Pythagorean triple

The Pythagorean theorem: a2 + b2 = c2

A Pythagorean triple consists of three positive integers a, b, and c, such that a2 + b2 = c2. Such a triple is commonly written (a, b, c), and a well-known example is (3, 4, 5). If (a, b, c) is a Pythagorean triple, then so is (ka, kb, kc) for any positive integer k. A primitive Pythagorean triple is one in which a, b and c are pairwise coprime. A right triangle whose sides form a Pythagorean triple is called a Pythagorean triangle.

The name is derived from the Pythagorean theorem, stating that every right triangle has side lengths satisfying the formula a2 + b2 = c2; 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 a = b = 1 and c = √2 is right, but (1, 1, √2) is not a Pythagorean triple because √2 is not an integer. Moreover, 1 and √2 do not have an integer common multiple because √2 is irrational. There are 16 primitive Pythagorean triples with c ≤ 100:

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

Each one of this low-c points forms one of the more easily-recognizable radiating lines in the scatter plot.

A scatter plot of the legs (a,b) of the Pythagorean triples with c less than 6000. Negative values are included to illustrate the parabolic patterns in the plot more clearly.

Contents

Generating a triple

A plot of triples generated by Euclid's formula map out part of the z2 = x2 + y2 cone. A constant m or n trace out part of a parabola on the cone.

Euclid's formula[1] is a fundamental formula for generating Pythagorean triples given an arbitrary pair of positive integers m and n with m > n. 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's formula is primitive if and only if m and n are coprime and exactly one of them is even. If both n and m are odd, then a, b, and c will be even, and so the triple will not be primitive; however dividing a, b, and c by 2 will yield a primitive triple if m and n are coprime and both odd.[2] Every primitive triple (possibly after exchanging a and b) arises from a unique pair of coprime numbers m, n, one of which is even. It follows that there are infinitely many primitive Pythagorean triples. This relationship of a and b to m and n 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. This can be remedied by inserting an additional parameter k to the formula. The following will generate all Pythagorean triples (although not uniquely):

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

where m, n, and k are positive integers with m > n.

That these formulae generate Pythagorean triples can be verified by expanding a2 + b2 using elementary algebra and verifying that the result coincides with c2 (although this does not prove that it generates all such triples). Many other formulae for generating triples have been developed since the time of Euclid.

Elementary properties of primitive Pythagorean triples

The properties of a Primitive Pythagorean Triple (PPT) include:

a + b = c + 2\sqrt{\frac{(c - a)(c - b)}{2}}

Some relationships

Right Triangle with inscribed circle of radius r

The radius, r, of the inscribed circle can be found by:

r = \frac{ab}{a+b+c} = \frac{a+b-c}{2} = n(m-n).\

The unknown sides of a triple can be calculated directly from the radius of the incircle, r, and the value of a single known side, b.

k = b − 2r
a = 2r + (2 r2/k)
c = a+ k = 2r + (2r2 /k) + k

The solution to the 'Incircle' problem shows that, for any circle whose radius is a whole number r, setting k = 1, we are guaranteed at least one right angled triangle containing this circle as its inscribed circle where the lengths of the sides of the triangle are a primitive Pythagorean triple:

a=2r + 2r2
b=2r + 1
c=2r + 2r2 + 1

The perimeter P and area L of the right triangle corresponding to a primitive Pythagorean triple triangle are

P = a + b + c = 2m(m + n)
L = ab/2 = mn(m2 − n2)

Additional relationships:[4]

\frac{c+a}{b}=\frac{m}{n}\,
\frac{c+b+a}{c+b-a}= \frac{m}{n}\,
b/(c-a)= \frac{m}{n} \,
(a+c-b)/(a+b-c)= \frac{m}{n} \,
\cos\theta\ = {m^2-n^2 \over m^2+n^2} = {1-t^2 \over 1+t^2}= {a \over c}
\sin\theta\ = {2mn \over m^2+n^2} = {2t \over 1+t^2} = {b \over c}
\tan\theta\ = {2mn \over m^2-n^2} = {2t \over 1-t^2} = {b \over a}
\tan\left({\theta \over 2}\right) = {n \over m}
\tan\left({\beta \over 2}\right) = {m-n \over m+n}

If two numbers of a triple are known, the third can be found using the Pythagorean theorem.

