# Numbers not Expressible as Sum of Fewer than 19 Fourth Powers

## Theorem

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

$79, 159, 239, 319, 399, 479, 559$

## Proof

On a case-by-case basis:

$79 = 15 \times 1^4 + 4 \times 2^4$
$159 = 14 \times 1^4 + 4 \times 2^4 + 3^4$
$239 = 13 \times 1^4 + 4 \times 2^4 + 2 \times 3^4$
$319 = 15 \times 1^4 + 3 \times 2^4 + 4^4$

or:

$319 = 12 \times 1^4 + 4 \times 2^4 + 3 \times 3^4$
$399 = 14 \times 1^4 + 3 \times 2^4 + 3^4 + 4^4$

or:

$399 = 11 \times 1^4 + 4 \times 2^4 + 4 \times 3^4$
$479 = 13 \times 1^4 + 3 \times 2^4 + 2 \times 3^4 + 4^4$

or:

$479 = 10 \times 1^4 + 4 \times 2^4 + 5 \times 3^4$
$559 = 15 \times 1^4 + 2 \times 2^4 + 2 \times 4^4$

or:

$559 = 9 \times 1^4 + 4 \times 2^4 + 6 \times 3^4$

## Historical Note

It was noted by Leonard Eugene Dickson that there are no other positive integers less than $4100$ needing $19$ fourth powers to express them.

This limit was reported by David Wells in his Curious and Interesting Numbers of $1986$.

Jean-Marc Deshouillers, François Hennecart and Bernard Landreau extended this to $10^{245}$ in $2000$.

In $2005$, Koichi Kawada, Trevor Dion Wooley and Jean-Marc Deshouillers showed that the sequence is complete beyond $10^{220}$.