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; with this interpretation, Pythagorean quadruples 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
All four primitive Pythagorean quadruples with only single-digit values
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
\( {\displaystyle {\begin{aligned}a&=m^{2}+n^{2}-p^{2}-q^{2},\\b&=2(mq+np),\\c&=2(nq-mp),\\d&=m^{2}+n^{2}+p^{2}+q^{2},\end{aligned}}} \)
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
\( (m^2 + n^2 + p^2 + q^2)^2 = (2mq + 2np)^2 + (2nq - 2mp)^2 + (m^2 + n^2 - p^2 - q^2)^2. \)
Alternate parametrization
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/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
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
\( {\displaystyle E(\alpha )={\begin{pmatrix}m^{2}+n^{2}-p^{2}-q^{2}&2np-2mq&2mp+2nq\\2mq+2np&m^{2}-n^{2}+p^{2}-q^{2}&2pq-2mn\\2nq-2mp&2mn+2pq&m^{2}-n^{2}-p^{2}+q^{2}\\\end{pmatrix}},}\)
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
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
Beal's conjecture
Euler brick
Euler's sum of powers conjecture
Fermat cubic
Jacobi–Madden equation
Prouhet–Tarry–Escott problem
Quaternions and spatial rotation
Euler-Rodrigues formula for 3D rotations
Taxicab number
References
R. Spira, The diophantine equation x2 + y2 + z2 = m2, Amer. Math. Monthly Vol. 69 (1962), No. 5, 360–365.
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.
Sierpiński, Wacław, Pythagorean Triangles, Dover, 2003 (orig. 1962), p.102–103.
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.
External links
Weisstein, Eric W. "Pythagorean Quadruple". MathWorld.
Weisstein, Eric W. "Lebesgue's Identity". MathWorld.
Carmichael. Diophantine Analysis at Project Gutenberg
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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