# Numbers not Expressible as Sum of Fewer than 19 Fourth Powers It has been suggested that this page or section be merged into Positive Integers not Expressible as Sum of Fewer than 19 Fourth Powers. (Discuss)

## 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 $10^{220}$.