Pythagorean triple

File:Pythagorean theorem - Ani.gif

Animation demonstrating the smallest Pythagorean triple, Template:Math.

A Pythagorean triple consists of three positive integers Template:Math, Template:Math, and Template:Math, such that Template:Math. Such a triple is commonly written Template:Math, a well-known example is Template:Math. If Template:Math is a Pythagorean triple, then so is Template:Math for any positive integer Template:Math. A triangle whose side lengths are a Pythagorean triple is a right triangle and called a Pythagorean triangle.

A primitive Pythagorean triple is one in which Template:Math, Template:Math and Template:Math are coprime (that is, they have no common divisor larger than 1).<ref>Template:Harvtxt</ref> For example, Template:Math is a primitive Pythagorean triple whereas Template:Math is not. Every Pythagorean triple can be scaled to a unique primitive Pythagorean triple by dividing Template:Math by their greatest common divisor. Conversely, every Pythagorean triple can be obtained by multiplying the elements of a primitive Pythagorean triple by a positive integer (the same for the three elements).

The name is derived from the Pythagorean theorem, stating that every right triangle has side lengths satisfying the formula <math>a^2+b^2=c^2</math>; 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 with sides <math>a=b=1</math> and <math>c=\sqrt2</math> is a right triangle, but <math>(1,1,\sqrt2)</math> is not a Pythagorean triple because the square root of 2 is not an integer. Moreover, <math>1</math> and <math>\sqrt2</math> do not have an integer common multiple because <math>\sqrt2</math> is irrational.

Pythagorean triples have been known since ancient times. The oldest known record comes from Plimpton 322, a Babylonian clay tablet from about 1800 BC, written in a sexagesimal number system.<ref>Template:Citation</ref>

When searching for integer solutions, the equation Template:Math is a Diophantine equation. Thus Pythagorean triples are among the oldest known solutions of a nonlinear Diophantine equation.

Examples

File:Pythagorean Triples Scatter Plot.png

Scatter plot of the legs (Template:Math, Template:Math) of the first Pythagorean triples with Template:Math and Template:Math less than 6000. Negative values are included to illustrate the parabolic patterns. The "rays" are a result of the fact that if Template:Math is a Pythagorean triple, then so is Template:Math, Template:Math and, more generally, Template:Math for any positive integer Template:Math.

There are 16 primitive Pythagorean triples of numbers up to 100:

(3, 4, 5)	(5, 12, 13)	(8, 15, 17)	(7, 24, 25)
(20, 21, 29)	(12, 35, 37)	(9, 40, 41)	(28, 45, 53)
(11, 60, 61)	(16, 63, 65)	(33, 56, 65)	(48, 55, 73)
(13, 84, 85)	(36, 77, 85)	(39, 80, 89)	(65, 72, 97)

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:

(20, 99, 101)	(60, 91, 109)	(15, 112, 113)	(44, 117, 125)
(88, 105, 137)	(17, 144, 145)	(24, 143, 145)	(51, 140, 149)
(85, 132, 157)	(119, 120, 169)	(52, 165, 173)	(19, 180, 181)
(57, 176, 185)	(104, 153, 185)	(95, 168, 193)	(28, 195, 197)
(84, 187, 205)	(133, 156, 205)	(21, 220, 221)	(140, 171, 221)
(60, 221, 229)	(105, 208, 233)	(120, 209, 241)	(32, 255, 257)
(23, 264, 265)	(96, 247, 265)	(69, 260, 269)	(115, 252, 277)
(160, 231, 281)	(161, 240, 289)	(68, 285, 293)

Generating a triple

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File:PrimitivePythagoreanTriplesRev08.svg

The primitive Pythagorean triples. The odd leg Template:Math is plotted on the horizontal axis, the even leg Template:Math on the vertical. The curvilinear grid is composed of curves of constant Template:Math and of constant Template:Math in Euclid's formula.

File:Pythagorean Triples from Grapher.png

A plot of triples generated by Euclid's formula maps out part of the Template:Math cone. A constant Template:Math or Template:Math traces out part of a parabola on the cone.

Euclid's formula<ref>Template:Citation</ref> is a fundamental formula for generating Pythagorean triples given an arbitrary pair of integers Template:Math and Template:Math with Template:Math. The formula states that the integers

<math> a = m^2 - n^2 ,\ \, b = 2mn ,\ \, c = m^2 + n^2 </math>

form a Pythagorean triple. For example, given

<math> m = 2 ,\ \, n = 1 </math>

generate the primitive triple (3,4,5):

<math> a = 2^2 - 1^2 = 3 ,\ \, b = 2 \times 2 \times 1 = 4 ,\ \, c = 2^2 + 1^2 = 5. </math>

The triple generated by Euclid's formula is primitive if and only if Template:Math and Template:Math are coprime and exactly one of them is even. When both Template:Math and Template:Math are odd, then Template:Math, Template:Math, and Template:Math will be even, and the triple will not be primitive; however, dividing Template:Math, Template:Math, and Template:Math by 2 will yield a primitive triple when Template:Math and Template:Math are coprime.<ref>Template:Citation</ref>

Every primitive triple arises (after the exchange of Template:Math and Template:Math, if Template:Math is even) from a unique pair of coprime numbers Template:Math, Template:Math, one of which is even. It follows that there are infinitely many primitive Pythagorean triples. This relationship of Template:Math, Template:Math and Template:Math to Template:Math and Template:Math 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 Template:Math and Template:Math. This can be remedied by inserting an additional parameter Template:Math to the formula. The following will generate all Pythagorean triples uniquely:

<math> a = k\cdot(m^2 - n^2) ,\ \, b = k\cdot(2mn) ,\ \, c = k\cdot(m^2 + n^2)</math>

where Template:Math, Template:Math, and Template:Math are positive integers with Template:Math, and with Template:Math and Template:Math coprime and not both odd.

That these formulas generate Pythagorean triples can be verified by expanding Template:Math using elementary algebra and verifying that the result equals Template:Math. Since every Pythagorean triple can be divided through by some integer Template:Math to obtain a primitive triple, every triple can be generated uniquely by using the formula with Template:Math and Template:Math to generate its primitive counterpart and then multiplying through by Template:Math as in the last equation.

