Does there exist a natural number $n$ for which there is a supremum of a set $$A = \{a\in\mathbb{Q}^+ | a^3+a\leq n^2\}$$ in the set of rational numbers?
Since $f(a) = a^3+a$ is increasing function for $a\in \mathbb{Q}^+$, set $A$ has upper bound $n^2$, so we need to solve $a^3+a=n^2$? Is there such $n$?