In mathematics, the modular group is the projective special linear group of matrices with integer coefficients and determinant 1. The matrices and are identified. The modular group acts on the upper-half of the complex plane by fractional ...
Γ is the set of 2×2 matrices with integer entries :A = \begin\alpha&\beta\\ \gamma&\delta\end with determinant equal to one: . This set forms a
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 ...
, since the inverse of a matrix in Γ is again in Γ, as is the product of two matrices in Γ. The modular group
acts The Acts of the Apostles ( grc-koi, Πράξεις Ἀποστόλων, ''Práxeis Apostólōn''; la, Actūs Apostolōrum) is the fifth book of the New Testament; it tells of the founding of the Christian Church and the spread of its message ...
on the collection of all integer spinors. Furthermore, the group is transitive on the collection of integer spinors with relatively prime entries. For if has relatively prime entries, then :\beginm&-v\\n&u\end\begin1\\0\end = \beginm\\n\end where and are selected (by the
Euclidean algorithm In mathematics, the Euclidean algorithm,Some widely used textbooks, such as I. N. Herstein's ''Topics in Algebra'' and Serge Lang's ''Algebra'', use the term "Euclidean algorithm" to refer to Euclidean division or Euclid's algorithm, is an ef ...
) so that . By acting on the spinor ξ in (), the action of Γ goes over to an action on Pythagorean triples, provided one allows for triples with possibly negative components. Thus if is a matrix in , then gives rise to an action on the matrix in (). This does not give a well-defined action on primitive triples, since it may take a primitive triple to an imprimitive one. It is convenient at this point (per ) to call a triple standard if and either are relatively prime or are relatively prime with odd. If the spinor has relatively prime entries, then the associated triple determined by () is a standard triple. It follows that the action of the modular group is transitive on the set of standard triples. Alternatively, restrict attention to those values of and for which is odd and is even. Let the
subgroup In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgrou ...
Γ(2) of Γ be the
kernel Kernel may refer to: Computing * Kernel (operating system), the central component of most operating systems * Kernel (image processing), a matrix used for image convolution * Compute kernel, in GPGPU programming * Kernel method, in machine learni ...
of the
group homomorphism In mathematics, given two groups, (''G'', ∗) and (''H'', ·), a group homomorphism from (''G'', ∗) to (''H'', ·) is a function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' it holds that : h(u*v) = h(u) \cdot h(v) ...
:\Gamma=\mathrm(2,\mathbf)\to \mathrm(2,\mathbf_2) where is the
special linear group In mathematics, the special linear group of degree ''n'' over a field ''F'' is the set of matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix inversion. This is the normal subgroup of the gener ...
over the
finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...
of integers modulo 2. Then Γ(2) is the group of unimodular transformations which preserve the parity of each entry. Thus if the first entry of ξ is odd and the second entry is even, then the same is true of for all . In fact, under the action (), the group Γ(2) acts transitively on the collection of primitive Pythagorean triples . The group Γ(2) is the
free group In mathematics, the free group ''F'S'' over a given set ''S'' consists of all words that can be built from members of ''S'', considering two words to be different unless their equality follows from the group axioms (e.g. ''st'' = ''suu''−1 ...
whose generators are the matrices :U=\begin1&2\\0&1\end,\qquad L=\begin1&0\\2&1\end. Consequently, every primitive Pythagorean triple can be obtained in a unique way as a product of copies of the matrices and .


Parent/child relationships

By a result of , all primitive Pythagorean triples can be generated from the (3, 4, 5) triangle by using the three
linear transformation In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a Map (mathematics), mapping V \to W between two vect ...
s T1, T2, T3 below, where , , are sides of a triple: In other words, every primitive triple will be a "parent" to three additional primitive triples. Starting from the initial node with , , and , the operation produces the new triple :(3 − (2×4) + (2×5), (2×3) − 4 + (2×5), (2×3) − (2×4) + (3×5)) = (5, 12, 13), and similarly and produce the triples (21, 20, 29) and (15, 8, 17). The linear transformations T1, T2, and T3 have a geometric interpretation in the language of
quadratic form In mathematics, a quadratic form is a polynomial with terms all of degree two (" form" is another name for a homogeneous polynomial). For example, :4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong t ...
s. They are closely related to (but are not equal to) reflections generating the
orthogonal group In mathematics, the orthogonal group in dimension , denoted , is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations. ...
of over the integers.


