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It is a well-known and open problem to determine whether there exists a rectangular cuboid where the distance from any corner to any other corner is an integer. Such a beast, if it exists, is called an Euler brick.

What happens if we work in larger dimensions? It is easy to generalize the definition of a rectangular cuboid to higher dimensions, and it appears that this question is open! Thus, my questions is:

Do we know if there is no 4-dimensional Euler brick? How about 10-dimensional?

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    $\begingroup$ Aren't the faces of an Euler 4-brick themselves Euler 3-bricks? (I may be being stupid here) $\endgroup$
    – Yemon Choi
    Commented Dec 1, 2016 at 16:35
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    $\begingroup$ According to erpublication.org/admin/vol_issue1/upload%20Image/… , there are no 4-dimensional Euler bricks. (Also, the ABC conjecture is true.) $\endgroup$
    – Emil Jeřábek
    Commented Dec 1, 2016 at 16:59
  • $\begingroup$ @EmilJeřábek That claim does throw some questionability on their work, doesn't it. :-) $\endgroup$
    – Pace Nielsen
    Commented Dec 1, 2016 at 18:29
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    $\begingroup$ @YemonChoi Yes, so this problem should be easier to solve (given it is likely that there is no Euler brick). $\endgroup$
    – Pace Nielsen
    Commented Dec 1, 2016 at 18:30
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    $\begingroup$ I was hoping that there would be a local obstruction at some prime, making this problem easy. Unfortunately there is not - for each $p$ there are bricks where all diagonals are $p$-adic integers. For $p>2$ one can take the side lengths to be $1,p,p^2,\dots$, and then every diagonal will be of the form $p^n \sqrt{ 1 + m p}$ for integers $n,m$ and so will be a $p$-adic integer. For $p=2$ the same thing works but with powers of $4$. $\endgroup$
    – Will Sawin
    Commented Dec 6, 2016 at 18:38

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