Positive Integers not Expressible as Sum of Fewer than 19 Fourth Powers

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Theorem

The following positive integers are the only ones which cannot be expressed as the sum of fewer than $19$ fourth powers:


$79$ as Sum of $19$ Fourth Powers

$79 = 15 \times 1^4 + 4 \times 2^4$


$159$ as Sum of $19$ Fourth Powers

$159 = 14 \times 1^4 + 4 \times 2^4 + 3^4$


$319$ as Sum of $19$ Fourth Powers

$319 = 15 \times 1^4 + 3 \times 2^4 + 4^4$

or:

$319 = 12 \times 1^4 + 4 \times 2^4 + 3 \times 3^4$


$399$ as Sum of $19$ 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