Choosing Template:Math and Template:Math from certain integer sequences gives interesting results. For example, if Template:Math and Template:Math are consecutive Pell numbers, Template:Math and Template:Math will differ by 1.<ref>Template:Cite OEIS</ref>

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 Template:Math and Template:Math, Template:Math, the Template:Math, Template:Math, and Template:Math given by the formula are all positive integers, and from the fact that

<math> a^2+b^2 = (m^2 - n^2)^2 + (2mn)^2 = (m^2 + n^2)^2 = c^2. </math>

A proof of the necessity that a, b, c be expressed by Euclid's formula for any primitive Pythagorean triple is as follows.<ref>Template:Citation</ref> All such primitive triples can be written as Template:Math where Template:Math and Template:Math, Template:Math, Template:Math are coprime. Thus Template:Math, Template:Math, Template:Math are pairwise coprime (if a prime number divided two of them, it would be forced also to divide the third one). As Template:Math and Template:Math are coprime, at least one of them is odd. If we suppose that Template:Math is odd, then Template:Math is even and Template:Math is odd (if both Template:Math and Template:Math were odd, Template:Math would be even, and Template:Math would be a multiple of 4, while Template:Math would be congruent to 2 modulo 4, as an odd square is congruent to 1 modulo 4).

From <math>a^2+b^2=c^2,</math> assume Template:Math is odd. We obtain <math>c^2-a^2=b^2</math> and hence <math>(c-a)(c+a)=b^2.</math> Then <math>\tfrac{(c+a)}{b}=\tfrac{b}{(c-a)}.</math> Since <math>\tfrac{(c+a)}{b}</math> is rational, we set it equal to <math>\tfrac{m}{n}</math> in lowest terms. Thus <math>\tfrac{(c-a)}{b}=\tfrac{n}{m},</math> being the reciprocal of <math>\tfrac{(c+a)}{b}.</math> Then solving

<math>\frac{c}{b}+\frac{a}{b}=\frac{m}{n}, \quad \quad \frac{c}{b}-\frac{a}{b}=\frac{n}{m}</math>

for <math>\tfrac{c}{b}</math> and <math>\tfrac{a}{b}</math> gives

<math>\frac{c}{b}=\frac{1}{2}\left(\frac{m}{n}+\frac{n}{m}\right)=\frac{m^2+n^2}{2mn}, \quad \quad \frac{a}{b}=\frac{1}{2}\left(\frac{m}{n}-\frac{n}{m}\right)=\frac{m^2-n^2}{2mn}.</math>

As <math>\tfrac{m}{n}</math> is fully reduced, Template:Math and Template:Math are coprime, and they cannot both be even. If they were both odd, the numerator of <math>\tfrac{m^2-n^2}{2mn}</math> would be a multiple of 4 (because an odd square is congruent to 1 modulo 4), and the denominator 2mn 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 Template:Math to be even despite defining it as odd. Thus one of Template:Math and Template:Math is odd and the other is even, and the numerators of the two fractions with denominator 2mn are odd. Thus these fractions are fully reduced (an odd prime dividing this denominator divides one of Template:Math and Template:Math but not the other; thus it does not divide Template:Math). One may thus equate numerators with numerators and denominators with denominators, giving Euclid's formula

<math> a = m^2 - n^2 ,\ \, b = 2mn ,\ \, c = m^2 + n^2</math> with Template:Math and Template:Math coprime and of opposite parities.

A longer but more commonplace proof is given in Maor (2007)<ref>Maor, Eli, The Pythagorean Theorem, Princeton University Press, 2007: Appendix B.</ref> and Sierpiński (2003).<ref name=Sierpinski>Template:Citation</ref> Another proof is given in Template:Slink, as an instance of a general method that applies to every homogeneous Diophantine equation of degree two.

Interpretation of parameters in Euclid's formula

Suppose the sides of a Pythagorean triangle have lengths Template:Math, Template:Math, and Template:Math, and suppose the angle between the leg of length Template:Math and the hypotenuse of length Template:Math is denoted as Template:Math. Then <math>\tan{\tfrac{\beta}{2}}=\tfrac{n}{m}</math> and the full-angle trigonometric values are <math>\sin{\beta}=\tfrac{2mn}{m^2+n^2}</math>, <math>\cos{\beta}=\tfrac{m^2-n^2}{m^2+n^2}</math>, and Template:Tmath.<ref>Template:Citation</ref>

A variant

The following variant of Euclid's formula is sometimes more convenient, as being more symmetric in Template:Math and Template:Math (same parity condition on Template:Mvar and Template:Mvar).

If Template:Math and Template:Math are two odd integers such that Template:Math, then

<math> a = mn ,\quad b =\frac {m^2 - n^2}{2} ,\quad c = \frac{m^2 + n^2}{2} </math>

are three integers that form a Pythagorean triple, which is primitive if and only if Template:Math and Template:Math are coprime. Conversely, every primitive Pythagorean triple arises (after the exchange of Template:Math and Template:Math, if Template:Math is even) from a unique pair Template:Math of coprime odd integers.

Not exchanging a and b

In the presentation above, it is said that all Pythagorean triples are uniquely obtained from Euclid's formula "after the exchange of a and b, if a is even". Euclid's formula and the variant above can be merged as follows to avoid this exchange, leading to the following result.

Every primitive Pythagorean triple can be uniquely written

<math> a = 2\varepsilon mn ,\quad b =\varepsilon (m^2 - n^2) ,\quad c = \varepsilon (m^2 + n^2), </math>

where Template:Mvar and Template:Mvar are positive coprime integers, and <math>\varepsilon=\frac12</math> if Template:Mvar and Template:Mvar are both odd, and <math>\varepsilon=1</math> otherwise. Equivalently, <math>\varepsilon=\frac12</math> if Template:Mvar is odd, and <math>\varepsilon=1</math> if Template:Mvar is even.

