# Hilbert-Waring Theorem

## 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 $\map g k$ $k$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 Theoremâ€Ž:

- $\map g 2 = 4$

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:

- $g \left({5}\right) = 37$

### 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:

- $g \left({7}\right) = 143$

## 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 $1909$.

## 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 $1909$.

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

- 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next):**Waring's problem** - 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**