A Pythagorean quadruple is a tuple of integers a, b, c and d, such that d > 0 and $a^2 + b^2 + c^2 = d^2$, and is often denoted $(a,b,c,d)$. Geometrically, a Pythagorean quadruple $(a,b,c,d)$ defines a cuboid with side lengths |a|, |b|, and |c|, whose space diagonal has integer length d. Pythagorean quadruples are thus also called Pythagorean Boxes.[1]

The set of all primitive Pythagorean quadruples, i.e., those for which gcd(a,b,c) = 1, where gcd denotes the greatest common divisor, is parametrized by,[2][3][4]

$a = m^2+n^2-p^2-q^2,\,$
$b = 2(mq+np),\,$
$c = 2(nq-mp),\,$
$d = m^2+n^2+p^2+q^2,\,$

where m, n, p, q are non-negative integers and gcd(m, n, p, q) = 1 and m + n + p + q ≡ 1 (mod 2). Thus, all primitive Pythagorean quadruples are characterized by the Lebesgue Identity

$(m^2 + n^2 + p^2 + q^2)^2 = (2mq + 2np)^2 + (2nq - 2mp)^2 + (m^2 + n^2 - p^2 - q^2)^2.$

## 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 $c = (a^2 + b^2 - p^2)/(2p)$ and $d = (a^2 + b^2 + p^2)/(2p)$. Note that $p = {d - c}$.

A similar method exists[5] for $a, b$ both even, with the further restriction that $2p$ must be an even factor of $a^2 + b^2$. No such method exists if both a and b are odd.

## Properties

The biggest 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 $E(\alpha)$ of conjugation $\alpha(\cdot)\overline{\alpha}$ by the Hurwitz quaternion $\alpha = m + ni + pj + qk$ restricted to the subspace of $\mathbb{H}$ spanned by $i, j, k$, which is given by

$E(\alpha) = \begin{pmatrix} m^2+n^2-p^2-q^2&2np-2mq &2mp+2nq \\ 2mq+2np &m^2-n^2+p^2-q^2&2pq-2mn \\ 2nq-2mp &2mn+2pq &m^2-n^2-p^2+q^2\\ \end{pmatrix},$

where the columns are pairwise orthogonal and each has norm d. Furthermore, we have $\frac{1}{d}E(\alpha)$ $\in \text{SO}(3, \mathbb{Q})$, and, in fact, all 3 × 3 orthogonal matrices with rational coefficients arise in this manner.[7]

## Pythagorean quadruples with small norm

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

4. ^ R. Spira, The diophantine equation $x^2 + y^2 + z^2 = m^2$, Amer. Math. Monthly 69 (1962), 360–365.