Elementary properties of primitive Pythagorean triples

General properties

The properties of a primitive Pythagorean triple Template:Math with Template:Math (without specifying which of Template:Math or Template:Math is even and which is odd) include:

• <math>\tfrac{(c-a)(c-b)}{2}</math> is always a perfect square.<ref>Template:Citation.</ref> 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. For example, the triples Template:Math and Template:Math each pass the test that Template:Math is a perfect square, but neither is a Pythagorean triple.
• When a triple of numbers Template:Math, Template:Math and Template:Math forms a primitive Pythagorean triple, then Template:Math and one-half of Template:Math are both perfect squares; however this is not a sufficient condition, as the numbers Template:Math pass the perfect squares test but are not a Pythagorean triple since Template:Math.
• At most one of Template:Math, Template:Math, Template:Math is a square.<ref>For the nonexistence of solutions where Template:Math and Template:Math are both square, originally proved by Fermat, see Template:Citation. For the other case, in which Template:Math is one of the squares, see Template:Citation.</ref>
• The area of a Pythagorean triangle cannot be the square<ref name="Carmichael"/>Template:Rp or twice the square<ref name="Carmichael"/>Template:Rp of an integer.
• Exactly one of Template:Math, Template:Math is divisible by 2 (is even), and the hypotenuse Template:Math is always odd.<ref name=Sierpinski6>Template:Harvnb</ref>
• Exactly one of Template:Math, Template:Math is divisible by 3, but never Template:Math.<ref>Template:Citation</ref><ref name=Sierpinski />Template:Rp
• Exactly one of Template:Math, Template:Math is divisible by 4,<ref name=Sierpinski/> but never Template:Math (because Template:Math is never even).
• Exactly one of Template:Math, Template:Math, Template:Math is divisible by 5.<ref name=Sierpinski/>
• The largest number that always divides abc is 60.<ref name=MacHale/>
• Any odd number of the form Template:Math, where Template:Math is an integer and Template:Math, can be the odd leg of a primitive Pythagorean triple. See almost-isosceles primitive Pythagorean triples section below. However, only even numbers divisible by 4 can be the even leg of a primitive Pythagorean triple. This is because Euclid's formula for the even leg given above is Template:Math and one of Template:Math or Template:Math must be even.
• The hypotenuse Template:Math (which is always odd) is the sum of two squares. This requires all of its prime factors to be [[Pythagorean prime|primes of the form Template:Math]].<ref>Template:Citation.</ref> Therefore, c is of the form Template:Math. A sequence of possible hypotenuse numbers for a primitive Pythagorean triple can be found at (sequence A008846 in the OEIS).
• The area Template:Math is a congruent number<ref>This follows immediately from the fact that ab is divisible by twelve, together with the definition of congruent numbers as the areas of rational-sided right triangles. See e.g. Template:Citation.</ref> divisible by 6.
• In every Pythagorean triangle, the radius of the incircle and the radii of the three excircles are positive integers. Specifically, for a primitive triple the radius of the incircle is Template:Math, and the radii of the excircles opposite the sides Template:Math, 2mn, and the hypotenuse Template:Math are respectively Template:Math, Template:Math, and Template:Math.<ref>Template:Citation</ref>
• As for any right triangle, the converse of Thales' theorem says that the diameter of the circumcircle equals the hypotenuse; hence for primitive triples the circumdiameter is Template:Math, and the circumradius is half of this and thus is rational but non-integer (since Template:Math and Template:Math have opposite parity).
• When the area of a Pythagorean triangle is multiplied by the curvatures of its incircle and 3 excircles, the result is four positive integers Template:Math, respectively. Integers Template:Math satisfy Descartes's Circle Equation.<ref name=Bernhart>Template:Citation</ref> Equivalently, the radius of the outer Soddy circle of any right triangle is equal to its semiperimeter. The outer Soddy center is located at Template:Math, where Template:Math is a rectangle, Template:Math the right triangle and Template:Math its hypotenuse.<ref name=Bernhart/>Template:Rp
• 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.<ref>Template:Cite OEIS</ref> However, only one side can be an integer of perfect power <math>p \ge 2</math> because if two sides were integers of perfect powers with equal exponent <math>p</math> it would contradict the fact that there are no integer solutions to the Diophantine equation <math>x^{2p} \pm y^{2p}=z^2</math>, with <math>x</math>, <math>y</math> and <math>z</math> being pairwise coprime.<ref name=DM>H. Darmon and L. Merel. Winding quotients and some variants of Fermat’s Last Theorem, J. Reine Angew. Math. 490 (1997), 81–100.</ref>
• 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.<ref name="Carmichael">Template:Citation</ref>Template:Rp
• Each primitive Pythagorean triangle has a ratio of area, Template:Math, to squared semiperimeter, Template:Math, that is unique to itself and is given by<ref>Template:Citation</ref>

<math>\frac{K}{s^2} = \frac{n(m-n)}{m(m+n)} = 1-\frac{c}{s}.</math>

• No primitive Pythagorean triangle has an integer altitude from the hypotenuse; that is, every primitive Pythagorean triangle is indecomposable.<ref name=Yiu/>
• The set of all primitive Pythagorean triples forms a rooted ternary tree in a natural way; see Tree of primitive Pythagorean triples.
• Neither of the acute angles of a Pythagorean triangle can be a rational number of degrees.<ref>Template:Mathworld</ref> (This follows from Niven's theorem.)

