<s>[INST] solve $x^4 + y^4 = x^3 + y^3 + 10$ for $x,y \in \mathbb{Z}$ solve $x^4 + y^4 = x^3 + y^3 + 10$  for  $x,y \in \mathbb{Z}$.
I tried solving this by trying to find upper bounds for $|x|$ and $|y|$, therefor it is quite useful to write:
$x^4 + y^4 - x^3 - y^3 = .....$
where ... is in the form of a square.
I tried to write $x^4 - x^3 = x^2(x^2+x) = x^2((x+\frac{1}{2})^2 - \frac{1}{4})$, but the $'\frac{1}{4}'$ is kind of troublesome, any tips or hints on how to get a good square, so i can estimate my polynomial?
Kees
 [/INST] $x^4-x^3$, and $y^4-y^3$, are never negative and rarely less than $10$.  There must be very few possibilities to check.
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