I have the following question:
What is the range of the sum of three distinct natural numbers to the fifth power than are equal to a known natural number?
Mathematically speaking:
$$n=x^5+y^5+z^5\tag1$$
When $n\in\mathbb{N}$ is known, what is the range where $x\space\wedge\space y\space\wedge\space z\in\mathbb{N}$ can be in when we know that $x\ne y\ne z$?
I think that the range should be: $1\le x,y,z\le\left(\left\lceil\sqrt{n}\right\rceil\right)^5$ but I am not sure why that should be true.
If $x$ is maximal among them, then $y\leq x-1$ and $z\leq x-2$ (or vice versa).
So $$(x-2)^5+(x-1)^5+x^5\geq n\implies 3(x-1)^5\geq n$$
so $$x\geq \sqrt[5]{n\over 3}+1$$ So $$\sqrt[5]{n\over 3}+1\leq x\leq \sqrt[5]{n-33}$$
if $n\geq 107313$.