Smallest Fifth Power which is Sum of 6 Fifth Powers

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Theorem

The smallest fifth power which is the sum of $6$ fifth powers is $12^5 = 248 \, 832$:

$12^5 = 4^5 + 5^5 + 6^5 + 7^5 + 9^5 + 11^5$


Proof

We have:

\(\ds 12^5\) \(=\) \(\ds 248 \, 832\)
\(\ds \) \(=\) \(\ds 1024 + 3125 + 7776 + 16 \, 807 + 59 \, 049 + 161 \, 051\)
\(\ds \) \(=\) \(\ds 4^5 + 5^5 + 6^5 + 7^5 + 9^5 + 11^5\)


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Sources

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