Pythagorean quadruple

A set of four positive integers "a", "b", "c" and "d" such that "a"² + "b"²+ "c"² = "d"² is called a Pythagorean quadruple.

The set of all primitive Pythagorean quadruples, i.e., those for which gcd("a","b","c","d") = 1, where gcd denotes the greatest common divisor of "a", "b", "c", and "d", is parameterized by [R.D. Carmichael, Diophantine Analysis, New York: John Wiley & Sons, 1915.]

:$a\; =\; m^2+n^2-p^2-q^2,,$

:$b\; =\; 2(mp+nq),,$

:$c\; =\; 2(np-mq),,$

:$d\; =\; m^2+n^2+p^2+q^2,,$

where "m", "n", "p", "q" are integers.

If we set "q" = 0, then we get the simpler parameterization

:$a\; =\; m^2+n^2-p^2,,$

:$b\; =\; 2mp,,$

:$c\; =\; 2np,,$

:$d\; =\; m^2+n^2+p^2,,$

which does not generate all quadruples. For example (3,36,8,37) is a quadruple that is generated by the first parameterization by taking "m" = 4, "n" = 2, "p" = 4, and "q" = 1, but is not generated by the second.

Using the first case above, the set of all primitive Pythagorean quadruples for {"a", "b", "c", "d"} > 0 and "d" < 30 is
:$(1,2,2,3),\; (2,3,6,7),\; (1,4,8,9),\; (4,4,7,9),,$
:$(2,6,9,11),\; (6,6,7,11),\; (3,4,12,13),,$
:$(1,12,12,17),\; (8,9,12,17),\; (1,6,18,19),\; (6,6,17,19),\; (6,10,15,19),,$
:$(4,5,20,21),\; (4,8,19,21),\; (4,13,16,21),\; (8,11,16,21),,$
:$(3,6,22,23),\; (3,14,18,23),\; (6,13,18,23),,$
:$(2,7,26,27),\; (2,10,25,27),\; (2,14,23,27),\; (7,14,22,27),\; (10,10,23,27),,$
:$(3,16,24,29),\; (11,12,24,29),\; (12,16,21,29),$

References

ee also

*Pythagorean triples

External links

* [http://mathworld.wolfram.com/PythagoreanQuadruple.html Wolfram write-up] (does not include the complete parameterization)

* [http://www.gutenberg.org/etext/20073 Carmichael's Diophantine Analysis at Project Gutenburg]

* [http://www.math.siu.edu/kocik/pracki/44Cliffpdf.pdf The complete parametrisation derived using a Minkowskian Clifford Algebra]