283. Perfect Square. Find all integers $x, y, z$ for which the value $4^{x}+4^{y}+4^{z}$ is a perfect square.
283. First, note that $x, y, z$ cannot be negative, as otherwise $4^{x}+4^{y}+4^{z}$ would not be an integer. If $x \leqslant y \leqslant z$, then from the fact that $4^{x}+4^{y}+4^{z}$ is a perfect square, it follows that there exist a positive integer $m$ and a positive odd number $t$ such that

$$
1+4^{y-x}+4^{z-x}=\left(1+2^{m} t\right)^{2}
$$

Therefore,

$$
4^{y-x}\left(1+4^{z-y}\right)=2^{m+1} t\left(1+2^{m-1} t\right)
$$

from which $m=2 y-2 x-1$. Consequently,

$$
\begin{gathered}
t-1=4^{y-x-1}\left(4^{z-2 y+z+1}-t\right)= \\
=4^{y-x-1}\left(2^{z-2 y+z+1}+t\right)\left(2^{z-2 y+x+1}-t\right)
\end{gathered}
$$

from which $t=1$ and $z=2 y-x-1 *$. Thus, all integer solutions of the given equation can be represented in the form $\{x, y, 2 y-x-1\}$, where $x, y$ are arbitrary. For these values of $x, y, z$, our square is equal to $\left(2^{x}+2^{2 y-x-1}\right)^{2}$.

[E. Jones, A.M.M., 76, 308 (March 1969).]