Relation to Gaussian integers

Alternatively, Euclid's formulae can be analyzed and proved using the
Gaussian integers In number theory, a Gaussian integer is a complex number whose real and imaginary parts are both integers. The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain, usually written as \mathbf /ma ...
. Gaussian integers are complex numbers of the form , where and are ordinary
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 o ...
s and is the square root of negative one. The
units Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or beat) in a theatrical presentation Music * ''Unit'' (al ...
of Gaussian integers are ±1 and ±i. The ordinary integers are called the
rational 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 of ...
s and denoted as ''. The Gaussian integers are denoted as . The right-hand side of 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 ...
may be factored in Gaussian integers: :c^2 = a^2+b^2 = (a+bi)\overline = (a+bi)(a-bi). A primitive Pythagorean triple is one in which and are
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 equival ...
, i.e., they share no prime factors in the integers. For such a triple, either or is even, and the other is odd; from this, it follows that is also odd. The two factors and of a primitive Pythagorean triple each equal the square of a Gaussian integer. This can be proved using the property that every Gaussian integer can be factored uniquely into Gaussian primes
up to Two mathematical objects ''a'' and ''b'' are called equal up to an equivalence relation ''R'' * if ''a'' and ''b'' are related by ''R'', that is, * if ''aRb'' holds, that is, * if the equivalence classes of ''a'' and ''b'' with respect to ''R'' a ...
units Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or beat) in a theatrical presentation Music * ''Unit'' (al ...
. See also ''Werke'', 2:67–148. (This unique factorization follows from the fact that, roughly speaking, a version of the
Euclidean algorithm In mathematics, the Euclidean algorithm,Some widely used textbooks, such as I. N. Herstein's ''Topics in Algebra'' and Serge Lang's ''Algebra'', use the term "Euclidean algorithm" to refer to Euclidean division or Euclid's algorithm, is an ef ...
can be defined on them.) The proof has three steps. First, if and share no prime factors in the integers, then they also share no prime factors in the Gaussian integers. (Assume and with Gaussian integers , and and not a unit. Then and lie on the same line through the origin. All Gaussian integers on such a line are integer multiples of some Gaussian integer . But then the integer ''gh'' ≠ ±1 divides both and .) Second, it follows that and likewise share no prime factors in the Gaussian integers. For if they did, then their common divisor would also divide and . Since and are coprime, that implies that divides . From the formula , that in turn would imply that is even, contrary to the hypothesis of a primitive Pythagorean triple. Third, since is a square, every Gaussian prime in its factorization is doubled, i.e., appears an even number of times. Since and share no prime factors, this doubling is also true for them. Hence, and are squares. Thus, the first factor can be written :a+bi = \varepsilon\left(m + ni \right)^2, \quad \varepsilon\in\. The real and imaginary parts of this equation give the two formulas: :\begin\varepsilon = +1, & \quad a = +\left( m^2 - n^2 \right),\quad b = +2mn; \\ \varepsilon = -1, & \quad a = -\left( m^2 - n^2 \right),\quad b = -2mn; \\ \varepsilon = +i, & \quad a = -2mn,\quad b = +\left( m^2 - n^2 \right); \\ \varepsilon = -i, & \quad a = +2mn,\quad b = -\left( m^2 - n^2 \right).\end For any primitive Pythagorean triple, there must be integers and such that these two equations are satisfied. Hence, every Pythagorean triple can be generated from some choice of these integers.


As perfect square Gaussian integers

If we consider the square of a Gaussian integer we get the following direct interpretation of Euclid's formula as representing the perfect square of a Gaussian integer. :(m+ni)^2 = (m^2-n^2)+2mni. Using the facts that the Gaussian integers are a Euclidean domain and that for a Gaussian integer p , p, ^2 is always a square it is possible to show that a Pythagorean triple corresponds to the square of a prime Gaussian integer if the hypotenuse is prime. If the Gaussian integer is not prime then it is the product of two Gaussian integers p and q with , p, ^2 and , q, ^2 integers. Since magnitudes multiply in the Gaussian integers, the product must be , p, , q, , which when squared to find a Pythagorean triple must be composite. The contrapositive completes the proof.