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 Template:Math) is part of a primitive Pythagorean triple. (If the integer has the form Template:Math, one may take Template:Math and Template:Math in Euclid's formula; if the integer is Template:Math, one may take Template:Math and Template:Math.)
• 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 Template:Math, Template:Math and Template:Math.
• 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 Template:Math. This results from Euclid's formula by remarking that the condition implies that the triple is primitive and must verify Template:Math. This implies Template:Math, and thus Template:Math. The above form of the triples results thus of substituting Template:Math for Template:Math 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 Template:Math in Euclid's formula. More generally, for every integer Template:Math, there exist infinitely many primitive Pythagorean triples in which the hypotenuse and the odd leg differ by Template:Math. They are obtained by putting Template:Math in Euclid's formula.
• There exist infinitely many Pythagorean triples in which the two legs differ by exactly one. For example, 20² + 21² = 29²; these are generated by Euclid's formula when <math>\tfrac{m-n}{n}</math> is a convergent to <math>\sqrt2.</math>
• For each positive integer Template:Math, there exist Template:Math Pythagorean triples with different hypotenuses and the same area.
• For each positive integer Template:Math, there exist at least Template:Math different primitive Pythagorean triples with the same leg Template:Math, where Template:Math is some positive integer (the length of the even leg is 2mn, and it suffices to choose Template:Math with many factorizations, for example Template:Math, where Template:Math is a product of Template:Math different odd primes; this produces at least Template:Math different primitive triples).<ref name=Sierpinski/>Template:Rp
• For each positive integer Template:Math, there exist at least Template:Math different Pythagorean triples with the same hypotenuse.<ref name=Sierpinski/>Template:Rp
• If Template:Math is a prime power, there exists a primitive Pythagorean triple Template:Math if and only if the prime Template:Math has the form Template:Math; this triple is unique up to the exchange of a and b.
• More generally, a positive integer Template:Mvar is the hypotenuse of a primitive Pythagorean triple if and only if each prime factor of Template:Mvar is congruent to Template:Math modulo Template:Math; that is, each prime factor has the form Template:Math. In this case, the number of primitive Pythagorean triples Template:Math with Template:Math is Template:Math, where Template:Mvar is the number of distinct prime factors of Template:Mvar.<ref>Template:Citation</ref>
• There exist infinitely many Pythagorean triples with square numbers for both the hypotenuse Template:Math and the sum of the legs Template:Math. According to Fermat, the smallest such triple<ref>Template:Citation</ref> has sides Template:Math; Template:Math; and Template:Math. Here Template:Math and Template:Math. This is generated by Euclid's formula with parameter values Template:Math and Template:Math.
• There exist non-primitive Pythagorean triangles with integer altitude from the hypotenuse.<ref>Template:Citation</ref><ref name="Richinik">Template:Citation</ref> Such Pythagorean triangles are known as decomposable since they can be split along this altitude into two separate and smaller Pythagorean triangles.<ref name=Yiu>Template:Citation</ref>

Geometry of Euclid's formula

Rational points on a unit circle

File:Pythagorean triple and rational point on unit triangle 1.svg

3,4,5 maps to x,y point (4/5,3/5) on the unit circle

File:Stereographic projection of rational points.svg

The rational points on a circle correspond, under stereographic projection, to the rational points of the line.

Euclid's formula for a Pythagorean triple

<math>a = m^2-n^2,\quad b=2mn,\quad c=m^2+n^2</math>

can be understood in terms of the geometry of rational points on the unit circle Template:Harv.

In fact, a point in the Cartesian plane with coordinates Template:Math belongs to the unit circle if Template:Math. The point is rational if Template:Math and Template:Math are rational numbers, that is, if there are coprime integers Template:Math such that

<math>\biggl(\frac{a}{c}\biggr)^2\! + \biggl(\frac{b}{c}\biggr)^2=1.</math>

By multiplying both members by Template:Math, 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

<math>x=\frac{1-t^2}{1+t^2},\quad y=\frac{2t}{1+t^2}.</math>

Euclid's formula for Pythagorean triples and the inverse relationship Template:Math mean that, except for Template:Math, a point Template:Math on the circle is rational if and only if the corresponding value of Template:Math is a rational number. Note that Template:Math is also the tangent of half of the angle that is opposite the triangle side of length Template:Mvar.

Stereographic approach

File:Stereoprojzero.svg

Stereographic projection of the unit circle onto the Template:Math-axis. Given a point Template:Math on the unit circle, draw a line from Template:Math to the point Template:Math (the north pole). The point Template:Math′ where the line intersects the Template:Math-axis is the stereographic projection of Template:Math. Inversely, starting with a point Template:Math′ on the Template:Math-axis, and drawing a line from Template:Math′ to Template:Math, the inverse stereographic projection is the point Template:Math where the line intersects the unit circle.

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 or equivalently by using the stereographic projection.

For the stereographic approach, suppose that Template:Math′ is a point on the Template:Math-axis with rational coordinates

<math>P' = \left(\frac{m}{n},0\right).</math>

Then, it can be shown by basic algebra that the point Template:Math has coordinates

<math>

P = \left(

\frac{2\left(\frac{m}{n}\right)}{\left(\frac{m}{n}\right)^2+1},
\frac{\left(\frac{m}{n}\right)^2-1}{\left(\frac{m}{n}\right)^2+1}

\right) = \left(

\frac{2mn}{m^2+n^2},
\frac{m^2-n^2}{m^2+n^2}

\right).</math>

This establishes that each rational point of the Template:Math-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 Template:Math-axis, follows by applying the inverse stereographic projection. Suppose that Template:Math is a point of the unit circle with Template:Math and Template:Math rational numbers. Then the point Template:Math′ obtained by stereographic projection onto the Template:Math-axis has coordinates

<math>\left(\frac{x}{1-y},0\right)</math>

which is rational.

In terms of algebraic geometry, the algebraic variety of rational points on the unit circle is birational to the affine line over the rational numbers. The unit circle is thus called a rational curve, 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 Template:Math where Template:Math and Template:Math range over all positive and negative integers. Any Pythagorean triangle with triple Template:Math can be drawn within a 2D lattice with vertices at coordinates Template:Math, Template:Math and Template:Math. The count of lattice points lying strictly within the bounds of the triangle is given by   <math>\tfrac{(a-1)(b-1)-\gcd{(a,b)}+1}{2};</math><ref>Template:Citation</ref> for primitive Pythagorean triples this interior lattice count is  <math>\tfrac{(a-1)(b-1)}{2}.</math> The area (by Pick's theorem equal to one less than the interior lattice count plus half the boundary lattice count) equals  <math>\tfrac{ab}{2}</math> .

