Let's determine the prime numbers $p$ and $q$ (both positive) if

$$
p+p^{2}+p^{4}-q-q^{2}-q^{4}=83805
$$
Solution. One of $p$ and $q$ is even, the other is odd, because otherwise the value of the sixth-degree expression could not be odd. It is also clear that $p>q$, so only $q=2$ is possible. Then $p+p^{2}+p^{4}=83$827. If we increase the value of $p$, then $p+p^{2}+p^{4}$ also increases, so if there is a solution, there can only be one. Since

$$
17+17^{2}+17^{4}=83827
$$

therefore the solution to the problem is: $p=17, q=2$.