Pythagorean quadruple
A Pythagorean quadruple is a tuple of integers , , , and , such that . 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 . In this setting, a Pythagorean quadruple defines a cuboid with integer side lengths , , and , whose space diagonal has integer length ; 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 is odd can be generated by the formulas formula_1 where , , , are non-negative integers with greatest common divisor 1 such that is odd. Thus, all primitive Pythagorean quadruples are characterized by the identity formula_2 Alternate parametrization. All Pythagorean quadruples (including non-primitives, and with repetition, though , , and do not appear  in all possible orders) can be generated from two positive integers and as follows: If and have different parity, let be any factor of such that . Then and . Note that . A similar method exists for generating all Pythagorean quadruples for which and are both even. Let and and let be a factor of such that . Then and . This method generates all Pythagorean quadruples exactly once each when and run through all pairs of natural numbers and runs through all permissible values for each pair. No such method exists if both and 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 is 12. The quadruple with the minimal product is (1, 2, 2, 3). Relationship with quaternions and rational orthogonal matrices. A primitive Pythagorean quadruple parametrized by corresponds to the first column of the matrix representation of conjugation by the Hurwitz quaternion restricted to the subspace of quaternions spanned by , , , which is given by formula_3 where the columns are pairwise orthogonal and each has norm . Furthermore, we have that belongs to the orthogonal group formula_4, 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.