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 oddintegers
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).
, 
, 
, and 
that satisfy
and 
, there exist such integers 
and 
; for positive odd 
and 
, no such integers exist (Oliverio
1996).
, 
, 
, 
, 
, and 
.
. Thus it records an exact three-dimensional use
of the Pythagorean theorem long before the
surviving Greek treatments (Friberg 2007).
, where 
are integers, and let 
be the number of odd integers
in 
.
Then iff 
(mod 4), there exist integers 
and 
such that
,
,
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.