May be following information helps you...
In the figure, yellow color conditionally shows half of the sweep of the Euler brick abc. We show that the length of the space diagonal in the Euler brick is not an integer. So. Simply look at the last equation on the figure:www.wolframalpha.com_the equation solutions
You may also find an y
coordinate which possible for a point A
, and then find the lenght CA
by dint of coordinates.
Hence the coordinate x
of the point A
, which is the end of the distance CA
has no solutions in integers for the Euler cuboid. Accordingly, after substitution, the coordinate y
also/or has no integer solutions, that is: x, y∉N
. Since the point C (-a; -b)
, which is the end of the same distance CA
has integer coordinates, then the length CA
, and, consequently, the length of the equal space diagonal of a given Euler brick can not be expressed as an integer.