I have the next equation: $x^2+y^3=n$. Where n is a positive integer constant.
I want to know the exact number of non-negative integer solutions.
Also I want to know what are those solutions. How can I find them?
I have the next equation: $x^2+y^3=n$. Where n is a positive integer constant.
I want to know the exact number of non-negative integer solutions.
Also I want to know what are those solutions. How can I find them?
As mentioned in the comments this is essentially the classic problem of finding integer points of the Mordell curve, and a lot of work has gone into it (for example towards bounding the number of solutions, see this paper).
If you want to get understand the basics of the process of finding integers solutions, this other paper works out the $|n|\leq10^4$ range completely (and partially the $|n|\leq10^5$ range).