Distribution of triples

There are a number of results on the distribution of Pythagorean triples. In the scatter plot, a number of obvious patterns are already apparent. Whenever the legs of a primitive triple appear in the plot, all integer multiples of must also appear in the plot, and this property produces the appearance of lines radiating from the origin in the diagram. Within the scatter, there are sets of parabolic patterns with a high density of points and all their foci at the origin, opening up in all four directions. Different parabolas intersect at the axes and appear to reflect off the axis with an incidence angle of 45 degrees, with a third parabola entering in a perpendicular fashion. Within this quadrant, each arc centered on the origin shows that section of the parabola that lies between its tip and its intersection with its
semi-latus rectum In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a specia ...
. These patterns can be explained as follows. If a^2/4n is an integer, then (, , n-a^2/4n, , n+a^2/4n) is a Pythagorean triple. (In fact every Pythagorean triple can be written in this way with integer , possibly after exchanging and , since n=(b+c)/2 and and cannot both be odd.) The Pythagorean triples thus lie on curves given by b = , n-a^2/4n, , that is, parabolas reflected at the -axis, and the corresponding curves with and interchanged. If is varied for a given (i.e. on a given parabola), integer values of occur relatively frequently if is a square or a small multiple of a square. If several such values happen to lie close together, the corresponding parabolas approximately coincide, and the triples cluster in a narrow parabolic strip. For instance, , , , and ; the corresponding parabolic strip around is clearly visible in the scatter plot. The angular properties described above follow immediately from the functional form of the parabolas. The parabolas are reflected at the -axis at , and the derivative of with respect to at this point is –1; hence the incidence angle is 45°. Since the clusters, like all triples, are repeated at integer multiples, the value also corresponds to a cluster. The corresponding parabola intersects the -axis at right angles at , and hence its reflection upon interchange of and intersects the -axis at right angles at , precisely where the parabola for is reflected at the -axis. (The same is of course true for and interchanged.) Albert Fässler and others provide insights into the significance of these parabolas in the context of conformal mappings.


Special cases and related equations


The Platonic sequence

The case of the more general construction of Pythagorean triples has been known for a long time.
Proclus Proclus Lycius (; 8 February 412 – 17 April 485), called Proclus the Successor ( grc-gre, Πρόκλος ὁ Διάδοχος, ''Próklos ho Diádokhos''), was a Greek Neoplatonist philosopher, one of the last major classical philosophers ...
, in his commentary to the 47th Proposition of the first book of Euclid's ''Elements'', describes it as follows:
Certain methods for the discovery of triangles of this kind are handed down, one which they refer to Plato, and another to
Pythagoras Pythagoras of Samos ( grc, Πυθαγόρας ὁ Σάμιος, Pythagóras ho Sámios, Pythagoras the Samian, or simply ; in Ionian Greek; ) was an ancient Ionian Greek philosopher and the eponymous founder of Pythagoreanism. His politi ...
. (The latter) starts from odd numbers. For it makes the odd number the smaller of the sides about the right angle; then it takes the square of it, subtracts unity and makes half the difference the greater of the sides about the right angle; lastly it adds unity to this and so forms the remaining side, the hypotenuse.
...For the method of Plato argues from even numbers. It takes the given even number and makes it one of the sides about the right angle; then, bisecting this number and squaring the half, it adds unity to the square to form the hypotenuse, and subtracts unity from the square to form the other side about the right angle. ... Thus it has formed the same triangle that which was obtained by the other method.
In equation form, this becomes: is odd (Pythagoras, c. 540 BC): :\texta : \textb = : \textc = . is even (Plato, c. 380 BC): :\texta : \textb = \left(\right)^2 - 1 : \textc = \left(\right)^2 + 1 It can be shown that all Pythagorean triples can be obtained, with appropriate rescaling, from the basic Platonic sequence (, and ) by allowing to take non-integer rational values. If is replaced with the fraction in the sequence, the result is equal to the 'standard' triple generator (2''mn'', ,) after rescaling. It follows that every triple has a corresponding rational value which can be used to generate a similar triangle (one with the same three angles and with sides in the same proportions as the original). For example, the Platonic equivalent of is generated by as . The Platonic sequence itself can be derived by following the steps for 'splitting the square' described in Diophantus II.VIII.


