$1 \cdot 55$ Find all non-negative integers $n$, such that $2^{2^{n}}+5$ is a prime number.

Find all non-negative integers $n$, such that $2^{2^{n}}+5$ is a prime number.
［Solution］When $n=0$,
$$
2^{2^{11}}+5=2^{2^{0}}+5=2^{1}+5=2+5=7
$$

is a prime number. When $n>1$, since $2 \equiv-1(\bmod 3)$,
so
$$
2^{2^{n}}+5 \equiv(-1)^{2^{n}}+2=1+2 \equiv 0(\bmod 3),
$$

and
$$
2^{2^{n}}+5>3,
$$

thus
$$
2^{2^{\prime \prime}}+5
$$

is a multiple of 3, and not equal to 3, hence it must be a composite number.
In conclusion, the only non-negative integer is $n=0$.