# Integer as Sum of Seven Positive Cubes

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## Contents

## Theorem

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

## Proof

## 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)**: 218 – 224)

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $7$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $7$