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:

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

Work In Progress In particular: It has not been confirmed that these are all of those. A deep literature search is needed. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by completing it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{WIP}} from the code.

Proof

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Also see

• Hilbert-Waring Theorem