Pythagorean quadruple

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]


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