Hilbert-Waring Theorem/Partial Resolution

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 $\map g k$ positive $k$th powers.

Partial Resolution

It was determined in $1990$ by Jeffrey M. Kubina and Marvin Charles Wunderlich that for every $k \le 471 \, 600 \, 000$, the value of $\map g k$ is given by the formula:

$\map g k = \floor {\paren {\dfrac 3 2}^k} + 2^k - 2$

It is suspected that it is true for all $k$, but this still remains to be proved.

Proof

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Examples

Example: $k = 5$

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 $\map g k$ positive $k$th powers.

The case where $k = 5$ is:

Every positive integer can be expressed as the sum of at most $37$ positive fifth powers.

That is:

$\map g 5 = 37$

Proof

From the of the Hilbert-Waring theorem:

It was determined in $1990$ by Jeffrey M. Kubina and Marvin Charles Wunderlich that for every $k \le 471 \, 600 \, 000$, the value of $\map g k$ is given by the formula:

$\map g k = \floor {\paren {\dfrac 3 2}^k} + 2^k - 2$

It is suspected that it is true for all $k$, but this still remains to be proved.

For $k = 5$, we have:

which is the result established by Jingrun Chen in $1964$.

Source of Name

This entry was named for David Hilbert and Edward Waring.

Sources

• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Waring's problem