In mathematics, an Euler brick, named after Leonhard Euler, is a rectangular cuboid whose edges and face diagonals all have integer lengths. A primitive Euler brick is an Euler brick whose edge lengths are relatively prime. A perfect Euler brick is one whose space diagonal is also an integer but such a brick has not yet been found. Definition The definition of an Euler brick in geometric terms is equivalent to a solution to the following system of Diophantine equations: where are the edges and are the diagonals. Properties If is a solution, then is also a solution for any . Consequently, the solutions in rational numbers are all rescalings of integer solutions. Given an Euler brick with edge-lengths , the triple constitutes an Euler brick as well. Exactly one edge and two face diagonals of a primitive Euler brick are odd. At least two edges of an Euler brick are divisible by 3. At least two edges of an Euler brick are divisible by 4. At least one edge of an Euler brick is divisible by 11. Examples The smallest Euler brick, discovered by Paul Halcke in 1719, has edges and face diagonals . Some other small primitive solutions, given as edges — face diagonals , are below: {| style="border-collapse:collapse;text-align:right;white-space:nowrap;" |(|| 85,|| 132,|| 720||) — (|| 157,|| 725,|| 732||) |- |(||140,|| 480,|| 693||) — (|| 500,|| 707,|| 843||) |- |(||160,|| 231,|| 792||) — (|| 281,|| 808,|| 825||) |- |(||187,||1020,||1584||) — (||1037,||1595,||1884||) |- |(||195,|| 748,||6336||) — (|| 773,||6339,||6380||) |- |(||240,|| 252,|| 275||) — (|| 348,|| 365,|| 373||) |- |(||429,|| 880,||2340||) — (|| 979,||2379,||2500||) |- |(||495,||4888,||8160||) — (||4913,||8175,||9512||) |- |(||528,||5796,||6325||) — (||5820,||6347,||8579||) |} Generating formula Euler found at least two parametric solutions to the problem, but neither gives all solutions. An infinitude of Euler bricks can be generated with Saunderson's parametric formula. Let be a Pythagorean triple (that is, .) Then the edges give face diagonals There are many Euler bricks which are not parametrized as above, for instance the Euler brick with edges and face diagonals . Perfect cuboid A perfect cuboid (also called a perfect Euler brick or 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: where is the space diagonal. , no example of a perfect cuboid had been found and no one has proven