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

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## Theorem

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

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

This sequence is A046050 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

## Proof

On a case-by-case basis:

From Smallest Number not Expressible as Sum of Fewer than 19 Fourth Powers:

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

From 159 is not Expressible as Sum of Fewer than 19 Fourth Powers:

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

From 239 is not Expressible as Sum of Fewer than 19 Fourth Powers:

- $239 = 13 \times 1^4 + 4 \times 2^4 + 2 \times 3^4$

From 319 is not Expressible as Sum of Fewer than 19 Fourth Powers

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

or:

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

From 399 is not Expressible as Sum of Fewer than 19 Fourth Powers

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

or:

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

From 479 is not Expressible as Sum of Fewer than 19 Fourth Powers

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

or:

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

From 559 is not Expressible as Sum of Fewer than 19 Fourth Powers

- $559 = 15 \times 1^4 + 2 \times 2^4 + 2 \times 4^4$

or:

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

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## Also see

## 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 $10^{220}$.

## Sources

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $559$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $559$