## Task A-2.2.

Determine all pairs $(a, n)$ of natural numbers for which

$$
3 a^{2}+2^{n}=a^{4}
$$
## First solution.

From the initial condition, we read $2^{n}=a^{4}-3 a^{2}$, that is, $2^{n}=a^{2}\left(a^{2}-3\right)$.

Notice that $a^{2}$ and $a^{2}-3$ have opposite parity. On the other hand, since their product equals $2^{n}$, the only prime factor that can appear is 2. Therefore, we conclude that the smaller of the two factors is equal to 1, and the larger one is $2^{n}$.

In other words, we have

$$
a^{2}-3=1 \quad \text { and } \quad a^{2}=2^{n}
$$

From the first equation, since $a>0$, we get $a=2$, and from the second, we get $n=2$.

We conclude that the only solution is $(a, n)=(2,2)$.