Diophantine equation (three variables)

Question

Show that the equation $$x^3+2y^2+4z=n$$ has an integer solution $(x,y,z)$ for all integers $n$.

This seems to be an Diophantine equation with three variables. How can I restrict $n$ in order to find the solution?

Answer 1

Let $k \in \mathbb{Z}$.

If $n=4k$, then $(x,y,z)=(0,0,k)$ can work.

If $n=4k+1$, then $(x,y,z)=(1,0,k)$.

If $n=4k+2$, then $(x,y,z)=(0,1,k)$.

If $n=4k+3$, then $(x,y,z)=(1,1,k)$.

Since any integer $n$ will be in one of the above forms ($4k+r$), hence the equation has integer solutions for all $n$.