Integer as Sum of Seven Positive Cubes
Contents
Theorem
Every sufficiently large integer can be expressed as the sum of no more than $7$ positive cubes.
Proof
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Also see
Historical Note
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.
Sources
- 1943: U.V. Linnik: On the representation of large numbers as sums of seven cubes (Mat. Sb. N.S. Vol. 12 (54): pp. 218 – 224)
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $7$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $7$
