Euler brick

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In mathematics, an Euler brick, named after the famous mathematician Leonhard Euler, is a cuboid whose edges and face diagonals are all integers. A primitive Euler brick is an Euler brick whose edges are relatively prime.

Alternatively stated, an Euler Brick is a solution to the following system of diophantine equations:

$a^2 + b^2 = d^2\,$

$b^2 + c^2 = e^2\,$

$a^2 + c^2 = f^2.\,$

The smallest Euler brick has edges

(a, b, c) = (240, 117, 44)

and face polyhedron diagonals

267, 244, and 125.

Paul Halcke discovered it in 1719.

Other solutions are: Given as: length (a, b, c)

(275, 252, 240),
(693, 480, 140),
(720, 132, 85), and
(792, 231, 160).

Euler found at least two parametric solutions to the problem, but neither give all solutions.

Given an Euler brick with edges (a, b, c), the triple (bc, ac, ab) constitutes an Euler brick as well.

[edit] Perfect cuboid

Unsolved problems in mathematics: Does a perfect cuboid exist?

A perfect cuboid (also called a perfect box) is an Euler brick whose body diagonal is also an integer.

In other words the following equation is added to the above Diophantine equations:

$a^2 + b^2 + c^2 = g^2.\,$

Some interesting facts about a perfect cuboid.

2 edges must be even and 1 edge must be odd (for a primitive perfect cuboid).
1 edge must be divisible by 4 and 1 edge must be divisible by 16
1 edge must be divisible by 3 and 1 edge must be divisible by 9
1 edge must be divisible by 5
1 edge must be divisible by 11

As of 2009^[update], no example of a perfect cuboid had been found and no one had proven that it cannot exist. Exhaustive computer searches have proven that the smallest edge of the perfect box is at least 4.3 billion. Solutions have been found where the body diagonal and two of the three face diagonals are integers, such as:

$(a, b, c) = (672, 153, 104).\,$