# Hilbert-Waring Theorem/Variant Form

## Contents

## 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.