Positive Integers , , , and which satisfy

 (1)

For Positive Even and , there exist such Integers and ; for Positive Odd and , no such Integers exist (Oliverio 1996). 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,
 (7)

and
 (8)

(Mordell 1969). This does not, however, generate all solutions. For instance, it excludes (36, 8, 3, 37). Another set of solutions can be obtained from
 (9) (10) (11) (12)

(Carmichael 1915).

References

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

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

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