1. (28th Moscow Mathematical Olympiad, 1965) Find all prime numbers of the form $p^{p}+1$ (where $p$ is a natural number) with no more than 19 digits.
1. Clearly $p1000^{6} \cdot 16=16 \cdot 10^{18} \text {, }
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Therefore $16^{16}+1$ has more than 19 digits.
If $p=8$, then
$8^{8}+1=2^{24}+1=\left(2^{8}\right)^{3}+1=\left(2^{8}+1\right)\left(2^{15}-2^{8}+1\right)$ is a composite number.
If $p=4$, then
$4^{4}+1=257$ is a prime number.
If $p=2$, then
$2^{2}+1=5$ is a prime number.
Thus, the prime numbers sought are $2, 5, 257$.