The Jacobi–Madden equation

The equation, :a^4+b^4+c^4+d^4 = (a+b+c+d)^4 is equivalent to the special Pythagorean triple, :(a^2+ab+b^2)^2+(c^2+cd+d^2)^2 = ((a+b)^2+(a+b)(c+d)+(c+d)^2)^2 There is an infinite number of solutions to this equation as solving for the variables involves 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 ...
. Small ones are, :a, b, c, d = -2634, 955, 1770, 5400 :a, b, c, d = -31764, 7590, 27385, 48150


Equal sums of two squares

One way to generate solutions to a^2+b^2=c^2+d^2 is to parametrize ''a, b, c, d'' in terms of integers ''m, n, p, q'' as follows: :(m^2+n^2)(p^2+q^2)=(mp-nq)^2+(np+mq)^2=(mp+nq)^2+(np-mq)^2.


Equal sums of two fourth powers

Given two sets of Pythagorean triples, :(a^2-b^2)^2+(2a b)^2 = (a^2+b^2)^2 :(c^2-d^2)^2+(2c d)^2 = (c^2+d^2)^2 the problem of finding equal products of a non-hypotenuse side and the hypotenuse, :(a^2 -b^2)(a^2+b^2) = (c^2 -d^2)(c^2+d^2) is easily seen to be equivalent to the equation, :a^4 -b^4 = c^4 -d^4 and was first solved by Euler as a, b, c, d = 133,59,158,134. Since he showed this is a rational point in 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 ...
, then there is an infinite number of solutions. In fact, he also found a 7th degree polynomial parameterization.


Descartes' Circle Theorem

For the case of Descartes' circle theorem where all variables are squares, :2(a^4+b^4+c^4+d^4) = (a^2+b^2+c^2+d^2)^2 Euler showed this is equivalent to three simultaneous Pythagorean triples, :(2ab)^2+(2cd)^2 = (a^2+b^2-c^2-d^2)^2 :(2ac)^2+(2bd)^2 = (a^2-b^2+c^2-d^2)^2 :(2ad)^2+(2bc)^2 = (a^2-b^2-c^2+d^2)^2 There is also an infinite number of solutions, and for the special case when a+b=c, then the equation simplifies to, :4(a^2+a b+b^2) = d^2 with small solutions as a, b, c, d = 3, 5, 8, 14 and can be solved as binary quadratic forms.


Almost-isosceles Pythagorean triples

No Pythagorean triples are
isosceles In geometry, an isosceles triangle () is a triangle that has two sides of equal length. Sometimes it is specified as having ''exactly'' two sides of equal length, and sometimes as having ''at least'' two sides of equal length, the latter versio ...
, because the ratio of the hypotenuse to either other side is , but cannot be expressed as the ratio of 2 integers. There are, however, right-angled triangles with integral sides for which the lengths of the non-hypotenuse sides differ by one, such as, :3^2+4^2 = 5^2 :20^2+21^2 = 29^2 and an infinite number of others. They can be completely parameterized as, :\left(\tfrac\right)^2+\left(\tfrac\right)^2 = y^2 where are the solutions to the
Pell equation Pell's equation, also called the Pell–Fermat equation, is any Diophantine equation of the form x^2 - ny^2 = 1, where ''n'' is a given positive nonsquare integer, and integer solutions are sought for ''x'' and ''y''. In Cartesian coordinates ...
x^2-2y^2 = -1. If , , are the sides of this type of primitive Pythagorean triple (PPT) then the solution to the Pell equation is given by the recursive formula :a_n=6a_-a_+2 with a_1=3 and a_2=20 :b_n=6b_-b_-2 with b_1=4 and b_2=21 :c_n=6c_-c_ with c_1=5 and c_2=29. This sequence of PPTs forms the central stem (trunk) of the rooted ternary tree of PPTs. When it is the longer non-hypotenuse side and hypotenuse that differ by one, such as in :5^2+12^2 = 13^2 :7^2+24^2 = 25^2 then the complete solution for the PPT , , is :a=2m+1, \quad b=2m^2+2m, \quad c=2m^2+2m+1 and :(2m+1)^2+(2m^2+2m)^2=(2m^2+2m+1)^2 where integer m>0 is the generating parameter. It shows that all
odd numbers In mathematics, parity is the property of an integer of whether it is even or odd. An integer is even if it is a multiple of two, and odd if it is not.. For example, −4, 0, 82 are even because \begin -2 \cdot 2 &= -4 \\ 0 \cdot 2 &= 0 \\ 41 ...
(greater than 1) appear in this type of almost-isosceles PPT. This sequence of PPTs forms the right hand side outer stem of the rooted ternary tree of PPTs. Another property of this type of almost-isosceles PPT is that the sides are related such that :a^b+b^a=Kc for some integer K. Or in other words a^b+b^a is divisible by c such as in :(5^+12^5)/13 = 18799189.


