# 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 ProgressIn 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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