A special case: the Platonic sequence

The case n = 1 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:

a is odd (Pythagoras, c. 540 BC):

\mbox{side }a�: \mbox{side }b = {a^2 - 1 \over 2}�: \mbox{side }c = {a^2 + 1 \over 2}.

a is even (Plato, c. 380 BC):

\mbox{side }a�: \mbox{side }b = \left({a \over 2}\right)^2 - 1�: \mbox{side }c = \left({a \over 2}\right)^2 + 1

It can be shown that all Pythagorean triples are derivatives of the basic Platonic sequence (x,y,z) = p, (p2 - 1)/2 and (p2 + 1)/2 by allowing p to take non-integer rational values. If p is replaced with the rational fraction m/n in the sequence, the 'standard' triple generator 2mn, m2 - n2 and m2 + n2 results. It follows that every triple has a corresponding rational p value which can be used to generate a similar (i.e. equiangular) triangle with rational sides in the same proportion as the original. For example, the Platonic equivalent of (6, 8,10) is (3/2; 2, 5/2). The Platonic sequence itself can be derived by following the steps for 'splitting the square' described in Diophantus II.VIII.

Geometry of Euclid's formula

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

Euclid's formulae for a Pythagorean triple

a = 2mn,\quad b=m^2-n^2,\quad c=m^2+n^2

can be understood in terms of the geometry of rational number points on the unit circle (Trautman 1998). To motivate this, consider a right triangle with legs a and b, and hypotenuse c, where a, b, and c are positive integers. By the Pythagorean theorem, a2 + b2 = c2 or, dividing both sides by c2,

\left(\frac{a}{c}\right)^2 + \left(\frac{b}{c}\right)^2=1.

Geometrically, the point in the Cartesian plane with coordinates

x=\frac{a}{c},\quad y=\frac{b}{c}

is on the unit circle x2 + y2 = 1. In this equation, the coordinates x and y are given by rational numbers. Conversely, any point on the unit circle whose coordinates x, y are rational numbers gives rise to a primitive Pythagorean triple. Indeed, write x and y as fractions in lowest terms:

x=\frac{a}{c},\quad y=\frac{b}{c}

where the greatest common divisor of a, b, and c is 1. Then, since x and y are on the unit circle,

\left(\frac{a}{c}\right)^2 + \left(\frac{b}{c}\right)^2=1\implies a^2+b^2=c^2,

as claimed.

Stereographic projection of the unit circle onto the x-axis. Given a point P on the unit circle, draw a line from P to the point N = (0, 1) (the north pole). The point P′ where the line intersects the x-axis is the stereographic projection of P. Inversely, starting with a point P′ on the x-axis, and drawing a line from P′ to N, the inverse stereographic projection is the point P where the line intersects the unit circle.

There is therefore 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 this, suppose that P′ is a point on the x-axis with rational coordinates P′(m/n,0). Then, it can be shown by basic algebra that the point P has coordinates

\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).

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

\left(\frac{x}{1-y},0\right)

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.

Spinors and the modular group

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

X = \begin{bmatrix} c+b & a\\ a & c-b \end{bmatrix}.

A matrix of this form is symmetric. Furthermore, the determinant of X is

\det X = c^2 - a^2 - b^2\,

which is zero precisely when (a,b,c) is a Pythagorean triple. If X corresponds to a Pythagorean triple, then as a matrix it must have rank 1. Since X is symmetric, it follows from a result in linear algebra that there is a vector ξ = [m n]T such that the outer product

X = 2\begin{bmatrix}m\\n\end{bmatrix}[m\ n] = 2\xi\xi^T\,

 

 

 

 

(1)

holds, where the T denotes the matrix transpose. The vector ξ is called 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 (1).

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

A = \begin{bmatrix}\alpha&\beta\\ \gamma&\delta\end{bmatrix}

with determinant equal to one: αδ − βγ = 1. 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 [m n]T has relatively prime entries, then

\begin{bmatrix}m&-v\\n&u\end{bmatrix}\begin{bmatrix}1\\0\end{bmatrix} = \begin{bmatrix}m\\n\end{bmatrix}

where u and v are selected (by the Euclidean algorithm) so that mu + nv = 1.

By acting on the spinor ξ in (1), the action of Γ goes over to an action on Pythagorean triples, provided one allows for triples with possibly negative components. Thus if A is a matrix in Γ, then

2(A\xi)(A\xi)^T = A X A^T\,

 

 

 

 

(2)

gives rise to an action on the matrix X in (1). 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 Trautman 1998) to call a triple (a,b,c) standard if c > 0 and either (a,b,c) are relatively prime or (a/2,b/2,c/2) are relatively prime with a/2 odd. If the spinor [m n]T has relatively prime entries, then the associated triple (a,b,c) determined by (1) 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 m and n for which m is odd and n is even. Let the subgroup Γ(2) of Γ be the kernel of the group homomorphism

\Gamma=SL(2,\mathbf{Z})\to SL(2,\mathbf{Z}_2)

where SL(2,Z2) is the special linear group over the finite field Z2 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 Aξ for all A ∈ Γ(2). In fact, under the action (2), the group Γ(2) acts transitively on the collection of primitive Pythagorean triples (Alperin 2005).

