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 
.
An Old Babylonian tablet dated to roughly 1900-1600 BC computes the space diagonal of a rectangular gate whose scaled thickness, width,
height, and diagonal form
the Pythagorean quadruple 
. Thus it records an exact three-dimensional use
of the Pythagorean theorem long before the
surviving Greek treatments (Friberg 2007).
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).
-tuples." Fib. Quart. 34, 98-101, 1996.