A Pythagorean quadruple is a tuple of integers a, b, c and d, such that d > 0 and , and is often denoted . Geometrically, a Pythagorean quadruple defines a cuboid with side lengths |a|, |b|, and |c|, whose space diagonal has integer length d. Pythagorean quadruples are thus also called Pythagorean Boxes.
Parametrization of primitive quadruples
All Pythagorean quadruples (including non-primitives, and with repetition, though a, b and c do not appear in all possible orders) can be generated from two positive integers a and b as follows:
If and have different parity, let p be any factor of such that . Then and . Note that .
A similar method exists for both even, with the further restriction that must be an even factor of . No such method exists if both a and b are odd.
The biggest number that always divides the product abcd is 12. The quadruple with the minimal product is (1, 2, 2, 3).
Relationship with quaternions and rational orthogonal matrices
A primitive Pythagorean quadruple parametrized by corresponds to the first column of the matrix representation of conjugation by the Hurwitz quaternion restricted to the subspace of spanned by , which is given by
Pythagorean quadruples with small norm
- (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), (2,5,14,15), (2, 10, 11, 15), (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), (9, 12, 20, 25), (12, 15, 16, 25), (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)
- Pythagorean triple
- Quaternions and spatial rotation
- Euler-Rodrigues formula for 3D rotations
- Euler's sum of powers conjecture
- Beal's conjecture
- Jacobi–Madden equation
- Prouhet–Tarry–Escott problem
- Taxicab number
- Fermat cubic
- R.A. Beauregard and E. R. Suryanarayan, Pythagorean boxes, Math. Magazine 74 (2001), 222–227.
- R.D. Carmichael, Diophantine Analysis, New York: John Wiley & Sons, 1915.
- L.E. Dickson, Some relations between the theory of numbers and other branches of mathematics, in Villat (Henri), ed., Conférence générale, Comptes rendus du Congrès international des mathématiciens, Strasbourg, Toulouse, 1921, pp. 41–56; reprint Nendeln/Liechtenstein: Kraus Reprint Limited, 1967; Collected Works 2, pp. 579–594.
- R. Spira, The diophantine equation , Amer. Math. Monthly Vol. 69 (1962), No. 5, 360–365.
- Sierpiński, Wacław, Pythagorean Triangles, Dover, 2003 (orig. 1962), p.102.
- MacHale, Des, and van den Bosch, Christian, "Generalising a result about Pythagorean triples", Mathematical Gazette 96, March 2012, pp. 91-96.
- J. Cremona, Letter to the Editor, Amer. Math. Monthly 94 (1987), 757–758.