Pythagorean quadruple

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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 a2 + b2 + c2 = d2. They are solutions of a Diophantine equation and often only positive integer values are considered.[1] However, to provide a more complete geometric interpretation, the integer values can be allowed to be negative and zero (thus allowing Pythagorean triples to be included) with the only condition being that d > 0. In this setting, 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, with this interpretation are thus also called Pythagorean boxes.[2] In this article we will assume, unless otherwise stated, that the values of a Pythagorean quadruple are all positive integers.

Parametrization of primitive quadruples[edit]

A Pythagorean quadruple is called primitive if the greatest common divisor of its entries is 1. Every Pythagorean quadruple is an integer multiple of a primitive quadruple. The set of primitive Pythagorean quadruples for which a is odd can be generated by the formulas

where m, n, p, q are non-negative integers with greatest common divisor 1 such that m + n + p + q is odd.[3][4][1] Thus, all primitive Pythagorean quadruples are characterized by Lebesgue's identity[clarification needed]

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 a2 + b2 such that p2 < a2 + b2. Then c = a2 + b2p2/2p and d = a2 + b2 + p2/2p. Note that p = dc.

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 + m2n2/n and d = l2 + m2 + n2/n. 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[edit]

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[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(α) of conjugation α(⋅)α by the Hurwitz quaternion α = m + ni + pj + qk restricted to the subspace of spanned by i, j, k, which is given by

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[edit]

There are 31 primitive Pythagorean quadruples in which all entries are less than 30.

( 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[edit]

References[edit]

  1. ^ a b R. Spira, The diophantine equation x2 + y2 + z2 = m2, Amer. Math. Monthly Vol. 69 (1962), No. 5, 360–365.
  2. ^ R.A. Beauregard and E. R. Suryanarayan, Pythagorean boxes, Math. Magazine 74 (2001), 222–227.
  3. ^ R.D. Carmichael, Diophantine Analysis, New York: John Wiley & Sons, 1915.
  4. ^ 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.
  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.

External links[edit]