Hilbert-Waring Theorem/Variant Form

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:

$4 \le \map G 3 \le 7$

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 $16$ $4$th powers.

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

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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 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.