Pythagorean quadruple

All four primitive Pythagorean quadruples with only single-digit values

A Pythagorean quadruple is a tuple of integers Template:Math, Template:Math, Template:Math, and Template:Math, such that Template:Math. 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 Template:Math. In this setting, a Pythagorean quadruple Template:Math defines a cuboid with integer side lengths Template:Math, Template:Math, and Template:Math, whose space diagonal has integer length Template:Math; 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 Template:Math is odd can be generated by the formulas $$\begin{array}{cc}a& =m^2+n^2-p^2-q^2,\\ b& =2(mq+np),\\ c& =2(nq-mp),\\ d& =m^2+n^2+p^2+q^2,\end{array}$$ where Template:Math, Template:Math, Template:Math, Template:Math are non-negative integers with greatest common divisor 1 such that Template:Math 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 Template:Math, Template:Math, and Template:Math do not appear in all possible orders) can be generated from two positive integers Template:Math and Template:Math as follows:

If Template:Math and Template:Math have different parity, let Template:Math be any factor of Template:Math such that Template:Math. Then Template:Math and Template:Math. Note that Template:Math.

A similar method exists for generating all Pythagorean quadruples for which Template:Math and Template:Math are both even. Let Template:Math and Template:Math and let Template:Math be a factor of Template:Math such that Template:Math. Then Template:Math and Template:Math. This method generates all Pythagorean quadruples exactly once each when Template:Math and Template:Math run through all pairs of natural numbers and Template:Math runs through all permissible values for each pair.

No such method exists if both Template:Math and Template:Math 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 Template:Math 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 Template:Math is a Pythagorean quadruple with $a^2+b^2+c^2=d^2$ it will generate a Heronian triangle with sides Template:Math as follows: $$\begin{array}{cc}x& =d^2-a^2\\ y& =d^2-b^2\\ z& =d^2-c^2\end{array}$$ It will have a semiperimeter $s=d^2$, an area $A=abcd$ and an inradius $r=abc/d$.

The exradii will be: $$\begin{array}{cc}{r}_x& =bcd/a,\\ r_y& =acd/b,\\ r_z& =abd/c.\end{array}$$ 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 Template:OEIS.

Relationship with quaternions and rational orthogonal matrices

A primitive Pythagorean quadruple Template:Math parametrized by Template:Math corresponds to the first column of the matrix representation Template:Math of conjugation Template:Math by the Hurwitz quaternion Template:Math restricted to the subspace of quaternions spanned by Template:Math, Template:Math, Template:Math, which is given by $$E(\alpha )=\left(\begin{array}{ccc}{m}^2+n^2-p^2-q^2& 2np-2mq& 2mp+2nq\\ 2mq+2np& m^2-n^2+p^2-q^2& 2pq-2mn\\ 2nq-2mp& 2mn+2pq& m^2-n^2-p^2+q^2\end{array}\right),$$ where the columns are pairwise orthogonal and each has norm Template:Math. Furthermore, we have that Template:Math 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	)

See also

• Beal conjecture
• Euler brick
• Euler's sum of powers conjecture
• Euler-Rodrigues formula for 3D rotations
• Fermat cubic
• Jacobi–Madden equation
• Lagrange's four-square theorem (every natural number can be represented as the sum of four integer squares)
• Legendre's three-square theorem (which natural numbers cannot be represented as the sum of three squares of integers)
• Prouhet–Tarry–Escott problem
• Quaternions and spatial rotation
• Taxicab number

External links

• Template:MathWorld
• Template:MathWorld