in mathematics, an euler brick, named after leonhard euler, is a cuboid whose edges and face diagonals all have integer lengths. a primitive euler brick is an euler brick whose edge lengths are relatively prime. == definition and properties == the definition of an euler brick in geometric terms is equivalent to a solution to the following system of diophantine equations: :\begin{cases} a^2 + b^2 = d^2\\ a^2 + c^2 = e^2\\ b^2 + c^2 = f^2\end{cases} euler found at least two parametric solutions to the problem, but neither give all solutions. if (a, b, c) is a solution, then (ka, kb, kc) is also a solution for any k. consequently, the solutions in rational numbers are all rescalings of integer solutions. given an euler brick with edge-lengths (a, b, c), the triple (bc, ac, ab) constitutes an euler brick as well. == examples == the smallest euler brick, discovered by paul halcke in 1719, has edges (a, b, c) = (44, 117, 240) and face diagonals 125, 244, and 267. some other small primitive solutions, given as edges (a, b, c) â face diagonals (d, e, f), are below: * (85, 132, 720) â (157, 725, 732); * (140, 480, 693) â (500, 707, 843); * (160, 231, 792) â (281, 808, 825); * (240, 252, 275) â (348, 365, 373). == perfect cuboid ==  a perfect cuboid (also called a perfect box) is an euler brick whose space diagonal also has integer length. in other words, the following equation is added to the system of diophantine equations defining an euler brick: :a^2 + b^2 + c^2 = g^2.\, , 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 3Â·1012.durango bill. the âinteger brickâ problem furthermore, its smallest edge must be longer than 1010.randall rathbun, perfect cuboid search to 1e10 completed - none found. nmbrthry maillist, november 28, 2010. some facts are known about properties that must be satisfied by a primitive perfect cuboid, if one exists, based on modular arithmetic:  * 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 * 2 edges must have length divisible by 3 and at least 1 of those edges must have length divisible by 9 * 1 edge must have length divisible by 5. * 1 edge must have length divisible by 7. * 1 edge must have length divisible by 11. * 1 edge must have length divi
Diophantine_equations Unsolved_problems_in_mathematics