I was recently given a math problem by a friend; the challenge was to find a rectangular prism whose side lengths (including diagonals and space diagonal) were all natural numbers. I found that for this to be true, some number $\sigma^2$ (the space diagonal) must be equal to the sums of the face diagonals squared and divided by two $((\alpha^2+\beta^2+\gamma^2)/2$) I know that any number of the form $4^n \left({8 m + 7}\right)|m,n\in\mathbb{N}$ cannot be expressed as the sum of three squares, which leaves a lot of room for numbers that might work for this problem.

However, there is one more condition I found though that might make this an impossible problem. Since the face diagonals are not components of a right triangle, $\alpha^2+\beta^2\neq\gamma^2$. Finding numbers before this condition was known was an easy task. For instance, $3^2+4^2+5^2=5^2+5^2=2*5^2$. Now it seems impossible to find ones that work. Knowing that all squares are either 0, 1 or 4 mod 8 (the information needed to show that $4^n \left({8 m + 7}\right)|m,n\in\mathbb{N}$) also yields no proof or disproof as you can still get mod 0, 1, 4 numbers out if you add three squares and half the result ($[0+4+4]/2=4$.

It feels like don't know enough about modular arithmetic to continue, is my assumption true that $\neg\exists \sigma | \sigma^2=(\alpha^2+\beta^2+\gamma^2)/2$ and $\alpha,\beta,\gamma,\sigma \in \mathbb{N}$ if $\alpha^2+\beta^2\neq\gamma^2$?