Hilbert-Waring Theorem/Historical Note

Historical Note on Hilbert-Waring Theorem

The Hilbert-Waring Theorem was conjectured for $k = 3$ and $k = 4$ by Edward Waring in $1770$, in his Meditationes Algebraicae, and was generally referred to as Waring's problem.

It was proved by David Hilbert in $1909$.


The assertion is that for each $k$ there exist such a number $\map g k$.

The problem remains to determine what that $\map g k$ actually is.


Partial Resolution

It was determined in $1990$ by Jeffrey M. Kubina and Marvin Charles Wunderlich that for every $k \le 471 \, 600 \, 000$, the value of $\map g k$ is given by the formula:

$\map g k = \floor {\paren {\dfrac 3 2}^k} + 2^k - 2$

It is suspected that it is true for all $k$, but this still remains to be proved.


Sources

  • 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $9$
  • 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $9$
  • 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Waring's problem
  • 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Waring's problem
  • 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): Waring's problem
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