The first occurrence of two primitive Pythagorean triples sharing the same area occurs with triangles with sides Template:Math and common area 210 (sequence A093536 in the OEIS). The first occurrence of two primitive Pythagorean triples sharing the same interior lattice count occurs with Template:Math and interior lattice count 2287674594 (sequence A225760 in the OEIS). Three primitive Pythagorean triples have been found sharing the same area: Template:Math, Template:Math, Template:Math 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 <math>m</math> and <math>n</math> with <math>m>n>0</math>, <math>m+n</math> odd and <math>\gcd(m, n)=1.</math> Hence there is a 1 to 1 mapping of rationals (in lowest terms) to primitive Pythagorean triples where <math>\tfrac{n}{m}</math> is in the interval <math>(0,1)</math> and <math>m+n</math> odd.

The reverse mapping from a primitive triple <math>(a , b , c)</math> where <math>c>b>a>0</math> to a rational <math>\tfrac{n}{m}</math> is achieved by studying the two sums <math>a+c</math> and <math>b+c.</math> One of these sums will be a square that can be equated to <math>(m+n)^2</math> and the other will be twice a square that can be equated to <math>2m^2.</math> It is then possible to determine the rational <math>\tfrac{n}{m}.</math>

In order to enumerate primitive Pythagorean triples the rational can be expressed as an ordered pair <math>(n,m)</math> and mapped to an integer using a pairing function such as Cantor's pairing function. An example can be seen at (sequence A277557 in the OEIS). It begins

<math>8,18,19,32,33,34,\dots</math> and gives rationals

<math>\tfrac{1}{2},\tfrac{2}{3},\tfrac{1}{4},\tfrac{3}{4},\tfrac{2}{5},\tfrac{1}{6},\dots</math> these, in turn, generate primitive triples

<math>(3,4,5),(5,12,13),(8,15,17),(7,24,25),(20,21,29),(12,35,37),\dots</math>

Spinors and the modular group

Pythagorean triples can likewise be encoded into a square matrix of the form

<math>X = \begin{bmatrix}

c+b & a\\ a & c-b \end{bmatrix}. </math> A matrix of this form is symmetric. Furthermore, the determinant of Template:Math is

<math>\det X = c^2 - a^2 - b^2\,</math>

which is zero precisely when Template:Math is a Pythagorean triple. If Template:Math corresponds to a Pythagorean triple, then as a matrix it must have rank 1.

Since Template:Math is symmetric, it follows from a result in linear algebra that there is a column vector Template:Math such that the outer product

holds, where the Template:Math denotes the matrix transpose. Since ξ and -ξ produce the same Pythagorean triple, the vector ξ can be considered a spinor (for the Lorentz group 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 (Template:EquationNote).

The modular group Γ is the set of 2×2 matrices with integer entries

<math>A = \begin{bmatrix}\alpha&\beta\\ \gamma&\delta\end{bmatrix}</math>

with determinant equal to one: Template:Math. This set forms a group, 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 on the collection of all integer spinors. Furthermore, the group is transitive on the collection of integer spinors with relatively prime entries. For if Template:Math has relatively prime entries, then

<math>\begin{bmatrix}m&-v\\n&u\end{bmatrix}\begin{bmatrix}1\\0\end{bmatrix} = \begin{bmatrix}m\\n\end{bmatrix}</math>

where Template:Math and Template:Math are selected (by the Euclidean algorithm) so that Template:Math.

By acting on the spinor ξ in (Template:EquationNote), the action of Γ goes over to an action on Pythagorean triples, provided one allows for triples with possibly negative components. Thus if Template:Math is a matrix in Template:Math, then

gives rise to an action on the matrix Template:Math in (Template:EquationNote). 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 Template:Harvnb) to call a triple Template:Math standard if Template:Math and either Template:Math are relatively prime or Template:Math are relatively prime with Template:Math odd. If the spinor Template:Math has relatively prime entries, then the associated triple Template:Math determined by (Template:EquationNote) 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 Template:Math and Template:Math for which Template:Math is odd and Template:Math is even. Let the subgroup Γ(2) of Γ be the kernel of the group homomorphism

<math>\Gamma=\mathrm{SL}(2,\mathbf{Z})\to \mathrm{SL}(2,\mathbf{Z}_2)</math>

where Template:Math is the special linear group over the finite field Template:Math 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 Template:Math for all Template:Math. In fact, under the action (Template:EquationNote), the group Γ(2) acts transitively on the collection of primitive Pythagorean triples Template:Harv.

The group Γ(2) is the free group whose generators are the matrices

<math>U=\begin{bmatrix}1&2\\0&1\end{bmatrix},\qquad L=\begin{bmatrix}1&0\\2&1\end{bmatrix}.</math>

Consequently, every primitive Pythagorean triple can be obtained in a unique way as a product of copies of the matrices Template:Math and Template:Math.

Parent/child relationships

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}}

By a result of Template:Harvtxt, all primitive Pythagorean triples can be generated from the (3, 4, 5) triangle by using the three linear transformations T₁, T₂, T₃ below, where Template:Math, Template:Math, Template:Math are sides of a triple:

	new side Template:Math	new side Template:Math	new side Template:Math
Template:Math:	Template:Math	Template:Math	Template:Math
Template:Math:	Template:Math	Template:Math	Template:Math
Template:Math:	Template:Math	Template:Math	Template:Math

In other words, every primitive triple will be a "parent" to three additional primitive triples. Starting from the initial node with Template:Math, Template:Math, and Template:Math, the operation Template:Math 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 Template:Math and Template:Math 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 forms. They are closely related to (but are not equal to) reflections generating the orthogonal group of Template:Math over the integers.<ref>Template:Harv</ref>

Relation to Gaussian integers

Alternatively, Euclid's formulae can be analyzed and proved using the Gaussian integers.<ref>Template:Citation</ref> Gaussian integers are complex numbers of the form Template:Math, where Template:Math and Template:Math are ordinary integers and Template:Math is the square root of negative one. The units of Gaussian integers are ±1 and ±i. The ordinary integers are called the rational integers and denoted as 'Template:Math'. The Gaussian integers are denoted as Template:Math. The right-hand side of the Pythagorean theorem may be factored in Gaussian integers:

<math>c^2 = a^2+b^2 = (a+bi)\overline{(a+bi)} = (a+bi)(a-bi).</math>

A primitive Pythagorean triple is one in which Template:Math and Template:Math are coprime, i.e., they share no prime factors in the integers. For such a triple, either Template:Math or Template:Math is even, and the other is odd; from this, it follows that Template:Math is also odd.

