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
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.
