Pythagorean quadruple

From Wikipedia, the free encyclopedia

A set of four positive integers a, b, c and d such that a2 + b2+ c2 = d2 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[1]

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 paramaterization

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.

[edit] References

  1. ^ R.D. Carmichael, Diophantine Analysis, New York: John Wiley & Sons, 1915.

[edit] See also

[edit] External links