4. Solve the equation $2^{x}+2^{y}=6^{t}$ in integers.
Solution. Let $x=y$ at first. In this case, the original equation will take the form

$$
2^{x+1}=6^{t}
$$

If $t>0$, then the right-hand side (1) is divisible by three, while the left-hand side is not. Therefore, $t \leq 0$. If we assume that $t<0$, then the left-hand side (5) is divisible by 4. If $t$ is odd, then the right-hand side (5) is not divisible by 4. Therefore, $t=2m, m \in \mathbb{Z}, m \geq 0$. From (5) it follows that $2^{n}=\left(3^{m}-1\right)\left(3^{m}+1\right)$. This means that the numbers $3^{m}-1$ and $3^{m}+1$ are powers of two. Note also that on the number line, these numbers are 2 units apart from each other. This is only possible if $3^{m}-1=2$ and $3^{m}+1=4$. From this, $m=1$ and then $t=2, n=3$. Substituting the found values into (2) and (4), we get the solution:

$$
x=2, y=5, t=2
$$

Answer: $(-1,-1,0),(1,2,1),(2,5,2)$ (under the condition $x \leq y$).