Fibonacci numbers in Pythagorean triples

Starting with 5, every second
Fibonacci number In mathematics, the Fibonacci numbers, commonly denoted , form a sequence, the Fibonacci sequence, in which each number is the sum of the two preceding ones. The sequence commonly starts from 0 and 1, although some authors start the sequence from ...
is the length of the hypotenuse of a right triangle with integer sides, or in other words, the largest number in a Pythagorean triple, obtained from the formula (F_nF_)^2 + (2F_F_)^2 = F_^2. The sequence of Pythagorean triangles obtained from this formula has sides of lengths :(3,4,5), (5,12,13), (16,30,34), (39,80,89), ... The middle side of each of these triangles is the sum of the three sides of the preceding triangle.


Generalizations

There are several ways to generalize the concept of Pythagorean triples.


Pythagorean -tuple

The expression :\left(m_1^2 - m_2^2 - \ldots - m_n^2\right)^2 + \sum_^n (2 m_1 m_k)^2 = \left(m_1^2 + \ldots + m_n^2\right)^2 is a Pythagorean -tuple for any tuple of positive integers with . The Pythagorean -tuple can be made primitive by dividing out by the largest common divisor of its values. Furthermore, any primitive Pythagorean -tuple can be found by this approach. Use to get a Pythagorean -tuple by the above formula and divide out by the largest common integer divisor, which is . Dividing out by the largest common divisor of these values gives the same primitive Pythagorean -tuple; and there is a one-to-one correspondence between tuples of setwise coprime positive integers satisfying and primitive Pythagorean -tuples. Examples of the relationship between setwise coprime values \vec and primitive Pythagorean -tuples include: :\begin \vec = (1) & \leftrightarrow 1^2 = 1^2 \\ \vec = (2, 1) & \leftrightarrow 3^2 + 4^2 = 5^2 \\ \vec = (2, 1, 1) & \leftrightarrow 1^2 + 2^2 + 2^2 = 3^2 \\ \vec = (3, 1, 1, 1) & \leftrightarrow 1^2 + 1^2 + 1^2 + 1^2 = 2^2 \\ \vec = (5, 1, 1, 2, 3) & \leftrightarrow 1^2 + 1^2 + 1^2 + 2^2 + 3^2 = 4^2 \\ \vec = (4, 1, 1, 1, 1, 2) & \leftrightarrow 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 2^2 = 3^2 \\ \vec = (5, 1, 1, 1, 2, 2, 2) & \leftrightarrow 1^2 + 1^2 + 1^2 + 1^2 + 2^2 + 2^2 + 2^2 = 4^2 \end


