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| In [[mathematics]], an '''Euler brick''', named after [[Leonhard Euler]], is a [[cuboid]] whose [[Edge (geometry)|edges]] and [[face diagonal]]s all have integer lengths. A '''primitive Euler brick''' is an Euler brick whose edge lengths are [[relatively prime]].
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| == Properties ==
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| Alternatively stated, an Euler brick is a solution to the following system of [[Diophantine equation]]s:
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| :<math>\begin{cases} a^2 + b^2 = d^2\\ a^2 + c^2 = e^2\\ b^2 + c^2 = f^2\end{cases}</math>
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| Euler found at least two [[parametric solution]]s to the problem, but neither give all solutions.<ref>{{mathworld|urlname=EulerBrick|title=Euler Brick}}</ref>
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| Given an Euler brick with edges (''a'', ''b'', ''c''), the triple (''bc'', ''ac'', ''ab'') constitutes an Euler brick as well.
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| == Examples ==
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| The smallest Euler brick, discovered by [[Paul Halcke]] in 1719, has edges <math>(a, b, c) = (44, 117, 240)</math> and face diagonals 125, 244, and 267.
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| Some other small primitive solutions, given as edges (''a'', ''b'', ''c'') — face diagonals (''d'', ''e'', ''f''), are below:
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| * (85, 132, 720) — (157, 725, 732);
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| * (140, 480, 693) — (500, 707, 843);
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| * (160, 231, 792) — (281, 808, 825);
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| * (240, 252, 275) — (348, 365, 373).
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| == Perfect cuboid ==
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| {{unsolved|mathematics|Does a perfect cuboid exist?}}
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| A '''perfect cuboid''' (also called a '''perfect box''') is an Euler brick whose [[space diagonal]] also has integer length.
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| In other words, the following equation is added to the system of [[Diophantine equation]]s defining an Euler brick:
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| :<math>a^2 + b^2 + c^2 = g^2.\,</math> | |
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| {{Asof|2012|November}}, no example of a perfect cuboid had been found and no one has proven that none exist. Exhaustive computer searches show that, if a perfect cuboid exists, one of its edges must be greater than 3·10<sup>12</sup>.<ref>Durango Bill. [http://www.durangobill.com/IntegerBrick.html The “Integer Brick” Problem]</ref><ref>{{mathworld|urlname=PerfectCuboid|title=Perfect Cuboid}}</ref> Furthermore, its smallest edge must be longer than 10<sup>10</sup>.<ref>Randall Rathbun, [http://old.nabble.com/Perfect-Cuboid-search-to-1e10-completed---none-found-p30324321.html Perfect Cuboid search to 1e10 completed - none found]. NMBRTHRY maillist, November 28, 2010.</ref>
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| Some facts are known about properties that must be satisfied by a ''primitive'' perfect cuboid, if one exists, based on modular arithmetic: {{cn|date=January 2013}}
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| * One edge, two face diagonals and the body diagonal must be odd, one edge and the remaining face diagonal must be divisible by 4, and the remaining edge must be divisible by 16
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| * 2 edges must have length divisible by 3 and at least 1 of those edges must have length divisible by 9
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| * 1 edge must have length divisible by 5.
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| * 1 edge must have length divisible by 7.
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| * 1 edge must have length divisible by 11.
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| * 1 edge must have length divisible by 19.
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| * 1 edge or space diagonal must be divisible by 13.
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| * 1 edge, face diagonal or space diagonal must be divisible by 17.
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| * 1 edge, face diagonal or space diagonal must be divisible by 29.
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| * 1 edge, face diagonal or space diagonal must be divisible by 37.
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| Solutions have been found where the space diagonal and two of the three face diagonals are integers, such as:
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| :<math>(a, b, c) = (672, 153, 104).\,</math>
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| Solutions are also known where all four diagonals but only two of the three edges are integers, such as:
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| :<math>(a, b, c) = (18720, \sqrt{211773121}, 7800)</math>
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| and
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| :<math>(a, b, c) = (520, 576, \sqrt{618849}).</math>
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| ==Perfect parallelepiped==
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| A perfect [[parallelepiped]] is a parallelepiped with integer-length edges, face diagonals, and body diagonals, but not necessarily with all right angles; a perfect cuboid is a special case of a perfect parallelepiped. In 2009, a perfect parallelepiped was shown to exist,<ref>{{Cite journal|first1=Jorge F.|last1=Sawyer|first2=Clifford A.|last2=Reiter|year=2011|title=Perfect parallelepipeds exist|journal=[[Mathematics of Computation]]|volume=80|pages=1037–1040|arxiv=0907.0220}}.</ref> answering an open question of [[Richard K. Guy|Richard Guy]]. Solutions with only a single oblique angle have been found.
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| ==Notes==
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| {{reflist}}
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| == References ==
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| * {{cite journal | first=John | last=Leech | authorlink=John Leech (mathematician) | title=The Rational Cuboid Revisited | journal=American Mathematical Monthly | volume=84 | issue=7 | pages=518–533 | year=1977 | doi=10.2307/2320014 | jstor=2320014 }}
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| * {{cite book|last=Guy|first=Richard K.|authorlink=Richard K. Guy|title=Unsolved Problems in Number Theory|publisher=[[Springer-Verlag]]|year=2004|isbn=0-387-20860-7|pages=275–283}}
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| * {{cite journal | first=Tim | last=Roberts | title=Some constraints on the existence of a perfect cuboid | journal=Australian Mathematical Society Gazette | volume=37 | pages=29–31 | year=2010 | issn=1326-2297 }}
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| {{DEFAULTSORT:Euler Brick}}
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| [[Category:Articles with inconsistent citation formats]]
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| [[Category:Arithmetic problems of solid geometry]]
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| [[Category:Diophantine equations]]
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| [[Category:Unsolved problems in mathematics]]
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