It seems like the proof to why a perfect cuboid doesn't exist is super simple? Can someone find the flaw?

Question

I am just struggling, because I know chances are that I didn't find a solution to an unsolved problem in under 3 hours, but I can't find what's wrong with this. It's really bugging me.

https://www.scribd.com/document/332017929/Proof-that-a-perfect-cuboid-does-not-exist

Answer 1

Your error is assuming that no solution to $A^2 + B^2 + C^2 = 2D^2$ exists because $D\sqrt{2}$ is irrational.

See, for example, $2 \cdot 29^2 = 28^2 + 27^2 + 13^2$

Answer 2

You show that $A^2+B^2+C^2 = 2D^2 $.

But you have not shown that $A^2+B^2+C^2 $ is a perfect square, only that it is an integer.

Therefore, you have not shown that $2D^2$ is the square of an integer.

This is the error in your proof.