I'm trying to solve the following problem:

Given $\epsilon>0$, are there positive integers $a,b$ such that $0<|\sqrt a-\sqrt[3]b|<\epsilon$ ?

**My solution**: given $n\in\Bbb N$,
$$|\sqrt{n^2}-\sqrt[3]{n^3+1}|=\sqrt[3]{n^3+1}-n=\frac1{\sqrt[3]{(n^3+1)^2}+n\sqrt[3]{n^3+1}+n^2}<\frac1{3n}\to 0$$
Thus, the answer is yes.

But I was trying to find an "optimal" solution. That is, now the problem becomes

Given $\epsilon>0$, find the least $b\in \Bbb Z_+$ such that there exists $a\in\Bbb Z_+$ such that $0<|\sqrt a-\sqrt[3]b|<\epsilon$

and now I'm totally lost. Is there some theory about this? Perhaps has it to do with the diophantine equation $a^3-b^2=\pm1$, and hence, to Catalan's conjecture?

*Remark*: Please note the '$0<$' in the inequality. I'm aware that $\sqrt 1=\sqrt[3]1$.