# Smallest Differences between Fractional Parts of Square and Cube Roots

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## Contents

## 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

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $30,739$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $30,739$