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. 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. 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

{\begin{aligned}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{aligned}} where m, n, p, q are non-negative integers with greatest common divisor 1 such that m + n + p + q is odd. Thus, all primitive Pythagorean quadruples are characterized by Lebesgue's 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 a2 + b2 such that p2 < a2 + b2. Then c = a2 + b2p2/2p and d = a2 + b2 + p2/2p. Note that p = dc.

A similar method exists 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 + m2n2/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. 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 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 1/dE(α) ∈ SO(3,ℚ), and, in fact, all 3 × 3 orthogonal matrices with rational coefficients arise in this manner.

## 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 )