# Pythagorean quadruple

A **Pythagorean quadruple** is a tuple of integers *a*, *b*, *c* and *d*, such that *a*^{2} + *b*^{2} + *c*^{2} = *d*^{2}. 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.

## 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

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 Lebesgue's identity

## 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 *a*^{2} + *b*^{2} such that *p*^{2} < *a*^{2} + *b*^{2}. Then *c* = *a*^{2} + *b*^{2} − *p*^{2}/2*p* and *d* = *a*^{2} + *b*^{2} + *p*^{2}/2*p*. 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 *l*^{2} + *m*^{2} such that *n*^{2} < *l*^{2} + *m*^{2}. Then *c* = *l*^{2} + *m*^{2} − *n*^{2}/*n* and *d* = *l*^{2} + *m*^{2} + *n*^{2}/*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).

## 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 ℍ spanned by *i*, *j*, *k*, which is given by

where the columns are pairwise orthogonal and each has norm *d*. Furthermore, we have 1/*d**E*(*α*) ∈ SO(3,ℚ), and, in fact, *all* 3 × 3 orthogonal matrices with rational coefficients arise in this manner.[7]

## 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 | ) |

( | 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

## References

- R. Spira,
*The diophantine equation**x*^{2}+*y*^{2}+*z*^{2}=*m*^{2}, Amer. Math. Monthly**Vol. 69**(1962), No. 5, 360–365. - R. A. Beauregard and E. R. Suryanarayan,
*Pythagorean boxes*, Math. Magazine**74**(2001), 222–227. - R.D. Carmichael,
*Diophantine Analysis*, New York: John Wiley & Sons, 1915. - 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. - Sierpiński, Wacław,
*Pythagorean Triangles*, Dover, 2003 (orig. 1962), p.102–103. - MacHale, Des, and van den Bosch, Christian, "Generalising a result about Pythagorean triples",
*Mathematical Gazette*96, March 2012, pp. 91-96. - J. Cremona,
*Letter to the Editor*, Amer. Math. Monthly**94**(1987), 757–758.

## External links

- Weisstein, Eric W. "Pythagorean Quadruple".
*MathWorld*. - Weisstein, Eric W. "Lebesgue's Identity".
*MathWorld*.

- Carmichael.
*Diophantine Analysis*at Project Gutenberg