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
\(\ds \sqrt {30 \, 739}\) | \(\approx\) | \(\ds 175 \cdotp 32541 \, 17349\) | ||||||||||||
\(\ds \sqrt [3] {30 \, 739}\) | \(\approx\) | \(\ds 31 \cdotp 32539 \, 66116\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \sqrt {30 \, 739} - \sqrt [3] {30 \, 739}\) | \(\approx\) | \(\ds 144 \cdotp 00001 \, 5123\) |
\(\ds \sqrt {62 \, 324}\) | \(\approx\) | \(\ds 249 \cdotp 64775 \, 18425\) | ||||||||||||
\(\ds \sqrt [3] {62 \, 324}\) | \(\approx\) | \(\ds 39 \cdotp 64774 \, 02668\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \sqrt {62 \, 324} - \sqrt [3] {62 \, 324}\) | \(\approx\) | \(\ds 210 \cdotp 00001 \, 1576\) |
This needs considerable tedious hard slog to complete it. In particular: This could be turned into a page where the sequence of $n$ for which this difference is smaller than for any smaller $n$. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Finish}} from the code.If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page. |
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): $30,739$