Okay, I'm not sure if this is the right way to go about this question, could someone please check it's validity.

I conjecture that all perfect cubes are congruent to either

Let be an integer.

Then in we have:

Now let be any perfect cubes.

Thus

However the least absolute residue is unaccounted for. Since there are infinitely many numbers congruent to then there are there are infinitely many positive integers which cannot be represented as a sum of three perfect cubes.