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).

REFERENCES:

Carmichael, R. D. Diophantine Analysis. New York: Wiley, 1915.

Dutch, S. "Power Page: Pythagorean Quartets." https://www.uwgb.edu/dutchs/RECMATH/rmpowers.htm#pythquart.

Mordell, L. J. Diophantine Equations. London: Academic Press, 1969.

Oliverio, P. "Self-Generating Pythagorean Quadruples and -tuples." Fib. Quart. 34, 98-101, 1996.