Consecutive squares

Since the sum of consecutive squares beginning with is given by the formula, :F(k,m)=km(k-1+m)+\frac one may find values so that is a square, such as one by Hirschhorn where the number of terms is itself a square, :m=\tfrac,\; k=v^2,\; F(m,k)=\tfrac and is any integer not divisible by 2 or 3. For the smallest case , hence , this yields the well-known cannonball-stacking problem of Lucas, :0^2+1^2+2^2+\dots+24^2 = 70^2 a fact which is connected to the
Leech lattice In mathematics, the Leech lattice is an even unimodular lattice Λ24 in 24-dimensional Euclidean space, which is one of the best models for the kissing number problem. It was discovered by . It may also have been discovered (but not published) by ...
. In addition, if in a Pythagorean -tuple () all
addend Addition (usually signified by the plus symbol ) is one of the four basic operations of arithmetic, the other three being subtraction, multiplication and division. The addition of two whole numbers results in the total amount or '' sum'' of ...
s are consecutive except one, one can use the equation, :F(k,m) + p^ = (p+1)^ Since the second power of cancels out, this is only linear and easily solved for as p=\tfrac though , should be chosen so that is an integer, with a small example being , yielding, :1^2+2^2+3^2+4^2+5^2+27^2=28^2 Thus, one way of generating Pythagorean -tuples is by using, for various , :x^2+(x+1)^2+\cdots +(x+q)^2+p^2=(p+1)^2, where ''q = n''–2 and where :p=\frac.


Fermat's Last Theorem

A generalization of the concept of Pythagorean triples is the search for triples of positive integers , , and , such that , for some strictly greater than 2.
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 i ...
in 1637 claimed that no such triple exists, a claim that came to be known as
Fermat's Last Theorem In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers , , and satisfy the equation for any integer value of greater than 2. The cases and have been k ...
because it took longer than any other conjecture by Fermat to be proved or disproved. The first proof was given by Andrew Wiles in 1994.


or th powers summing to an th power

Another generalization is searching for sequences of positive integers for which the th power of the last is the sum of the th powers of the previous terms. The smallest sequences for known values of are: * = 3: . * = 4: * = 5: * = 7: * = 8: For the case, in which x^3+y^3+z^3=w^3, called the
Fermat cubic In 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 ...
, a general formula exists giving all solutions. A slightly different generalization allows the sum of th powers to equal the sum of th powers. For example: * (): 1 + 12 = 9 + 10, made famous by Hardy's recollection of a conversation with Ramanujan about the number 1729 being the smallest number that can be expressed as a sum of two cubes in two distinct ways. There can also exist positive integers whose th powers sum to an th power (though, by
Fermat's Last Theorem In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers , , and satisfy the equation for any integer value of greater than 2. The cases and have been k ...
, not for ; these are counterexamples to
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 is ...
. The smallest known counterexamples are * : (95800, 217519, 414560; 422481) * : (27, 84, 110, 133; 144)


Heronian triangle triples

A Heronian triangle is commonly defined as one with integer sides whose area is also an integer. The lengths of the sides of such a triangle form a Heronian triple for . Every Pythagorean triple is a Heronian triple, because at least one of the legs , must be even in a Pythagorean triple, so the area ''ab''/2 is an integer. Not every Heronian triple is a Pythagorean triple, however, as the example with area 24 shows. If is a Heronian triple, so is where is any positive integer; its area will be the integer that is times the integer area of the triangle. The Heronian triple is primitive provided ''a'', ''b'', ''c'' are setwise coprime. (With primitive Pythagorean triples the stronger statement that they are ''pairwise'' coprime also applies, but with primitive Heronian triangles the stronger statement does not always hold true, such as with .) Here are a few of the simplest primitive Heronian triples that are not Pythagorean triples: : (4, 13, 15) with area 24 : (3, 25, 26) with area 36 : (7, 15, 20) with area 42 : (6, 25, 29) with area 60 : (11, 13, 20) with area 66 : (13, 14, 15) with area 84 : (13, 20, 21) with area 126 By
Heron's formula In geometry, Heron's formula (or Hero's formula) gives the area of a triangle in terms of the three side lengths , , . If s = \tfrac12(a + b + c) is the semiperimeter of the triangle, the area is, :A = \sqrt. It is named after first-century ...
, the extra condition for a triple of positive integers with to be Heronian is that : or equivalently : be a nonzero perfect square divisible by 16.


