How to find out number of solutions for equations of type $a+b^2+c^3+d^4 \le X$?

Question

Given the value of 'X' how do we calculate the number of solutions of the equation $$a+b^2+c^3+d^4 \le X$$ where $a,b,c,d$ are non negative integer values? What are these type of equations called and the concept of mathematics that deals with these type of equations?

Answer 1

I don't know anything about the background of these equations, but finding general solutions can be done by induction on the degree of the "polynomial".

First consider $a\leq X$, clearly this has $X+1$ solutions. Now consider $a+b^{2}\leq X$ which is equivalent to $a\leq X-b^{2}$. Then $b\in\{0,...,\lfloor\sqrt{X}\rfloor\}$, so there are $$\sum^{\lfloor\sqrt{X}\rfloor}_{i=0}X-i^{2}+1$$ solutions.

Continue this process to find the solution to the "polynomial" of degree 4.