# Hilbert-Waring Theorem/Particular Cases

## Contents

## Particular Cases of the Hilbert-Waring Theorem

The Hilbert-Waring Theorem states that:

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.

### Hilbert-Waring Theorem: $k = 2$

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

- $g \left({2}\right) = 4$

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:

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