Integer as Sum of Seven Positive Cubes/Historical Note

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Historical Note on Integer as Sum of Seven Positive Cubes

Edward Waring knew that some integers required at least $9$ positive cubes to represent them as a sum:

\(\displaystyle 23\) \(=\) \(\displaystyle 2^3 + 2^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3\)
\(\displaystyle 239\) \(=\) \(\displaystyle 4^3 + 4^3 + 3^3 + 3^3 + 3^3 + 3^3 + 1^3 + 1^3 + 1^3\)

In fact these are the only two integers that need as many as $9$ positive cubes to express them.

All other integers need no more than $8$.

It was shown in $1943$ by Yuri Vladimirovich Linnik that only finitely many numbers do require $8$ positive cubes.

That is, from some point on, $7$ cubes are enough.

It is not known what that point is.