Abstract
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.
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Editors note—The paper was received by CALCOLO as indicated. However, the Editor is aware of the following circumstances: Professor E. Bombieri presented the paper for publication in Mathematics of Computation in January 1980; the arrival of the paper was acknowledged but no action followed; no referee's comment has ever been received since by the Author, although such comments were repeatedly solicited. finally the Editor has agreed to publish the paper in CALCOLO.
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Romani, F. Computations concerning Waring's problem for cubes. Calcolo 19, 415–431 (1982). https://doi.org/10.1007/BF02575769
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DOI: https://doi.org/10.1007/BF02575769