Euler brick

The definition of an Euler brick in geometric terms is equivalent to a solution to the following system of Diophantine equations: where a, b, c are the edges and d, e, f are the diagonals.

The smallest Euler brick, discovered by Paul Halcke in 1719, has edges (a, b, c) = (44, 117, 240) and face diagonals (d, e, f ) = (125, 244, 267).

[3] An infinitude of Euler bricks can be generated with Saunderson's[4] parametric formula.

Then[1]: 105  the edges give face diagonals There are many Euler bricks which are not parametrized as above, for instance the Euler brick with edges (a, b, c) = (240, 252, 275) and face diagonals (d, e, f ) = (348, 365, 373).

As of March 2020[update], no example of a perfect cuboid had been found and no one has proven that none exist.

[5] Exhaustive computer searches show that, if a perfect cuboid exists, Some facts are known about properties that must be satisfied by a primitive perfect cuboid, if one exists, based on modular arithmetic:[8] In addition: If a perfect cuboid exists and

, then Three cuboid conjectures are three mathematical propositions claiming irreducibility of three univariate polynomials with integer coefficients depending on several integer parameters.

the eighth degree polynomial is irreducible over the ring of integers

such that none of the conditions are fulfilled, the twelfth-degree polynomial is irreducible over the ring of integers

In 2009, dozens of perfect parallelepipeds were shown to exist,[19] answering an open question of Richard Guy.

In 2022, Aubrey de Grey published[20] an exploration of perfect isosceles rectangular frusta, which he termed "plinths".

In a subsequent paper,[21] de Grey, Philip Gibbs and Louie Helm built on this finding to explore classes of elliptic curve that correspond to perfect plinths, almost-perfect cuboids, and other generalisations of perfect cuboids.

By this means they dramatically increased the range up to which perfect cuboids can be sought computationally, and thereby derived strong circumstantial evidence that none exists.

They also showed that a large proportion of Pythagorean triples cannot form a face of a perfect cuboid, by identifying several families of elliptic curves that must have positive rank if a perfect cuboid exists.

Independently, Paulsen and West showed[22] that a perfect cuboid must correspond to a congruent number elliptic curve of rank at least 2.