Hilbert-Waring Theorem/Variant Form/Particular Cases/7

Particular Case of the Hilbert-Waring Theorem -- Variant Form: $k = 7$

The Hilbert-Waring Theorem -- Variant Form states that:

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.

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:

$8 \le \map G 3 \le 33$

Historical Note

David Wells reports in his Curious and Interesting Numbers of $1986$ that:

All sufficiently large numbers are the sum of at most $137$ seventh powers.

At time of writing that may have been adequate, but since then the subject has moved on considerably.

The statement has been removed from later editions.