399 is not Expressible as Sum of Fewer than 19 Fourth Powers

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Theorem

$399$ cannot be expressed as the sum of fewer than $19$ fourth powers:

$399 = 14 \times 1^4 + 3 \times 2^4 + 3^4 + 4^4$

or:

$399 = 11 \times 1^4 + 4 \times 2^4 + 4 \times 3^4$


Proof


Also see


Sources