For the Diophantine equation (x^4)+(5*y^3)=2008, either find all integer solutions, or show there are no integer solutions.
This means $\displaystyle x^4 - 2008 = 5(-y)^3$. Thus, $\displaystyle x^4\equiv 2008 (\bmod 5)$ and so $\displaystyle x^4 \equiv 3(\bmod 5)$ which is impossible.