Smallest Integer which is Sum of 3 Sixth Powers in 2 Ways

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Theorem

The smallest positive integer which can be expressed as the sum of $3$ sixth powers in $2$ different ways is:

\(\ds 160 \, 426 \, 514\) \(=\) \(\ds 3^6 + 19^6 + 22^6\)
\(\ds \) \(=\) \(\ds 10^6 + 15^6 + 23^6\)


Also note that:

\(\ds 854\) \(=\) \(\ds 3^2 + 19^2 + 22^2\)
\(\ds \) \(=\) \(\ds 10^2 + 15^2 + 23^2\)


Proof

We have that:

\(\ds 160 \, 426 \, 514\) \(=\) \(\ds 729 + 47 \, 045 \, 881 + 113 \, 379 \, 904\)
\(\ds \) \(=\) \(\ds 3^6 + 19^6 + 22^6\)


\(\ds 160 \, 426 \, 514\) \(=\) \(\ds 1 \, 000 \, 000 + 11 \, 390 \, 625 + 148 \, 035 \, 889\)
\(\ds \) \(=\) \(\ds 10^6 + 15^6 + 23^6\)


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In particular: It remains to be shown there are no smaller
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Sources

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