Pythagorean Quadruple

A Pythagorean quadruple is a set of positive integers , , , and that satisfy

(1)

For positive even and , there exist such integers and ; for positive odd and , no such integers exist (Oliverio 1996).

Examples of primitive Pythagorean quadruples include , , , , , and .

Oliverio (1996) gives the following generalization of this result. Let , where are integers, and let be the number of odd integers in . Then iff (mod 4), there exist integers and such that

(2)

A set of Pythagorean quadruples is given by

			(3)
			(4)
			(5)
			(6)

where , , and are integers (Mordell 1969). This does not, however, generate all solutions. For instance, it excludes (36, 8, 3, 37).