The question is whether there exist integers $x,y,z$ such that $$ 9x^3+y^3=z^2+3. $$ This is one of the nicest (if not the nicest one!) cubic equations for which I do not know whether integer solutions exist.

Heuristic argument predicts a lot of solutions, but the search retuned no integer solutions up to $|z|\leq 50,000$. It seems that there must be a reason for this.

The question is whether there exist integers $x,y,z$ such that $$ 9x^3+y^3=z^2+3. $$ This is one of the nicest (if not the nicest one!) cubic equations for which I do not know whether integer solutions exist.

Heuristic argument predicts a lot of solutions, but the search returned no integer solutions up to $|z|\leq 50,000$. It seems that there must be a reason for this. ${}{}{}{}$