Pythagorean triple

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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 coprime.

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:

A scatter plot of the legs (a,b) of the first Pythagorean triples with a and b less than 4500.

A scatter plot Same as above with a and b less than 6000.

[edit] 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 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

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. 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):

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

That these formulas 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 formulas for generating triples have been developed since the time of Euclid.

[edit] Elementary properties of primitive Pythagorean triples

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

• In a pythagorean triple a + b = c + 2[(c − a)(c − b)/2]^1/2
• (c − a)(c − b)/2 is always a perfect square. This is particularly useful in checking if a given triple of numbers is a pythagorean triple, but it is only a necessary condition, not a sufficient one. The triple {6, 12, 18} passes (c − a)(c − b)/2 but is not a PPT. A simpler, more powerful test is, (by naming the even leg a): (c − a) and (c − b)/2 are both perfect squares. This is both necessary and sufficient for the triple to be a PT^{[citation needed]}, but the PT may be derivative. If any two sides of a PT are relatively prime, it is a PPT.
• Exactly one of a, b is odd; c is odd.
• The area (A = ab/2) is an integer.
• Exactly one of a, b is divisible by 3.
• Exactly one of a, b is divisible by 4.
• Exactly one of a, b, c is divisible by 5.
• Exactly one of a, b, (a + b), (b − a) is divisible by 7.
• All prime factors of c are primes of the form 4n + 1.
• At most one of a, b is a square.
• Every integer greater than 2 that is not congruent to 2 mod 4 is part of a primitive Pythagorean triple. In other words, integers greater than 2 which are not of the form 4n + 2.
• 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 6, 8, 10; 14, 48, 50 and 18, 80, 82.
• There exist infinitely many primitive Pythagorean triples whose hypotenuses are squares of natural numbers.
• There exist infinitely many primitive Pythagorean triples in which one of the legs is the square of a natural number.
• There exist infinitely many Pythagorean triples in which the hypotenuse and the longer of the two legs differ by exactly one (such triples are necessarily primitive). Generalization: For every odd integer j, there exist infinitely many primitive Pythagorean triples in which the hypoteneuse and the even leg differ by j ².
• There exist infinitely many primitive Pythagorean triples in which the hypotenuse and the longer of the two legs differ by exactly two (such triples are necessarily primitive). Generalization: For every integer k > 0, there exist infinitely many primitive Pythagorean triples in which the hypotenuse and the odd leg differ by 2k².
• If j and k are odd positive integers, not necessarily unequal, there is exactly one primitive Pythagorean triple with a + j² = c = b + 2^k.
• The hypoteneuse of every primitive Pythagorean triangle exceeds the even leg by the square of an odd integer j, and exceeds the odd leg by twice the square of an integer k > 0, from which it follows that:
• There are no primitive Pythagorean triples in which the hypotenuse and a leg differ by a prime number greater than 2.
• The sum of the hypoteneuse and the even leg of a primitive Pythagorean triple is the square of an odd number, and the arithmetic mean of the hypoteneuse and the odd leg is a perfect square.
• For each natural number n, there exist n Pythagorean triples with different hypotenuses and the same area.
• For each natural number n, there exist at least n different Pythagorean triples with the same leg a, where a is some natural number
• For each natural number n, there exist at least n different triangles with the same hypotenuse.
• In every Pythagorean triple, the radius of the incircle and the radii of the three excircles are natural numbers. (Actually the radius of the incircle can be shown to be r = n(m − n) )
• There are no Pythagorean triplets in which the hypotenuse and one leg are the legs of another Pythagorean triple.

[edit] Some relationships

Right Triangle with inscribed circle of radius r

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

for primitive triples:

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:

see: http://www.geocities.com/fredlb37/node8.html

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

[edit] 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):

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

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.

[edit] Geometry of Euclid's formula

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

Euclid's formulas for a Pythagorean triple

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,

Geometrically, the point in the Cartesian plane with coordinates

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:

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

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 formulas 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

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

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.

[edit] Spinors and the modular group

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

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

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 follow from a result in linear algebra that there is a vector ξ = [m n]^T such that the outer product

(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

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

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)

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

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

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

[edit] Parent/child relationships

By a result of Barning (1963), 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:

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. example: Let a = 3, b = 4, c = 5.

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).

[edit] Relation with Gaussian integers

Alternatively, Euclid's formulas can be analyzed and proven using the Gaussian integers.^[1] 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. It can be proven that the two factors, a + bi and a − bi, of a primitive Pythagorean triple each equal the square of a Gaussian integer. For example, the first factor can be written

a + bi = (m + ni)²

The real and imaginary parts of this equation gives 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.

The proof that z = a + bi and z* = a − bi must be each squares of Gaussian integers follows from the property that every Gaussian integer can be factored uniquely into Gaussian primes.^[2] (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. 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.

