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 
.
A Pythagorean quadruple 
determines a Heronian
triangle with side lengths 
, 
, and 
, semiperimeter 
, and area 
(F. M. Jackson, pers. comm.,
Dec. 3, 2023).
The family
 |
(2)
|
for positive integers 
produces Soddyian triangles
under this construction (F. M. Jackson, pers. comm., Oct. 17, 2026).
For example, 
and 
give 
and the Soddyian triangle with side lengths
(13, 40, 45).
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
 |
(3)
|
A set of Pythagorean quadruples is given by
 |  |  |
(4)
|
 |  |  |
(5)
|
 |  |  |
(6)
|
 |  |  |
(7)
|
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.