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