Pythagorean quadruple

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A Pythagorean quadruple is a tuple of integers a, b, c and d, such that d > 0 and a^2 + b^2 + c^2 = d^2, and is often denoted (a,b,c,d). Geometrically, a Pythagorean quadruple (a,b,c,d) 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.[1]

Parametrization of primitive quadruples[edit]

The set of all primitive Pythagorean quadruples, i.e., those for which gcd(a,b,c) = 1, where gcd denotes the greatest common divisor, is parametrized by,[2][3][4]

 a = m^2+n^2-p^2-q^2,\,
 b = 2(mq+np),\,
 c = 2(nq-mp),\,
 d = m^2+n^2+p^2+q^2,\,

where m, n, p, q are non-negative integers and gcd(m, n, p, q) = 1 and m + n + p + q ≡ 1 (mod 2). Thus, all primitive Pythagorean quadruples are characterized by the Lebesgue Identity

(m^2 + n^2 + p^2 + q^2)^2 = (2mq + 2np)^2 + (2nq - 2mp)^2 + (m^2 + n^2 - p^2 - q^2)^2.

Alternate parametrization[edit]

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 a and b have different parity, let p be any factor of a^2 + b^2 such that p^2 < a^2 + b^2. Then c = (a^2 + b^2 - p^2)/(2p) and d = (a^2 + b^2 + p^2)/(2p). Note that p = {d - c}.

A similar method exists[5] for a, b both even, with the further restriction that 2p must be an even factor of a^2 + b^2. No such method exists if both a and b are odd.

Properties[edit]

The biggest number that always divides the product abcd is 12.[6] The quadruple with the minimal product is (1, 2, 2, 3).

Relationship with quaternions and rational orthogonal matrices[edit]

A primitive Pythagorean quadruple (a,b,c,d) parametrized by (m,n,p,q) corresponds to the first column of the matrix representation E(\alpha) of conjugation \alpha(\cdot)\overline{\alpha} by the Hurwitz quaternion \alpha = m + ni + pj + qk restricted to the subspace of \mathbb{H} spanned by i, j, k, which is given by


E(\alpha) =
\begin{pmatrix}
m^2+n^2-p^2-q^2&2np-2mq        &2mp+2nq        \\
2mq+2np        &m^2-n^2+p^2-q^2&2pq-2mn        \\
2nq-2mp        &2mn+2pq        &m^2-n^2-p^2+q^2\\
\end{pmatrix},

where the columns are pairwise orthogonal and each has norm d. Furthermore, we have \frac{1}{d}E(\alpha) \in \text{SO}(3, \mathbb{Q}), and, in fact, all 3 × 3 orthogonal matrices with rational coefficients arise in this manner.[7]

Pythagorean quadruples with small norm[edit]

(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)

See also[edit]

References[edit]

  1. ^ R.A. Beauregard and E. R. Suryanarayan, Pythagorean boxes, Math. Magazine 74 (2001), 222–227.
  2. ^ R.D. Carmichael, Diophantine Analysis, New York: John Wiley & Sons, 1915.
  3. ^ 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.
  4. ^ R. Spira, The diophantine equation x^2 + y^2 + z^2 = m^2, Amer. Math. Monthly 69 (1962), 360–365.
  5. ^ Sierpiński, Wacław, Pythagorean Triangles, Dover, 2003 (orig. 1962), p.102.
  6. ^ MacHale, Des, and van den Bosch, Christian, "Generalising a result about Pythagorean triples", Mathematical Gazette 96, March 2012, pp. 91-96.
  7. ^ J. Cremona, Letter to the Editor, Amer. Math. Monthly 94 (1987), 757–758.

External links[edit]