# 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 $62 \, 324$.

## 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.