Integer as Sum of Seven Positive Cubes/Historical Note
Jump to navigation
Jump to search
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:
\(\ds 23\) | \(=\) | \(\ds 2^3 + 2^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3\) | ||||||||||||
\(\ds 239\) | \(=\) | \(\ds 4^3 + 4^3 + 3^3 + 3^3 + 3^3 + 3^3 + 1^3 + 1^3 + 1^3\) |
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.