Infinitely many integer solutions for the equations $x^3+y^3+z^3=1$ and $x^3+y^3+z^3=2$ How do you show that the equation $x^3+y^3+z^3=1$ has infinitely many solutions in integers? How about $x^3+y^3+z^3=2$?

You can reduce the first equation to $$x^3 = -y^3, z = 1$$ with obvious infinite solutions. 
This paper details other families of solutions.
The second equation has solutions $(x,y,z)\equiv (6t^3+1, 1-6t^3, -6t^2)$ which (AFAIK) you find by construction (i.e you have to guess it). 
