$\displaystyle 2009-x^4=4,\ 3\ (mod\ 5)$
$\displaystyle 2009-x^4=5y^3$ has no integer solutions.
$\displaystyle x^4+5y^3=2009$ has no integer solutions.
$\displaystyle 2009-x^4=4,\ 3\ (mod\ 5)$
$\displaystyle 2009-x^4=5y^3$ has no integer solutions.
$\displaystyle x^4+5y^3=2009$ has no integer solutions.
That makes sense. Did you choose (mod 5) just because one of the terms including a variable was a multiple of 5?