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

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

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

Parent/child relationships

By a result of Berggren (1934), all primitive Pythagorean triples can be generated from the (3, 4, 5) triangle by using the 3 linear transformations T1, T2, T3 below, where a, b, c are sides of a triple:

new side a new side b new side c
T1: a − 2b + 2c 2a − b + 2c 2a − 2b + 3c
T2: a + 2b + 2c 2a + b + 2c 2a + 2b + 3c
T3: −a + 2b + 2c −2a + b + 2c −2a + 2b + 3c

If one begins with 3, 4, 5 then all other primitive triples will eventually be produced. In other words, every primitive triple will be a “parent” to 3 additional primitive triples. Starting from the initial node with a = 3, b = 4, and c = 5, the next generation of triples is

new side a new side b new side c
3 − (2×4) + (2×5) = 5 (2×3) − 4 + (2×5) = 12 (2×3) − (2×4) + (3×5) = 13
3 + (2×4) + (2×5) = 21 (2×3) + 4 + (2×5) = 20 (2×3) + (2×4) + (3×5) = 29
−3 + (2×4) + (2×5) = 15 −(2×3) + 4 + (2×5) = 8 −(2×3) + (2×4) + (3×5) = 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 x2 + y2 − z2 over the integers.

For further discussion of parent-child relationships in triples, see: http://mathworld.wolfram.com/PythagoreanTriple.html and (Alperin 2005).

Relation to Gaussian integers

Alternatively, Euclid's formulae can be analyzed and proven using the Gaussian integers.[5] Gaussian integers are complex numbers of the form α = u + vi, where u and v are ordinary integers and i is the square root of negative one. The right-hand side of the Pythagorean theorem may be factored in Gaussian integers:

c2 = a2 + b2 = (a + bi) (a − bi).

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

The two factors z := a + bi and z* := a − bi 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.[6] (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 a and b share no prime factors in the integers, then they also share no prime factors in the Gaussian integers. (Assume a = gu and b = gv with Gaussian integers g, u and v and g not a unit. Then u and v lie on the same line through the origin. All Gaussian integers on such a line are integer multiples of some Gaussian integer h. But then the integer gh ≠ ±1 divides both a and b.) Second, it follows that z and z* likewise share no prime factors in the Gaussian integers. For if they did, then their common divisor δ would also divide z + z* = 2a and z - z* = 2ib. Since a and b are coprime, that implies that δ divides 2. From the formula c2 = zz*, that in turn would imply that c is even, contrary to the hypothesis of a primitive Pythagorean triple. Third, since c2 is a square, every Gaussian prime in its factorization is doubled, i.e., appears an even number of times. Since z and z* share no prime factors, this doubling is also true for them. Hence, z and z* are squares.

Thus, for example, the first factor can be written

a + bi = (m + ni)2.

The real and imaginary parts of this equation give the two formulae

a = m2 − n2,
b = 2 mn.

For any primitive Pythagorean triple, there must be integers m and n such that these two equations are satisfied. Hence, every Pythagorean triple can be generated from some choice of these integers.

Distribution of triples

A scatter plot of the legs (a,b) of the first Pythagorean triples with a and b 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 (a,b) of a primitive triple appear in the plot, all integer multiples of (a,b) 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 around 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 a^2/4n is an integer, then (a, |n-a^2/4n|, n+a^2/4n) is a Pythagorean triple. (In fact every Pythagorean triple (a, b, c) can be written in this way with integer n, possibly after exchanging a and b, since n=(b+c)/2 and a and b cannot both be odd.) The Pythagorean triples thus lie on curves given by b = |n-a^2/4n|, that is, parabolas reflected at the a-axis, and the corresponding curves with a and b interchanged. If a is varied for a given n (i.e. on a given parabola), integer values of b occur relatively frequently if n 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, 382 = 1444, 2 * 272 = 1458, 3 * 222 = 1452, 5 * 172 = 1445 and 10 * 122 = 1440; the corresponding parabolic strip around n ≈ 1450 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 a-axis at a = 2n, and the derivative of b with respect to a at this point is –1; hence the incidence angle is 45°. Since the clusters, like all triples, are repeated at integer multiples, the value 2n also corresponds to a cluster. The corresponding parabola intersects the b-axis at right angles at b = 2n, and hence it's reflection upon interchange of a and b intersects the a-axis at right angles at a = 2n, precisely where the parabola for n is reflected at the a-axis. (The same is of course true for a and b interchanged.)

Albert Fässler and others provide insights into the significance of these parabolas in the context of conformal mappings.[7][8]

Generalizations

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

Pythagorean quadruple

A set of four positive integers a, b, c and d such that a2 + b2+ c2 = d2 is called a Pythagorean quadruple. The simplest example is (1, 2, 2, 3), since 12 + 22 + 22 = 32. The next simplest (primitive) example is (2, 3, 6, 7), since 22 + 32 + 62 = 72.

