3. Given the equation $x^{4}-p x^{3}+q=0$ has an integer root, find the prime numbers $p$ and $q$.

 untranslated text:
已知方程 $x^{4}-p x^{3}+q=0$ 有一整数根,求素数 $p 、 q$.

 translated text:
Given the equation $x^{4}-p x^{3}+q=0$ has an integer root, find the prime numbers $p$ and $q$.
3. Solution: Let the integer $x$ satisfy $q=x^{3}(p-x)$.

Then $x \mid q$.
Since $q$ is a prime number, it can only be that $x= \pm 1, x= \pm q$. When $x= \pm q$, by (1) we have
$q^{2} \mid 1$, which is impossible (since $q$ is a prime number).
When $x=-1$, we have $p+q+1=0$, which is also impossible.
When $x=1$, by (1) we get $p=q+1$,
Since $p$ and $q$ are both prime numbers, then $p=3, q=2$.