Hilbert-Waring Theorem/Variant Form

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Theorem

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.


Particular Cases

Hilbert-Waring Theorem -- Variant Form: $k = 2$

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

$G \left({2}\right) = 4$

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


Hilbert-Waring Theorem -- Variant Form: $k = 3$

The case where $k = 3$ is:

Every sufficiently large positive integer can be expressed as the sum of a number of positive cubes.


The exact number is the subject of ongoing research, but at the time of writing ($11$th February $2017$) it is known that it is between $4$ and $7$.


That is:

$4 \le G \left({3}\right) \le 7$


Hilbert-Waring Theorem -- Variant Form: $k = 4$

The case where $k = 4$ is:

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

That is:

$\map G 4 = 16$


Hilbert-Waring Theorem -- Variant Form: $k = 7$

The case where $k = 7$ is:

Every sufficiently large positive integer can be expressed as the sum of a number of positive $7$th powers.


The exact number is the subject of ongoing research, but at the time of writing ($20$th December $2018$) it is known that it is between $8$ and $33$.


That is:

$8 \le \map G 3 \le 33$


Proof


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 $1909$.


Source of Name

This entry was named for David Hilbert and Edward Waring.


Historical Note

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

It was proved by David Hilbert in $1909$.


Its variant form was investigated by Godfrey Harold Hardy and John Edensor Littlewood.

The assertion is that for each $k$ there exist such a number $G \left({k}\right)$.

The problem remains to determine what that $G \left({k}\right)$ actually is.