Computations Concerning Waring's Problem for Cubes

Waring's problem for cubes is investigated using numerical computations. The densities of number not representable as a sum of four and five cubes are computed in large intervals. Extrapolation of these data allows us to conjecture on the order of magnitude of last exceptions. The representability with four relative cubes and the validity of the theoretical asymptotic formula are investigated too. All the results reasonably confirm the conjecture that four relative cubes suffice to represent any integer and four nonnegative cubes suffice to represent any «large» integer.