Quadruples are given by the formula

(a^2-b^2-c^2)^2 + (2ab)^2 + (2ac)^2 = (a^2+b^2+c^2)^2.

Pythagorean n-tuple

Using the simple algebraic identity,

(x_1^2-x_0)^2 + (2x_1)^{2}x_0 = (x_1^2+x_0)^2

for arbitrary x0, x1, it is easy to prove that the square of the sum of n squares is itself the sum of n squares by letting x0 = x22+x32+...+xn2 and then distributing terms.[9] One can see how Pythagorean triples and quadruples are just the particular cases x0 = x22 and x0 = x22+x32, respectively, and so on for other n, with quintuples given by,

(a^2-b^2-c^2-d^2)^2 + (2ab)^2 + (2ac)^2 + (2ad)^2 = (a^2+b^2+c^2+d^2)^2.

In addition, Pythagorean n-tuples with the first n–2 members being successive positive integers can be generated by[10]

x^{2} + (x+1)^{2} + \dots + (x+q)^{2} + p^{2} = (p+1)^{2}

where q=n–3 ≥ 0 and where

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

Fermat's Last Theorem

Main article: Fermat's Last Theorem

A generalization of the concept of Pythagorean triples is the search for triples of positive integers a, b, and c, such that an + bn = cn, for some n 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 proven or disproven. The first proof was given by Andrew Wiles in 1994.

n–1 or n nth powers summing to an nth power

Another generalization is searching for sets of n+1 positive integers for which the nth power of the last is the sum of the nth powers of the previous terms. The smallest sets for known values of n are:

A slightly different generalization allows the sum of (k+1) nth powers to equal the sum of (n-k) nth powers. For example:

There can also exist n–1 positive integers whose nth powers sum to an nth power (though, by Fermat's last theorem, not for n=3); these are counterexamples to Euler's sum of powers conjecture. The smallest known counterexamples are

Heronian triangle triples

A Heronian triangle is commonly defined as one with integer sides whose area is also an integer, and we shall consider Heronian triangles with distinct integer sides. The lengths of the sides of such a triangle form a Heronian triple (a, b, c) provided a < b < c. Clearly, any Pythagorean triple is a Heronian triple, since in a Pythagorean triple at least one of the legs a, b must be even, so that the area ab/2 is an integer. Not every Heronian triple is a Pythagorean triple, however, as the example (4, 13, 15) with area 24 shows.

If (a, b, c) is a Heronian triple, so is (ma, mb, mc) where m is any positive integer greater than one. The Heronian triple (a, b, c) is primitive provided a, b, c are pairwise relatively prime (as with a Pythagorean triple). Here are a few of the simplest primitive Heronian triples which 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 (a, b, c) with a < b < c to be Heronian is that

(a2 + b2 + c2)2 - 2(a4 + b4 + c4)

or equivalently

2(a2b2 + a2c2 + b2c2) - (a4 + b4 + c4)

be a nonzero perfect square divisible by 16.

See also

References

  1. ↑ "Euclid's Elements: Book X , Proposition XXIX". http://babbage.clarku.edu/~djoyce/java/elements/bookX/propX29.html. 
  2. ↑ Mitchell, Douglas W., "An alternative characterisation of all primitive Pythagorean triples," Mathematical Gazette July 2001,.
  3. ↑ 3.0 3.1 3.2 Carmichael, R. D., 1914, "Diophantine analysis," in second half of R. D. Carmichael, The Theory of Numbers and Diophantine Analysis, Dover Publ., 1959.
  4. ↑ "Multiplying Pythagorean Triples". http://www.pythag.net/node9.html. 
  5. ↑ *Stillwell J (2003), Elements of Number Theory, New York: Springer-Verlag, pp. 110–112, ISBN 0-387-95587-9  .
  6. ↑ Gauss CF (1832), "Theoria residuorum biquadraticorum", Comm. Soc. Reg. Sci. Gött. Rec. 4.  See also Werke, 2:67–148.
  7. ↑ 1988 Preprint See Figure 2 on page 3., later published as Multiple Pythagorean number triples Albert Fässler, American Mathematical Monthly, v.98 n.6, pages 505-517, June/July 1991 ( 10.2307/2324870)
  8. ↑ Pythagorean triangles with legs less than n by Manuel Benito and Juan L. Varona, Journal of Computational and Applied Mathematics 143 (2002) 117–126
  9. ↑ "A Collection of Algebraic Identities: Sums of n Squares". http://sites.google.com/site/tpiezas/005b. 
  10. ↑ Goehl, John F. Jr., "Reader reflections," Mathematics Teacher 98(9), May 2005, 580.

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