Hilbert-Waring Theorem/Particular Cases/4/Historical Note

Particular Case of the Hilbert-Waring Theorem: $k = 4$: Historical Note

It is clear that some integers require at least $19$ powers of $4$ to represent them as a sum, as $79$ requires $19$ of them:

Thus Edward Waring knew that:

$\map g 4 \ge 19$

Joseph Liouville subsequently showed that:

$\map g 4 \le 53$

Hardy and Littlewood showed that all sufficiently large numbers are the sum of at most $19$ fourth powers.

Finally it was shown in $1986$ by Ramachandran Balasubramanian, Jean-Marc Deshouillers and François Dress that $\map g 4 \le 19$.

Thus:

$\map g 4 = 19$

and so Waring's claim has been shown to be true.

Source of Name

This entry was named for David Hilbert and Edward Waring.

Sources

• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $9$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $9$
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Waring's problem