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Is there some nice characterization of natural numbers that are sum of two nonnegative integer cubes? What about three cubes? Four?

This seems quite hopeless to me, surely something is known about these problems and even if there are no necessary and sufficient conditions for a number to be a sum of two, or three, or four cubes, I think that there are at least either necessary ones, or sufficient ones.

Thanks for help.

This question arose as a result of a little bit of thinking on additive bases and an open question can every sufficiently large natural number be written as the sum of four nonnegative integer cubes.

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No simple characterization according to OEIS:

A003325 Numbers that are the sum of 2 positive cubes.

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