9.1. Find all possible pairs of prime numbers \( m \) and \( n \), such that the equation \( x^{2} - m x - n = 0 \) has a prime root.
Answer: $m=2, n=3$.

Solution. Suppose that $x=p$ is a prime root, then $p^{2}-m p-n=0$, from which it follows that $n=p(p-m)$. Since $n$ and $p$ are prime numbers, then $n=p$ and $p-m=1$, that is, $p=m+1$. The number $m$ can only be 2, because in any other case $m+1$ would be even, and then $p$ would not be a prime number. Then $n=3$, and the desired equation is $x^{2}-2 x-$ $3=0$.

Comments. An example of such a polynomial - 1 point.