# Hilbert-Waring Theorem/Variant Form/Particular Cases/4

## Particular Case of the Hilbert-Waring Theorem -- Variant Form: $k = 4$

The Hilbert-Waring Theorem -- Variant Form states that:

For each $k \in \Z: k \ge 2$, there exists a positive integer $G \left({k}\right)$ such that every sufficiently large positive integer can be expressed as a sum of at most $G \left({k}\right)$ $k$th powers.

The case where $k = 4$ is:

Every sufficiently large positive integer can be expressed as the sum of at most $16$ $4$th powers.

That is:

- $\map G 4 = 16$

## Proof

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## Also see

## Historical Note

Harold Davenport showed in $1939$ that $G \left({4}\right) = 16$.

$13792$ is the largest number to require $17$ fourth powers.

Jean-Marc Deshouillers, François Hennecart and Bernard Landreau showed in $2000$ that every number between $13793$ and $10^{245}$ requires at most $16$.

Koichi Kawada, Trevor Dion Wooley and Jean-Marc Deshouillers extended Harold Davenport's $1939$ result to show that every number above $10^{220}$ requires no more than $16$.

## Sources

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $16$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $16$