Smallest Cube whose Sum of Divisors is Cube

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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:

\(\displaystyle N\) \(=\) \(\displaystyle 27 \, 418 \, 521 \, 963 \, 671 \, 501 \, 273 \, 905 \, 190 \, 135 \, 082 \, 692 \, 041 \, 730 \, 405 \, 303 \, 870 \, 249 \, 023 \, 209\)
\(\displaystyle \) \(=\) \(\displaystyle 3^9 \times 7^3 \times 11^3 \times 13^3 \times 17^3 \times 41^3 \times 43^3 \times 47^3 \times 443^3 \times 499^3 \times 3583^3\)
\(\displaystyle \) \(=\) \(\displaystyle 30 \, 154 \, 214 \, 043 \, 975 \, 990 \, 969^3\)


Then from $\sigma$ of $27 \, 418 \, 521 \, 963 \, 671 \, 501 \, 273 \, 905 \, 190 \, 135 \, 082 \, 692 \, 041 \, 730 \, 405 \, 303 \, 870 \, 249 \, 023 \, 209$:

\(\displaystyle \map \sigma N\) \(=\) \(\displaystyle 65 \, 400 \, 948 \, 817 \, 364 \, 742 \, 403 \, 487 \, 616 \, 930 \, 512 \, 213 \, 536 \, 407 \, 552 \, 000 \, 000 \, 000 \, 000 \, 000\)
\(\displaystyle \) \(=\) \(\displaystyle 2^{39} \times 3^6 \times 5^{15} \times 7^3 \times 11^3 \times 13^3 \times 17^3 \times 29^3 \times 37^3 \times 61^3 \times 157^3\)
\(\displaystyle \) \(=\) \(\displaystyle 40 \, 289 \, 760 \, 243 \, 532 \, 800 \, 000^3\)



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