Hilbert-Waring Theorem/Partial Resolution/Examples

Examples of Partial Resolution of Hilbert-Waring Theorem

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 partial resolution 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:

\(\ds \map g 5\)

\(=\)

\(\ds \floor {\paren {\dfrac 3 2}^5} + 2^5 - 2\)

\(\ds \)

\(=\)

\(\ds \floor {\dfrac {243} {32} } + 32 - 2\)

\(\ds \)

\(=\)

\(\ds 7 + 32 - 2\)

\(\ds \)

\(=\)

\(\ds 37\)

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