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| A '''[[Pythagoras|Pythagorean]] quadruple''' is a [[tuple]] of [[integer]]s ''a'', ''b'', ''c'' and ''d'', such that ''d > 0'' and <math>a^2 + b^2 + c^2 = d^2</math>, and is often denoted <math>(a,b,c,d)</math>. Geometrically, a Pythagorean quadruple <math>(a,b,c,d)</math> defines a [[cuboid]] with side [[Norm (mathematics)#Euclidean norm|lengths]] |''a''|, |''b''|, and |''c''|, whose [[space diagonal]] has integer length ''d''. Pythagorean quadruples are thus also called ''Pythagorean Boxes''.<ref>R.A. Beauregard and E. R. Suryanarayan, ''Pythagorean boxes'', Math. Magazine '''74''' (2001), 222–227.</ref>
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| == Parametrization of primitive quadruples ==
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| The [[Set (mathematics)|set]] of all '''primitive''' Pythagorean quadruples, i.e., those for which gcd(''a'',''b'',''c'') = 1, where gcd denotes the [[greatest common divisor]] of ''a'', ''b'', and ''c'', is parametrized by,<ref>R.D. Carmichael, ''Diophantine Analysis'', New York: John Wiley & Sons, 1915.</ref><ref>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.</ref><ref>R. Spira, ''The diophantine equation <math>x^2 + y^2 + z^2 = m^2</math>'', Amer. Math. Monthly '''69''' (1962), 360–365.</ref>
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| :<math> a = m^2+n^2-p^2-q^2,\,</math>
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| :<math> b = 2(mq+np),\,</math>
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| :<math> c = 2(nq-mp),\,</math> | |
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| :<math> d = m^2+n^2+p^2+q^2,\,</math> | |
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| where ''m'', ''n'', ''p'', ''q'' are non-negative [[integer]]s 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]]
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| :<math>(m^2 + n^2 + p^2 + q^2)^2 = (2mq + 2np)^2 + (2nq - 2mp)^2 + (m^2 + n^2 - p^2 - q^2)^2.</math>
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| ==Alternate parametrization==
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| 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:
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| If <math>a</math> and <math>b</math> have different parity, let ''p'' be any factor of <math>a^2 + b^2</math> such that <math>p^2 < a^2 + b^2</math>. Then <math>c = (a^2 + b^2 - p^2)/(2p)</math> and <math>d = (a^2 + b^2 + p^2)/(2p)</math>. Note that <math>p = {d - c}</math>.
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| A similar method exists<ref>[[Wacław Sierpiński|Sierpiński, Wacław]], ''Pythagorean Triangles'', Dover, 2003 (orig. 1962), p.102.</ref> for <math>a, b</math> both even, with the further restriction that <math>2p</math> must be an even factor of <math>a^2 + b^2</math>. No such method exists if both ''a'' and ''b'' are odd.
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| ==Properties==
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| The biggest number that always divides the product ''abcd'' is 12.<ref>MacHale, Des, and van den Bosch, Christian, "Generalising a result about Pythagorean triples", ''[[Mathematical Gazette]]'' 96, March 2012, pp. 91-96.</ref> The quadruple with the minimal product is (1, 2, 2, 3).
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| ==Relationship with quaternions and rational orthogonal matrices==
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| A primitive Pythagorean quadruple <math>(a,b,c,d)</math> [[Parametrization|parametrized]] by <math>(m,n,p,q)</math> corresponds to the first [[Column vector|column]] of the [[Linear transformation#Matrices|matrix representation]] <math>E(\alpha)</math> of [[Quaternion#Conjugation.2C_the_norm.2C_and_reciprocal|conjugation]] <math>\alpha(\cdot)\overline{\alpha}</math> by the [[Hurwitz quaternion]] <math>\alpha = m + ni + pj + qk</math> [[Function (mathematics)#Restrictions and extensions|restricted]] to the subspace of [[Quaternion|<math>\mathbb{H}</math>]] spanned by <math>i, j, k</math>, which is given by
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| :<math>
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| E(\alpha) =
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| \begin{pmatrix}
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| m^2+n^2-p^2-q^2&2np-2mq &2mp+2nq \\
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| 2mq+2np &m^2-n^2+p^2-q^2&2pq-2mn \\
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| 2nq-2mp &2mn+2pq &m^2-n^2-p^2+q^2\\
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| \end{pmatrix},</math>
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| where the columns are pairwise [[orthogonality|orthogonal]] and each has [[Norm_(mathematics)#Euclidean_norm|norm]] ''d''. Furthermore, we have <math>\frac{1}{d}E(\alpha)</math> [[Orthogonal group|<math>\in \text{SO}(3, \mathbb{Q})</math>]], and, in fact, ''all'' 3 × 3 orthogonal matrices with [[Rational number|rational]] coefficients arise in this manner.<ref>J. Cremona, ''Letter to the Editor'', Amer. Math. Monthly '''94''' (1987), 757–758.</ref>
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| ==Pythagorean quadruples with small norm==
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| : (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)
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| ==See also==
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| *[[Pythagorean triple]]
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| *[[Quaternions and spatial rotation]]
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| *[[SO(4)#The Euler–Rodrigues formula for 3D rotations|Euler-Rodrigues formula for 3D rotations]]
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| *[[Euler's sum of powers conjecture]]
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| *[[Beal's conjecture]]
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| *[[Jacobi–Madden equation]]
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| *[[Prouhet–Tarry–Escott problem]]
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| *[[Taxicab number]]
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| *[[Fermat cubic]]
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| ==References==
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| {{Reflist}}
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| ==External links==
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| *{{MathWorld|urlname=PythagoreanQuadruple|title=Pythagorean Quadruple}}
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| *{{MathWorld|urlname=LebesgueIdentity|title=Lebesgue's Identity}}
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| *{{Gutenberg|no=20073|name=Diophantine Analysis|author=Carmichael}}
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| *[http://www.math.siu.edu/kocik/papers/44Cliff.pdf The complete parametrization derived using a Minkowskian Clifford Algebra]
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| [[Category:Additive number theory]]
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| [[Category:Arithmetic problems of plane geometry]]
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| [[Category:Diophantine equations]]
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| [[Category:Diophantine geometry]]
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