Does a perfect cuboid exist?

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archive posted 6 weeks ago

Find a rectangular box whose three edges, three face diagonals and space diagonal all have integer lengths, or prove that none exists. In other words, find positive integers $a, b, c$ such that $a^{2} + b^{2}$ , $b^{2} + c^{2}$ , $a^{2} + c^{2}$ and $a^{2} + b^{2} + c^{2}$ are all perfect squares.

Posed by folklore (grew out of the study of Euler bricks)

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Still open as of 2026.

A box whose edges and face diagonals are all integers is called an Euler brick. The smallest, with edges 44, 117 and 240, was found by Paul Halcke in 1719, and Euler later gave parametric families of them. Requiring the space diagonal to be an integer as well gives a perfect cuboid. Exhaustive computer searches show that any perfect cuboid would have an odd edge longer than $2.5 \times 1 0^{13}$ and a smallest edge longer than $5 \times 1 0^{11}$ . Modular arithmetic forces many further conditions, for example a primitive perfect cuboid would have one edge divisible by 16 and one divisible by 5. No example is known and no proof of impossibility exists.

References

Euler brick (Wikipedia)

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