What is Pythagorean quadruple?

A Pythagorean quadruple is a tuple of integers a, b, c, and d, such that a + b + c = d. 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 with the only condition being that d > 0. In this setting, a Pythagorean quadruple 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.