Smallest Differences between Fractional Parts of Square and Cube Roots

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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 $62 \, 324$.


Proof

\(\displaystyle \sqrt {30 \, 739}\) \(\approx\) \(\displaystyle 175 \cdotp 32541 \, 17349\)
\(\displaystyle \sqrt [3] {30 \, 739}\) \(\approx\) \(\displaystyle 31 \cdotp 32539 \, 66116\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle \sqrt {30 \, 739} - \sqrt [3] {30 \, 739}\) \(\approx\) \(\displaystyle 144 \cdotp 00001 \, 5123\)


\(\displaystyle \sqrt {62 \, 324}\) \(\approx\) \(\displaystyle 249 \cdotp 64775 \, 18425\)
\(\displaystyle \sqrt [3] {62 \, 324}\) \(\approx\) \(\displaystyle 39 \cdotp 64774 \, 02668\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle \sqrt {62 \, 324} - \sqrt [3] {62 \, 324}\) \(\approx\) \(\displaystyle 210 \cdotp 00001 \, 1576\)



Historical Note

According to David Wells in his Curious and Interesting Numbers of $1986$, this result is reported by J.H. Baumwell and F. Rubin in volume $9$ of Journal of Recreational Mathematics, but this has not been corroborated.


Sources