Pythagorean quadruple
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
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
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.
Note that p = d − c. A similar method exists[5] for generating all Pythagorean quadruples for which a and b are both even.
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).
Given a Pythagorean quadruple
can be defined as the norm of the quadruple in that
and is analogous to the hypotenuse of a Pythagorean triple.
Every odd positive number other than 1 and 5 can be the norm of a primitive Pythagorean quadruple
are greater than zero and are coprime.
[7] All primitive Pythagorean quadruples with the odd numbers as norms up to 29 except 1 and 5 are given in the table below.
Similar to a Pythagorean triple which generates a distinct right triangle, a Pythagorean quadruple will generate a distinct Heronian triangle.
[8] If a, b, c, d is a Pythagorean quadruple with
it will generate a Heronian triangle with sides x, y, z as follows:
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
where the columns are pairwise orthogonal and each has norm d. Furthermore, we have that 1/dE(α) belongs to the orthogonal group
, and, in fact, all 3 × 3 orthogonal matrices with rational coefficients arise in this manner.
[9] There are 31 primitive Pythagorean quadruples in which all entries are less than 30.