The two factors Template:Math and Template:Math 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 units.<ref name="Gauss_1832">Template:Citation See also Werke, 2:67–148.</ref> (This unique factorization follows from the fact that, roughly speaking, a version of the Euclidean algorithm can be defined on them.) The proof has three steps. First, if Template:Math and Template:Math share no prime factors in the integers, then they also share no prime factors in the Gaussian integers. (Assume Template:Math and Template:Math with Gaussian integers Template:Math, Template:Math and Template:Math and Template:Math not a unit. Then Template:Math and Template:Math lie on the same line through the origin. All Gaussian integers on such a line are integer multiples of some Gaussian integer Template:Math. But then the integer gh ≠ ±1 divides both Template:Math and Template:Math.) Second, it follows that Template:Math and Template:Math likewise share no prime factors in the Gaussian integers. For if they did, then their common divisor Template:Math would also divide Template:Math and Template:Math. Since Template:Math and Template:Math are coprime, that implies that Template:Math divides Template:Math. From the formula Template:Math, that in turn would imply that Template:Math is even, contrary to the hypothesis of a primitive Pythagorean triple. Third, since Template:Math is a square, every Gaussian prime in its factorization is doubled, i.e., appears an even number of times. Since Template:Math and Template:Math share no prime factors, this doubling is also true for them. Hence, Template:Math and Template:Math are squares.

Thus, the first factor can be written

<math>a+bi = \varepsilon\left(m + ni \right)^2, \quad \varepsilon\in\{\pm 1, \pm i\}.</math>

The real and imaginary parts of this equation give the two formulas:

<math>\begin{cases}\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{cases}</math>

For any primitive Pythagorean triple, there must be integers Template:Math and Template:Math 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.

<math>(m+ni)^2 = (m^2-n^2)+2mni.</math>

Using the facts that the Gaussian integers are a Euclidean domain and that for a Gaussian integer p <math>|p|^2</math> 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 <math>|p|^2</math> and <math>|q|^2</math> integers. Since magnitudes multiply in the Gaussian integers, the product must be <math>|p||q|</math>, which when squared to find a Pythagorean triple must be composite. The contrapositive completes the proof.

Distribution of triples

File:Pythagorean triple scatterplot.svg

A scatter plot of the legs Template:Math of the first Pythagorean triples with Template:Math and Template:Math less than 4500.

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 Template:Math of a primitive triple appear in the plot, all integer multiples of Template:Math 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.

These patterns can be explained as follows. If <math>a^2/4n</math> is an integer, then (Template:Math, <math>|n-a^2/4n|</math>, <math>n+a^2/4n</math>) is a Pythagorean triple. (In fact every Pythagorean triple Template:Math can be written in this way with integer Template:Math, possibly after exchanging Template:Math and Template:Math, since <math>n=(b+c)/2</math> and Template:Math and Template:Math cannot both be odd.) The Pythagorean triples thus lie on curves given by <math>b = |n-a^2/4n|</math>, that is, parabolas reflected at the Template:Math-axis, and the corresponding curves with Template:Math and Template:Math interchanged. If Template:Math is varied for a given Template:Math (i.e. on a given parabola), integer values of Template:Math occur relatively frequently if Template:Math 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, Template:Math, Template:Math, Template:Math, Template:Math and Template:Math; the corresponding parabolic strip around Template:Math 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 Template:Math-axis at Template:Math, and the derivative of Template:Math with respect to Template:Math at this point is –1; hence the incidence angle is 45°. Since the clusters, like all triples, are repeated at integer multiples, the value Template:Math also corresponds to a cluster. The corresponding parabola intersects the Template:Math-axis at right angles at Template:Math, and hence its reflection upon interchange of Template:Math and Template:Math intersects the Template:Math-axis at right angles at Template:Math, precisely where the parabola for Template:Math is reflected at the Template:Math-axis. (The same is of course true for Template:Math and Template:Math interchanged.)

Albert Fässler and others provide insights into the significance of these parabolas in the context of conformal mappings.<ref>1988 Preprint Template:Webarchive See Figure 2 on page 3., later published as Template:Citation</ref><ref>Template:Citation as PDF</ref>

Special cases and related equations

The Platonic sequence

The case Template:Math of the more general construction of Pythagorean triples has been known for a long time. Proclus, 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. (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:

Template:Math is odd (Pythagoras, c. 540 BC):

<math>\text{side }a : \text{side }b = {a^2 - 1 \over 2} : \text{side }c = {a^2 + 1 \over 2}.</math>

Template:Math is even (Plato, c. 380 BC):

<math>\text{side }a : \text{side }b = \left({a \over 2}\right)^2 - 1 : \text{side }c = \left({a \over 2}\right)^2 + 1</math>

It can be shown that all Pythagorean triples can be obtained, with appropriate rescaling, from the basic Platonic sequence (Template:Math, Template:Math and Template:Math) by allowing Template:Math to take non-integer rational values. If Template:Math is replaced with the fraction Template:Math in the sequence, the result is equal to the 'standard' triple generator (2mn, Template:Math,Template:Math) after rescaling. It follows that every triple has a corresponding rational Template:Math 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 Template:Math is generated by Template:Math as Template:Math. The Platonic sequence itself can be derivedTemplate:Clarify by following the steps for 'splitting the square' described in Diophantus II.VIII.

