Smallest Differences between Fractional Parts of Square and Cube Roots

Theorem
Apart from $6$th powers, the value of $n$ less than $50 \, 000$ for which the difference between the fractional parts of $\sqrt n$ and $\sqrt [3] n$ is smallest is $30 \, 739$.

The next integer to produce a smaller difference above that is $62 \, 324$.