Pythagorean quadruple

All four primitive Pythagorean quadruples with only single-digit values

A Pythagorean quadruple is a tuple of integers a, b, c and d, such that d > 0 and a2 + b2 + c2 = d2, and is often denoted (a,b,c,d). Geometrically, a Pythagorean quadruple (a,b,c,d) defines a cuboid with integer 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

The set of all primitive Pythagorean quadruples, i.e., those for which gcd(a,b,c) = 1 where gcd denotes the greatest common divisor, and for which without loss of generality a is odd, is parametrized by,[2][3][4]

\begin{align} 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, \end{align}

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.

and Becker-Sievert Identity for special cases

A) n^2+n^2+(n/2)^2 = (3n/2)^2 ; for all even n

B) (2n^2+1)^2=(2n)^2n^2+(2n)^2+1^2

Alternate parametrization

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 a2 + b2 such that p2 < a2 + b2. Then c = a2 + b2 − p2/2p and d = a2 + b2 + p2/2p. Note that p = d − c.

A similar method exists[5] for generating all Pythagorean quadruples for which a and b are both even. Let l = a/2 and m = b/2 and let n be a factor of l2 + m2 such that n2 < l2 + m2. Then c = l2 + m2 − n2/n and d = l2 + m2 + n2/2n. This method generates all Pythagorean quadruples exactly once each when l and m run through all pairs of natural numbers and n runs through all permissible values for each pair.

No such method exists if both a and b are odd, in which case no solutions exist as can be seen by the parametrization in the previous section.

Properties

The largest 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

A primitive Pythagorean quadruple (a,b,c,d) parametrized by (m,n,p,q) corresponds to the first column of the matrix representation E(α) of conjugation α(⋅)α by the Hurwitz quaternion α = m + ni + pj + qk restricted to the subspace of 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 1/dE(α) ∈ SO(3,ℚ), and, in fact, all 3 × 3 orthogonal matrices with rational coefficients arise in this manner.[7]

Primitive Pythagorean quadruples with small norm

( 1, 2, 2, 3),   ( 2,10,11,15),   ( 4,13,16,21),   ( 2,10,25,27),
( 2, 3, 6, 7),   ( 1,12,12,17),   ( 8,11,16,21),   ( 2,14,23,27),
( 1, 4, 8, 9),   ( 8, 9,12,17),   ( 3, 6,22,23),   ( 7,14,22,27),
( 4, 4, 7, 9),   ( 1, 6,18,19),   ( 3,14,18,23),   (10,10,23,27),
( 2, 6, 9,11),   ( 6, 6,17,19),   ( 6,13,18,23),   ( 3,16,24,29),
( 6, 6, 7,11),   ( 6,10,15,19),   ( 9,12,20,25),   (11,12,24,29),
( 3, 4,12,13),   ( 4, 5,20,21),   (12,15,16,25),   (12,16,21,29) 
( 2, 5,14,15),   ( 4, 8,19,21),   ( 2, 7,26,27),

See also

References

  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 x2 + y2 + z2 = m2, Amer. Math. Monthly Vol. 69 (1962), No. 5, 360–365.
  5. ↑ Sierpiński, Wacław, Pythagorean Triangles, Dover, 2003 (orig. 1962), p.102–103.
  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.

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