Smallest Fourth Power expressible as Sum of 4 Fourth Powers

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Theorem

$15 \, 527 \, 402 \, 881$ is the smallest fourth power which can be expressed as the sum of $4$ fourth powers:

$15 \, 527 \, 402 \, 881 = 353^4 = 30^4 + 120^4 + 272^4 + 315^4$


Proof

\(\ds \) \(\) \(\ds 30^4 + 120^4 + 272^4 + 315^4\)
\(\ds \) \(=\) \(\ds 810 \, 000 + 207 \, 360 \, 000 + 5 \, 473 \, 632 \, 256 + 9 \, 845 \, 600 \, 625\)
\(\ds \) \(=\) \(\ds 15 \, 527 \, 402 \, 881\)
\(\ds \) \(=\) \(\ds 353^4\)


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Sources

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