# 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. 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.

## 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 ${\begin{aligned}a&=m^{2}+n^{2}-p^{2}-q^{2},\\b&=2(mq+np),\\c&=2(nq-mp),\\d&=m^{2}+n^{2}+p^{2}+q^{2},\end{aligned}}$ 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 the 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 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. The quadruple with the minimal product is (1, 2, 2, 3).
Given a Pythagorean quadruple $(a,b,c,d)$ where $d^{2}=a^{2}+b^{2}+c^{2}$ then $d$ can be defined as the norm of the quadruple in that $d={\sqrt {a^{2}+b^{2}+c^{2}}}$ 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 $d^{2}=a^{2}+b^{2}+c^{2}$ such that $a,b,c$ are greater than zero and are coprime. 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. If a, b, c, d is a Pythagorean quadruple with a$^2$+b$^2$+c$^2$=d$^2${\textstyle a^{2}+b^{2}+c^{2}=d^{2}} it will generate a Heronian triangle with sides x, y, z as follows: ${\begin{aligned}x&=d^{2}-a^{2}\\y&=d^{2}-b^{2}\\z&=d^{2}-c^{2}\end{aligned}}$ It will have a semiperimeter s=d$^2${\textstyle s=d^{2}}, an area A=abcd{\textstyle A=abcd} and an inradius r=abc/d{\textstyle r=abc/d}.
The exradii will be: ${\begin{aligned}r_{x}&=bcd/a,\\r_{y}&=acd/b,\\r_{z}&=abd/c.\end{aligned}}$ The circumradius will be: $R={\frac {(d^{2}-a^{2})(d^{2}-b^{2})(d^{2}-c^{2})}{4abcd}}={\frac {abcd(1/a^{2}+1/b^{2}+1/c^{2}-1/d^{2})}{4}}$
The ordered sequence of areas of this class of Heronian triangles can be found at (sequence A367737 in the OEIS).

## 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 quaternions 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 that ⁠1/d⁠E(α) belongs to the orthogonal group $SO(3,\mathbb {Q} )$, 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 | ) |