Pythagorean quadruple

Short description: Four integers where the sum of the squares of three equals the square of the fourth

All four primitive Pythagorean quadruples with only single-digit values

A Pythagorean quadruple is a tuple of integers a, b, c, and d, such that a2 \+ b2 \+ c2 = d2. They are solutions of a Diophantine equation and often only positive integer values are considered.[1] However, to provide a more complete geometric interpretation, the integer values can be allowed to be negative and zero (thus allowing Pythagorean triples to be included) with the only condition being that d > 0. In this setting, a Pythagorean quadruple (a, b, c, d) defines a cuboid with integer side lengths |a|, |b|, and |c|, whose space diagonal has integer length d; with this interpretation, Pythagorean quadruples are thus also called Pythagorean boxes.[2] In this article we will assume, unless otherwise stated, that the values of a Pythagorean quadruple are all positive integers.

## Contents

* 1 Parametrization of primitive quadruples
* 2 Alternate parametrization
* 3 Properties
* 4 Relationship with quaternions and rational orthogonal matrices
* 5 Primitive Pythagorean quadruples with small norm
* 6 See also
* 7 References
* 8 External links

## Parametrization of primitive quadruples

A Pythagorean quadruple is called primitive if the greatest common divisor of its entries is 1. Every Pythagorean quadruple is an integer multiple of a primitive quadruple. The set of primitive Pythagorean quadruples for which a is odd can be generated by the formulas [math]\displaystyle{ \begin{align} a &= m^2+n^2-p^2-q^2, \\\ b &= 2(mq+np), \\\ c &= 2(nq-mp), \\\ d &= m^2+n^2+p^2+q^2, \end{align} }[/math] where m, n, p, q are non-negative integers with greatest common divisor 1 such that m \+ n \+ p \+ q is odd.[3][4][1] Thus, all primitive Pythagorean quadruples are characterized by the identity [math]\displaystyle{ (m^2 + n^2 + p^2 + q^2)^2 = (2mq + 2np)^2 + (2nq - 2mp)^2 + (m^2 + n^2 - p^2 - q^2)^2. }[/math]

## Alternate parametrization

All Pythagorean quadruples (including non-primitives, and with repetition, though a, b, and c do not appear in all possible orders) can be generated from two positive integers a and b as follows:

If a and b have different parity, let p be any factor of a2 \+ b2 such that p2 < a2 \+ b2. Then c = a2 \+ b2 − p2/2p and d = a2 \+ b2 \+ p2/2p. Note that p = d − c.

A similar method exists[5] for generating all Pythagorean quadruples for which a and b are both even. Let l = a/2 and m = b/2 and let n be a factor of l2 \+ m2 such that n2 < l2 \+ m2. Then c = l2 \+ m2 − n2/n and d = l2 \+ m2 \+ n2/n. This method generates all Pythagorean quadruples exactly once each when l and m run through all pairs of natural numbers and n runs through all permissible values for each pair.

No such method exists if both a and b are odd, in which case no solutions exist as can be seen by the parametrization in the previous section.

## Properties

The largest number that always divides the product abcd is 12.[6] The quadruple with the minimal product is (1, 2, 2, 3).

Similar to a Pythagorean triple which generates a distinct right triangle, a Pythagorean quadruple will generate a distinct Heronian triangle.[7] If a, b, c, d is a Pythagorean quadruple with [math]\displaystyle{ a^2 + b^2 + c^2 = d^2 }[/math] it will generate a Heronian triangle with sides x, y, z as follows:-

[math]\displaystyle{ x = d^2 - a^2 }[/math]
[math]\displaystyle{ y = d^2 - b^2 }[/math]
[math]\displaystyle{ z = d^2 - c^2 }[/math].

It will have a semiperimeter [math]\displaystyle{ s = d^2 }[/math], an area [math]\displaystyle{ A = abcd }[/math] and an inradius [math]\displaystyle{ r = abc/d }[/math].

The exradii will be:-

[math]\displaystyle{ r_x = bcd/a }[/math]
[math]\displaystyle{ r_y = acd/b }[/math]
[math]\displaystyle{ r_z = abd/c }[/math].

The circumradius will be [math]\displaystyle{ R=(d^2 - a^2)(d^2 - b^2)(d^2 - c^2)/(4abcd) = abcd(1/a^2 + 1/b^2 + 1/c^2 -1/d^2)/4 }[/math].

The ordered sequence of areas of this class of Heronian triangles can be found at (sequence A367737 in the OEIS).

