Example 8. Find the prime number $p$ such that $8 p^{2}+1$ is also a prime number.



The text above is translated into English, preserving the original text's line breaks and format.
When $p=2$, $8 p^{2}+1=33$ is not a prime number.
When $p=3$, $8 p^{2}+1=73$ is a prime number.
For $p>3$, since $p$ is a prime, $p \in\{3 k+1\}$ or $p \in\{3 k+2\}$.
(i) If $p \in\{3 k+1\}$, then $p^{2} \in\{3 k+1\}$. $8 \in\{3 k+2\}$.

Thus $8 p^{2} \in\{3 k+2\}$, so $8 p^{2}+1$ $\in\{3 k\}$ is a composite number.
(ii) If $p \in\{3 k+2\}$, then $p^{2} \in\{3 k+1\}, 8 \in\{3 k+2\}. 8 p^{2} \in\{3 k+2\}$, $8 p^{2}+1 \in\{3 k\}$ is a composite number.

In conclusion, $8 p^{2}+1$ is a prime number if and only if $p=3$.