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.
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
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.
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 it will generate a Heronian triangle with sides x, y, z as follows:-
It will have a semiperimeter , an area and an inradius .
The exradii will be:-
The circumradius will be .
The ordered sequence of areas of this class of Heronian triangles can be found at (sequence A367737 in the OEIS).
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
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 | ) |