Smallest Cube whose Sum of Divisors is Cube
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This theorem requires a proof.
It remains to be shown that this is the smallest such number.
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof.
Contents
Theorem
The smallest cube $N$ such that $\map \sigma N$ is also a cube is:
- $27 \, 418 \, 521 \, 963 \, 671 \, 501 \, 273 \, 905 \, 190 \, 135 \, 082 \, 692 \, 041 \, 730 \, 405 \, 303 \, 870 \, 249 \, 023 \, 209$
where $\map \sigma N$ denotes the $\sigma$ function of $N$: the sum of the divisors of $N$
Proof
We have that:
| \(\displaystyle N\) | \(=\) | \(\displaystyle 27 \, 418 \, 521 \, 963 \, 671 \, 501 \, 273 \, 905 \, 190 \, 135 \, 082 \, 692 \, 041 \, 730 \, 405 \, 303 \, 870 \, 249 \, 023 \, 209\) | |||||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle 3^9 \times 7^3 \times 11^3 \times 13^3 \times 17^3 \times 41^3 \times 43^3 \times 47^3 \times 443^3 \times 499^3 \times 3583^3\) | |||||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle 30 \, 154 \, 214 \, 043 \, 975 \, 990 \, 969^3\) |
| \(\displaystyle \map \sigma N\) | \(=\) | \(\displaystyle 65 \, 400 \, 948 \, 817 \, 364 \, 742 \, 403 \, 487 \, 616 \, 930 \, 512 \, 213 \, 536 \, 407 \, 552 \, 000 \, 000 \, 000 \, 000 \, 000\) | |||||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle 2^{39} \times 3^6 \times 5^{15} \times 7^3 \times 11^3 \times 13^3 \times 17^3 \times 29^3 \times 37^3 \times 61^3 \times 157^3\) | |||||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle 40 \, 289 \, 760 \, 243 \, 532 \, 800 \, 000^3\) |
It remains to be shown that this is the smallest such number.
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof.
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Historical Note
According to David Wells in his Curious and Interesting Numbers, 2nd ed. of $1997$, this result is due to Rubin, and can be found in Journal of Recreational Mathematics, Volume $27$, on page $229$.
However, this has not been corroborated.
Sources
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $3^9 7^3 11^3 13^3 17^3 41^3 43^3 47^3 443^3 499^3 3583^3$