[edit] 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 appears in the plot, the 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. The points cluster along the parabolic arc in noisy manner as they do in Euler's totient function. 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 between its tip and its intersection with its semi-latus rectum. As mentioned earlier, the pattern repeats as you move away from or towards the origin. Albert Fässler, and others, provide insights into the significance of these real arcs and additional scatter plots that assist in recognizing those pattern in the diagram in this article.^[3][4]

[edit] Generalizations

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

[edit] 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

(a² + b² + c²)² = (a² − b² − c²)² + (2ab)² + (2ac)².

[edit] 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.

[edit] 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:

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

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

• (n=3): 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.

[edit] Integral triangle triples

An integral triangle is one with distinct positive integer sides whose area is also an integer. The lengths of the sides of such a triangle form an integral triangle triple (or ITT) (a, b, c) provided a < b < c. Clearly, any PT is an ITT, since in a PT at least one of the legs a, b must be even, so that the area ab/2 is an integer. Not every ITT is a PT, however, as the example (4, 13, 15) with area 24 shows.

If (a, b, c) is an ITT, so is (ma, mb, mc) where m is any positive integer greater than one. The ITT (a, b, c) is primitive provided a, b, c are relatively prime (as with a PT). Here are a few of the simplest primitive ITTs which are not PTs:

(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 on a triple of positive integers (a, b, c) with a < b < c 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.

[edit] See also

• Heronian triangle
• Pythagorean prime
• Nonhypotenuse number
• Modular arithmetic
• Trigonometric identity
• Tangent half-angle formula
• Plimpton 322
• Diophantus II.VIII
• Hilbert's theorem 90

[edit] Notes

[edit] References

• Alperin, Roger C. (2005), "The modular tree of Pythagoras", American Mathematical Monthly 112: 807–816, MR2179860, http://www.math.sjsu.edu/~alperin/pt.pdf 
• Barning, F.J.M. (1963), "On Pythagorean and quasi-Pythagorean triangles and a generation process with the help of unimodular matrices", Math. Centrum Amsterdam Afd. Zuivere Wisk. ZW-011: 37, http://repository.cwi.nl:8888/cwi_repository/zoekinoaienora/fullrecord.php?publnr=7151  (Dutch)
• Eckert, Ernest (1992), "Primitive Pythagorean triples", The College Mathematics Journal 23 (5): 413–417, doi:10.2307/2686417, http://www.jstor.org/stable/2686417 
• Elkies, Noam, Pythagorean triples and Hilbert's theorem 90, http://www.math.harvard.edu/~elkies/Misc/hilbert.pdf .
• Heath, Thomas (1956), The Thirteen Books of Euclid's Elements Vol. 1 (Books I and II) (2nd ed.), Dover Publications, ISBN 0-486-60088-2 .
• Teigen, M. G.; Hadwin, D. W. (1971), "On Generating Pythagorean Triples", The American Mathematical Monthly 78 (4): 378–379, doi:10.2307/2316903, http://www.jstor.org/stable/2316903 
• Martin, Artemas (1875), "Rational right angled triangles nearly isosceles", The Analyst 3 (2): 47–50, doi:10.2307/2635906, http://www.jstor.org/pss/2635906 
• McCullough, Darryl (2005), "Height and excess of Pythagorean triples", Mathematics Magazine 78 (1), http://www.math.ou.edu/~dmccullough/teaching/pythagoras2.pdf 
• Romik, Dan (2004), The dynamics of Pythagorean triples, arΧiv:math.DS/0406512 
• Sierpinski, Wacław (2003), Pythagorean Triangles, Dover Publications, ISBN 0-486-43278-5 .
• Trautman, Andrzej (1998), "Pythagorean spinors and Penrose twistors", in S.A. Hugget, L.J. Mason, K.P. Tod, S.T. Tsou, N.M.J. Woodhouse, Geometric universe, http://www.fuw.edu.pl/~amt/oxford.ps .

[edit] External links

• http://mathworld.wolfram.com/PythagoreanTriple.html has an extensive discussion of Pythagorean triples.
• Pythagorean Triples at cut-the-knot Interactive Applet showing unit circle relationships to Pythagorean Triples
• The Trinary Tree(s) underlying Primitive Pythagorean Triples at cut-the-knot
• http://www.math.rutgers.edu/~erowland/pythagoreantriples.html Theoretical properties of the Pythagorean Triples and connections to geometry
• Clifford Algebras and Euclid's Parameterization of Pythagorean triples
• Pythagorean Triplets
• http://www.mcs.surrey.ac.uk/Personal/R.Knott/Pythag/pythag.html Discussion of Properties of Pythagorean triples, Interactive Calculators, Puzzles and Problems
• http://people.wcsu.edu/sandifere/Academics/2007Spring/Mat342/PythagTrip02.pdf Generating Pythagorean Triples Using Arithmetic Progressions
• Parameterization of Pythagorean Triples by a single triple of polynomials.
• Curious Consequences of a Miscopied Quadratic
• Solutions to Quadratic Compatible Pairs in relation to Pythagorean Triples
• The negative Pell equation and Pythagorean triples
• The Remarkable Incircle of a Triangle
• Interactive Calculator for Pythagorean Triples
• The Pythagorean Tree: A New Species[1] Compares the Barning Tree to an entirely new one.