The Jacobi–Madden equation

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} The equation,

<math>a^4+b^4+c^4+d^4 = (a+b+c+d)^4</math>

is equivalent to the special Pythagorean triple,

<math>(a^2+ab+b^2)^2+(c^2+cd+d^2)^2 = ((a+b)^2+(a+b)(c+d)+(c+d)^2)^2</math>

There is an infinite number of solutions to this equation as solving for the variables involves an elliptic curve. Small ones are,

<math>a, b, c, d = -2634, 955, 1770, 5400</math>

<math>a, b, c, d = -31764, 7590, 27385, 48150</math>

Equal sums of two squares

One way to generate solutions to <math>a^2+b^2=c^2+d^2</math> is to parametrize a, b, c, d in terms of integers m, n, p, q as follows:<ref>Template:Citation</ref>

<math>(m^2+n^2)(p^2+q^2)=(mp-nq)^2+(np+mq)^2=(mp+nq)^2+(np-mq)^2.</math>

Equal sums of two fourth powers

Given two sets of Pythagorean triples,

<math>(a^2-b^2)^2+(2a b)^2 = (a^2+b^2)^2</math>

<math>(c^2-d^2)^2+(2c d)^2 = (c^2+d^2)^2</math>

the problem of finding equal products of a non-hypotenuse side and the hypotenuse,

<math>(a^2 -b^2)(a^2+b^2) = (c^2 -d^2)(c^2+d^2)</math>

is easily seen to be equivalent to the equation,

<math>a^4 -b^4 = c^4 -d^4</math>

and was first solved by Euler as <math>a, b, c, d = 133,59,158,134.</math> Since he showed this is a rational point in an elliptic curve, 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,

<math>2(a^4+b^4+c^4+d^4) = (a^2+b^2+c^2+d^2)^2</math>

Euler showed this is equivalent to three simultaneous Pythagorean triples,

<math>(2ab)^2+(2cd)^2 = (a^2+b^2-c^2-d^2)^2</math>

<math>(2ac)^2+(2bd)^2 = (a^2-b^2+c^2-d^2)^2</math>

<math>(2ad)^2+(2bc)^2 = (a^2-b^2-c^2+d^2)^2</math>

There is also an infinite number of solutions, and for the special case when <math>a+b=c</math>, then the equation simplifies to,

<math>4(a^2+a b+b^2) = d^2</math>

with small solutions as <math>a, b, c, d = 3, 5, 8, 14</math> and can be solved as binary quadratic forms.

Almost-isosceles Pythagorean triples

No Pythagorean triples are isosceles, because the ratio of the hypotenuse to either other side is Template:Radic, but [[Square root of 2#Proofs of irrationality|Template:Radic 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,

<math>3^2+4^2 = 5^2</math>

<math>20^2+21^2 = 29^2</math>

and an infinite number of others. They can be completely parameterized as,

<math>\left(\tfrac{x-1}{2}\right)^2+\left(\tfrac{x+1}{2}\right)^2 = y^2</math>

where {x, y} are the solutions to the Pell equation <math>x^2-2y^2 = -1.</math>

If Template:Math, Template:Math, Template:Math are the sides of this type of primitive Pythagorean triple then the solution to the Pell equation is given by the recursive formula

<math>a_n=6a_{n-1}-a_{n-2}+2</math> with <math>a_1=3</math> and <math>a_2=20</math>

<math>b_n=6b_{n-1}-b_{n-2}-2</math> with <math>b_1=4</math> and <math>b_2=21</math>

<math>c_n=6c_{n-1}-c_{n-2}</math> with <math>c_1=5</math> and Template:Tmath.<ref>Template:Cite OEIS; Template:Cite OEIS</ref>

This sequence of primitive Pythagorean triples forms the central stem (trunk) of the rooted ternary tree of primitive Pythagorean triples.

When it is the longer non-hypotenuse side and hypotenuse that differ by one, such as in

<math>5^2+12^2 = 13^2</math>

<math>7^2+24^2 = 25^2</math>

then the complete solution for the primitive Pythagorean triple Template:Math, Template:Math, Template:Math is

<math>a=2m+1, \quad b=2m^2+2m, \quad c=2m^2+2m+1</math>

and

<math>(2m+1)^2+(2m^2+2m)^2=(2m^2+2m+1)^2</math>

where integer <math>m>0</math> is the generating parameter.

It shows that all odd numbers (greater than 1) appear in this type of almost-isosceles primitive Pythagorean triple. This sequence of primitive Pythagorean triples forms the right hand side outer stem of the rooted ternary tree of primitive Pythagorean triples.

Another property of this type of almost-isosceles primitive Pythagorean triple is that the sides are related such that

<math>a^b+b^a=Kc</math>

for some integer <math>K</math>. Or in other words <math>a^b+b^a</math> is divisible by <math>c</math> such as in

<math>(5^{12}+12^5)/13 = 18799189</math>.<ref>Template:Cite OEIS</ref>

Fibonacci numbers in Pythagorean triples

Starting with 5, every second Fibonacci number 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 <math display=block>(F_nF_{n+3})^2 + (2F_{n+1}F_{n+2})^2 = F_{2n+3}^2.</math> 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.<ref>Template:Citation</ref>

Generalizations

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

Pythagorean Template:Math-tuple

The expression

<math>\left(m_1^2 - m_2^2 - \ldots - m_n^2\right)^2 + \sum_{k=2}^n (2 m_1 m_k)^2 = \left(m_1^2 + \ldots + m_n^2\right)^2</math>

is a Pythagorean Template:Mvar-tuple for any tuple of positive integers Template:Math with Template:Math. The Pythagorean Template:Mvar-tuple can be made primitive by dividing out by the largest common divisor of its values.

Furthermore, any primitive Pythagorean Template:Mvar-tuple Template:Math can be found by this approach. Use Template:Math to get a Pythagorean Template:Mvar-tuple by the above formula and divide out by the largest common integer divisor, which is Template:Math. Dividing out by the largest common divisor of these Template:Math values gives the same primitive Pythagorean Template:Mvar-tuple; and there is a one-to-one correspondence between tuples of setwise coprime positive integers Template:Math satisfying Template:Math and primitive Pythagorean Template:Mvar-tuples.