## Relationship with quaternions and rational orthogonal matrices

A primitive Pythagorean quadruple (a, b, c, d) parametrized by (m, n, p, q) corresponds to the first column of the matrix representation E(α) of conjugation α(⋅)α by the Hurwitz quaternion α = m \+ ni \+ pj \+ qk restricted to the subspace of quaternions spanned by i, j, k, which is given by [math]\displaystyle{ E(\alpha) = \begin{pmatrix} m^2+n^2-p^2-q^2&2np-2mq &2mp+2nq \\\ 2mq+2np &m^2-n^2+p^2-q^2&2pq-2mn \\\ 2nq-2mp &2mn+2pq &m^2-n^2-p^2+q^2\\\ \end{pmatrix}, }[/math] where the columns are pairwise orthogonal and each has norm d. Furthermore, we have that 1/dE(α) belongs to the orthogonal group [math]\displaystyle{ SO(3,\mathbb{Q}) }[/math], and, in fact, all 3 × 3 orthogonal matrices with rational coefficients arise in this manner.[8]

## Primitive Pythagorean quadruples with small norm

There are 31 primitive Pythagorean quadruples in which all entries are less than 30.

( |  1 | , |  2 | , | 2 | , | 3 | ) |  ( |  2 | , | 10 | , | 11 | , | 15 | ) |  ( | 4 | , | 13 | , | 16 | , | 21 | ) |  ( | 2 | , | 10 | , | 25 | , | 27 | )

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( | 2 | , | 3 | , | 6 | , | 7 | ) |  ( | 1 | , | 12 | , | 12 | , | 17 | ) |  ( | 8 | , | 11 | , | 16 | , | 21 | ) |  ( | 2 | , | 14 | , | 23 | , | 27 | )
( | 1 | , | 4 | , | 8 | , | 9 | ) |  ( | 8 | , | 9 | , | 12 | , | 17 | ) |  ( | 3 | , | 6 | , | 22 | , | 23 | ) |  ( | 7 | , | 14 | , | 22 | , | 27 | )
( | 4 | , | 4 | , | 7 | , | 9 | ) |  ( | 1 | , | 6 | , | 18 | , | 19 | ) |  ( | 3 | , | 14 | , | 18 | , | 23 | ) |  ( | 10 | , | 10 | , | 23 | , | 27 | )
( | 2 | , | 6 | , | 9 | , | 11 | ) |  ( | 6 | , | 6 | , | 17 | , | 19 | ) |  ( | 6 | , | 13 | , | 18 | , | 23 | ) |  ( | 3 | , | 16 | , | 24 | , | 29 | )
( | 6 | , | 6 | , | 7 | , | 11 | ) |  ( | 6 | , | 10 | , | 15 | , | 19 | ) |  ( | 9 | , | 12 | , | 20 | , | 25 | ) |  ( | 11 | , | 12 | , | 24 | , | 29 | )
( | 3 | , | 4 | , | 12 | , | 13 | ) |  ( | 4 | , | 5 | , | 20 | , | 21 | ) |  ( | 12 | , | 15 | , | 16 | , | 25 | ) |  ( | 12 | , | 16 | , | 21 | , | 29 | )
( | 2 | , | 5 | , | 14 | , | 15 | ) |  ( | 4 | , | 8 | , | 19 | , | 21 | ) |  ( | 2 | , | 7 | , | 26 | , | 27 | )

## See also

* Beal conjecture
* Euler brick
* Euler's sum of powers conjecture
* Euler-Rodrigues formula for 3D rotations
* Fermat cubic
* Jacobi–Madden equation
* Lagrange's four-square theorem (every natural number can be represented as the sum of four integer squares)
* Legendre's three-square theorem (which natural numbers cannot be represented as the sum of three squares of integers)
* Prouhet–Tarry–Escott problem
* Quaternions and spatial rotation
* Taxicab number

## References

1. ↑ 1.0 1.1 R. Spira, The diophantine equation x2 \+ y2 \+ z2 = m2, Amer. Math. Monthly Vol. 69 (1962), No. 5, 360–365.
2. ↑ R. A. Beauregard and E. R. Suryanarayan, Pythagorean boxes, Math. Magazine 74 (2001), 222–227.
3. ↑ R.D. Carmichael, Diophantine Analysis, New York: John Wiley & Sons, 1915.
4. ↑ L.E. Dickson, Some relations between the theory of numbers and other branches of mathematics, in Villat (Henri), ed., Conférence générale, Comptes rendus du Congrès international des mathématiciens, Strasbourg, Toulouse, 1921, pp. 41–56; reprint Nendeln/Liechtenstein: Kraus Reprint Limited, 1967; Collected Works 2, pp. 579–594.
5. ↑ Sierpiński, Wacław, Pythagorean Triangles, Dover, 2003 (orig. 1962), p.102–103.
6. ↑ MacHale, Des, and van den Bosch, Christian, "Generalising a result about Pythagorean triples", Mathematical Gazette 96, March 2012, pp. 91-96.
7. ↑ "OEIS A367737". The On-Line Encyclopedia of Integer Sequences. http://oeis.org/A367737.
8. ↑ J. Cremona, Letter to the Editor, Amer. Math. Monthly 94 (1987), 757–758.

## External links

* Weisstein, Eric W.. "Pythagorean Quadruple". http://mathworld.wolfram.com/PythagoreanQuadruple.html.
* Weisstein, Eric W.. "Lebesgue's Identity". http://mathworld.wolfram.com/LebesgueIdentity.html.

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