Hilbert-Waring Theorem

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Theorem

For each $k \in \Z: k \ge 2$, there exists a positive integer $\map g k$ such that every positive integer can be expressed as a sum of at most $\map g k$ positive $k$th powers.


Sequence

The integer sequence of values of $\map g k$ begins:

$1, 4, 9, 19, 37, 73, 143, 279, 548, 1079, 2132, 4223, 8384, 16673, 33203, 66190, 132055, \ldots$


Particular Cases

Hilbert-Waring Theorem: $k = 2$

The case where $k = 2$ is proved by Lagrange's Four Square Theorem‎:

$\map g 2 = 4$

That is, every positive integer can be expressed as the sum of at most $4$ squares.


Hilbert-Waring Theorem: $k = 3$

The case where $k = 3$ is:

Every positive integer can be expressed as the sum of at most $9$ positive cubes.

That is:

$\map g 3 = 9$


Hilbert-Waring Theorem: $k = 4$

The case where $k = 4$ is:

Every positive integer can be expressed as the sum of at most $19$ powers of $4$.

That is:

$\map g 4 = 19$


Hilbert-Waring Theorem: $k = 5$

The case where $k = 5$ is:

Every positive integer can be expressed as the sum of at most $37$ positive fifth powers.

That is:

$\map g 5 = 37$


Hilbert-Waring Theorem: $k = 6$

The case where $k = 6$ is:

Every positive integer can be expressed as the sum of at most $73$ positive sixth powers.

That is:

$\map g 6 = 73$


Hilbert-Waring Theorem: $k = 7$

The case where $k = 7$ is:

Every positive integer can be expressed as the sum of at most $143$ positive seventh powers.

That is:

$\map g 7 = 143$


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.


Proof


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Also known as

The Hilbert-Waring Theorem is often referred to as Waring's problem, which was how it was named before David Hilbert proved it in $1909$.

However, Waring's problem is properly used for the particular case $3$ and the particular case $4$.


Source of Name

This entry was named for David Hilbert and Edward Waring.


Historical Note

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.


Sources

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