# Integer as Sum of Seven Positive Cubes

## Theorem

Every sufficiently large integer can be expressed as the sum of no more than $7$ positive cubes.

## 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. LinnikOn the representation of large numbers as sums of seven cubes (Mat. Sb. N.S. Vol. 12 (54): pp. 218 – 224)