1. It is known that there is a prime number between $10^{17}$ and $10^{17}+10$. Try to find this prime number.
1. $10^{17}+3$.

It is easy to know that positive integers of at least two digits ending in even numbers or the digit 5 are all composite numbers.
$$
\begin{array}{l}
\text { By } 10^{17}+9=10 \times 100^{8}+9 \\
\equiv 3 \times 2^{8}+2 \equiv 0(\bmod 7), \\
10^{17}+1 \equiv(-1)^{17}+1 \equiv 0(\bmod 11), \\
10^{17}+7=10 \times 100^{8}+7 \\
\equiv 10(-2)^{8}+7 \\
\equiv 0(\bmod 17),
\end{array}
$$

We know that $10^{17}+1$, $10^{17}+7$, and $10^{17}+9$ are all composite numbers.
Therefore, only $10^{17}+3$ meets the requirement.