# Math Help - Sum of perfect cubes

1. ## Sum of perfect cubes

Prove there are infinitely many positive integers which cannot be represented as a sum of three perfect cubes

I conjecture that all perfect cubes are congruent to either $0, \pm 1 \pmod {9}$

Let $x$ be an integer.

Then in $\mathbb{Z}_9$ we have:

$x \equiv 0, \pm 1, \pm 2, \pm 3, \pm 4 \pmod {9}$

$\implies x^3 \equiv 0, \pm 1, \pm 8, \pm 27, \pm 64 \pmod {9}$

$\implies x^3 \equiv 0, \pm 1 \pmod {9}$

Now let $a^3, b^3, c^3$ be any $3$ perfect cubes.

Thus $a^3+b^3+c^3 \equiv 0, \pm 1, \pm 2, \pm 3 \pmod {9}$

However the least absolute residue $\pm 4$ is unaccounted for. Since there are infinitely many numbers congruent to $\pm 4 \pmod {9}$ then there are there are infinitely many positive integers which cannot be represented as a sum of three perfect cubes.

3. Usagi Killer is correct. Neat method!