TITLE: Diophantine: $x^3+y^3=z^3 \pm 1$
QUESTION [10 upvotes]: How many nontrivial integer solutions does $x^3+y^3=z^3 \pm 1$ have? 
The trivial solutions are $(\pm 1,z,z)$ and $(z,\pm 1,z)$.

REPLY [4 votes]: Note that Noam Elkies has given a complete parametrization of the integer solutions of $x^{3} + y^{3} +z^{3}+w^{3}=0,$ which may be found at http://www.math.harvard.edu/~elkies/4cubes.html .