Application to cryptography

Primitive Pythagorean triples have been used in cryptography as random sequences and for the generation of keys. Kak, S. and Prabhu, M. Cryptographic applications of primitive Pythagorean triples. Cryptologia, 38:215–222, 2014

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See also

* Boolean Pythagorean triples problem *
Congruum In number theory, a congruum (plural ''congrua'') is the difference between successive square numbers in an arithmetic progression of three squares. That is, if x^2, y^2, and z^2 (for integers x, y, and z) are three square numbers that are equall ...
* Diophantus II.VIII * Eisenstein triple *
Euler brick In mathematics, an Euler brick, named after Leonhard Euler, is a rectangular cuboid whose edges and face diagonals all have integer lengths. A primitive Euler brick is an Euler brick whose edge lengths are relatively prime. A perfect Euler bric ...
*
Heronian triangle In geometry, a Heronian triangle (or Heron triangle) is a triangle whose side lengths , , and and area are all integers. Heronian triangles are named after Heron of Alexandria, based on their relation to Heron's formula. Heron's formula implies ...
*
Hilbert's theorem 90 In abstract algebra, Hilbert's Theorem 90 (or Satz 90) is an important result on cyclic extensions of fields (or to one of its generalizations) that leads to Kummer theory. In its most basic form, it states that if ''L''/''K'' is an extension of f ...
*
Integer triangle An integer triangle or integral triangle is a triangle all of whose sides have lengths that are integers. A rational triangle can be defined as one having all sides with rational length; any such rational triangle can be integrally rescaled (ca ...
*
Modular arithmetic In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his bo ...
*
Nonhypotenuse number In mathematics, a nonhypotenuse number is a natural number whose square ''cannot'' be written as the sum of two nonzero squares. The name stems from the fact that an edge of length equal to a nonhypotenuse number ''cannot'' form the hypotenuse of ...
*
Plimpton 322 Plimpton 322 is a Babylonian clay tablet, notable as containing an example of Babylonian mathematics. It has number 322 in the G.A. Plimpton Collection at Columbia University. This tablet, believed to have been written about 1800 BC, has a tabl ...
*
Pythagorean prime A Pythagorean prime is a prime number of the Pythagorean primes are exactly the odd prime numbers that are the sum of two squares; this characterization is Fermat's theorem on sums of two squares. Equivalently, by the Pythagorean theorem, they ...
*
Pythagorean quadruple A Pythagorean quadruple is a tuple of integers , , , and , such that . They are solutions of a Diophantine equation and often only positive integer values are considered.R. Spira, ''The diophantine equation '', Amer. Math. Monthly Vol. 69 (1962), ...
*
Quadric In mathematics, a quadric or quadric surface (quadric hypersurface in higher dimensions), is a generalization of conic sections (ellipses, parabolas, and hyperbolas). It is a hypersurface (of dimension ''D'') in a -dimensional space, and it is d ...
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Tangent half-angle formula In trigonometry, tangent half-angle formulas relate the tangent of half of an angle to trigonometric functions of the entire angle. The tangent of half an angle is the stereographic projection of the circle onto a line. Among these formulas are the ...
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Trigonometric identity In trigonometry, trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables for which both sides of the equality are defined. Geometrically, these are identities involvi ...


Notes


References

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External links


Clifford Algebras and Euclid's Parameterization of Pythagorean triples

Curious Consequences of a Miscopied Quadratic


* ttps://web.archive.org/web/20160304023524/http://people.wcsu.edu/sandifere/Academics/2007Spring/Mat342/PythagTrip02.pdf Generating Pythagorean Triples Using Arithmetic Progressions*
Interactive Calculator for Pythagorean Triples



Parameterization of Pythagorean Triples by a single triple of polynomials
*
Pythagorean Triples and the Unit Circle
chap. 2–3, in

by Joseph H. Silverman, 3rd ed., 2006, Pearson Prentice Hall, Upper Saddle River, NJ,
Pythagorean Triples
at cut-the-knot Interactive Applet showing unit circle relationships to Pythagorean Triples
Pythagorean Triplets

The Remarkable Incircle of a Triangle

Solutions to Quadratic Compatible Pairs in relation to Pythagorean Triples


* ttp://www.cut-the-knot.org/pythagoras/PT_matrix.shtml The Trinary Tree(s) underlying Primitive Pythagorean Triplesat cut-the-knot * {{DEFAULTSORT:Pythagorean Triple Arithmetic problems of plane geometry Diophantine equations Triple Squares in number theory no:Pythagoras’ læresetning#Pytagoreiske tripler