Examples of the relationship between setwise coprime values <math>\vec{m}</math> and primitive Pythagorean Template:Mvar-tuples include:<ref>Template:Cite OEIS</ref>

<math>\begin{align}

                 \vec{m} = (1) & \leftrightarrow 1^2 = 1^2 \\
              \vec{m} = (2, 1) & \leftrightarrow 3^2 + 4^2 = 5^2 \\
           \vec{m} = (2, 1, 1) & \leftrightarrow 1^2 + 2^2 + 2^2 = 3^2 \\
        \vec{m} = (3, 1, 1, 1) & \leftrightarrow 1^2 + 1^2 + 1^2 + 1^2 = 2^2 \\
     \vec{m} = (5, 1, 1, 2, 3) & \leftrightarrow 1^2 + 1^2 + 1^2 + 2^2 + 3^2 = 4^2 \\
  \vec{m} = (4, 1, 1, 1, 1, 2) & \leftrightarrow 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 2^2 = 3^2 \\

\vec{m} = (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{align}</math>

Consecutive squares

Since the sum Template:Math of Template:Math consecutive squares beginning with Template:Math is given by the formula,<ref>Template:Citation</ref>

<math>F(k,m)=km(k-1+m)+\frac{k(k-1)(2k-1)}{6}</math>

one may find values Template:Math so that Template:Math is a square, such as one by Hirschhorn where the number of terms is itself a square,<ref>Template:Citation</ref>

<math>m=\tfrac{v^4-24v^2-25}{48},\; k=v^2,\; F(m,k)=\tfrac{v^5+47v}{48}</math>

and Template:Math is any integer not divisible by 2 or 3. For the smallest case Template:Math, hence Template:Math, this yields the well-known cannonball-stacking problem of Lucas,

<math>0^2+1^2+2^2+\dots+24^2 = 70^2</math>

a fact which is connected to the Leech lattice.

In addition, if in a Pythagorean Template:Math-tuple (Template:Math) all addends are consecutive except one, one can use the equation,<ref>Template:Citation</ref>

<math>F(k,m) + p^{2} = (p+1)^{2}</math>

Since the second power of Template:Math cancels out, this is only linear and easily solved for as <math>p=\tfrac{F(k,m)-1}{2}</math> though Template:Math, Template:Math should be chosen so that Template:Math is an integer, with a small example being Template:Math, Template:Math yielding,

<math>1^2+2^2+3^2+4^2+5^2+27^2=28^2</math>

Thus, one way of generating Pythagorean Template:Math-tuples is by using, for various Template:Math,<ref>Goehl, John F., Jr., "Triples, quartets, pentads", Mathematics Teacher 98, May 2005, p. 580.</ref>

<math>x^2+(x+1)^2+\cdots +(x+q)^2+p^2=(p+1)^2,</math>

where q = n–2 and where

<math>p=\frac{(q+1)x^2+q(q+1)x+\frac{q(q+1)(2q+1)}{6} -1}{2}.</math>

Fermat's Last Theorem

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}}

A generalization of the concept of Pythagorean triples is the search for triples of positive integers Template:Math, Template:Math, and Template:Math, such that Template:Math, for some Template:Math strictly greater than 2. Pierre de Fermat in 1637 claimed that no such triple exists, a claim that came to be known as Fermat's Last Theorem 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.

Template:Math or Template:Math Template:Mathth powers summing to an Template:Mathth power

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Another generalization is searching for sequences of Template:Math positive integers for which the Template:Mathth power of the last is the sum of the Template:Mathth powers of the previous terms. The smallest sequences for known values of Template:Math are:

• Template:Math = 3: {3, 4, 5; 6}.
• Template:Math = 4: {30, 120, 272, 315; 353}
• Template:Math = 5: {19, 43, 46, 47, 67; 72}
• Template:Math = 7: {127, 258, 266, 413, 430, 439, 525; 568}
• Template:Math = 8: {90, 223, 478, 524, 748, 1088, 1190, 1324; 1409}

For the Template:Math case, in which <math>x^3+y^3+z^3=w^3,</math> called the Fermat cubic, a general formula exists giving all solutions.

A slightly different generalization allows the sum of Template:Math Template:Mathth powers to equal the sum of Template:Math Template:Mathth powers. For example:

• (Template:Math): 1³ + 12³ = 9³ + 10³, 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 Template:Math positive integers whose Template:Mathth powers sum to an Template:Mathth power (though, by Fermat's Last Theorem, not for Template:Math; these are counterexamples to Euler's sum of powers conjecture. The smallest known counterexamples are<ref>Template:Citation</ref><ref>Template:Citation</ref><ref name=MacHale>Template:Citation</ref>

• Template:Math: (95800, 217519, 414560; 422481)
• Template:Math: (27, 84, 110, 133; 144)

Heronian triangle triples

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}}

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 Template:Math for Template:Math. Every Pythagorean triple is a Heronian triple, because at least one of the legs Template:Math, Template:Math 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 Template:Math with area 24 shows.

If Template:Math is a Heronian triple, so is Template:Math where Template:Math is any positive integer; its area will be the integer that is Template:Math times the integer area of the Template:Math triangle. The Heronian triple Template:Math 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 Template:Math.) 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, the extra condition for a triple of positive integers Template:Math with Template:Math to be Heronian is that

Template:Math

or equivalently

Template:Math

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.<ref>Kak, S. and Prabhu, M. Cryptographic applications of primitive Pythagorean triples. Cryptologia, 38:215–222, 2014. [1]</ref>

See also

• Boolean Pythagorean triples problem
• Brahmagupta triangle
• Congruum
• Diophantus II.VIII
• Eisenstein triple
• Euler brick
• Heronian triangle
• Hilbert's theorem 90
• Integer triangle
• Modular arithmetic
• Nonhypotenuse number
• Plimpton 322
• Pythagorean prime
• Pythagorean quadruple
• Quadric
• Tangent half-angle formula
• Trigonometric identity

References

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External links

• Clifford Algebras and Euclid's Parameterization of Pythagorean triples
• Curious Consequences of a Miscopied Quadratic
• Discussion of Properties of Pythagorean triples, Interactive Calculators, Puzzles and Problems
• Generating Pythagorean Triples Using Arithmetic Progressions
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• Interactive Calculator for Pythagorean Triples
• The negative Pell equation and Pythagorean triples
• Parameterization of Pythagorean Triples by a single triple of polynomials
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• Pythagorean Triples and the Unit Circle, chap. 2–3, in "A Friendly Introduction to Number Theory" by Joseph H. Silverman, 3rd ed., 2006, Pearson Prentice Hall, Upper Saddle River, NJ, Template:Isbn
• 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
• Theoretical properties of the Pythagorean Triples and connections to geometry
• The Trinary Tree(s) underlying Primitive Pythagorean Triples at cut-the-knot
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