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Is every integer the sum of four perfect cubes?
The sum of four cubes problem[1] asks whether every integer is the sum of four cubes of integers. It is conjectured the answer is affirmative, but this conjecture has been neither proven nor disproven.[2] Some of the cubes may be negative numbers, in contrast to Waring's problem on sums of cubes, where they are required to be positive.
Partial results edit
The substitutions , , and in the identity
Since every integer is congruent to its own cube modulo 6, it follows that every integer is the sum of five cubes of integers.
In 1966, V. A. Demjanenko proved that any integer that is congruent neither to 4 nor to −4 modulo 9 is the sum of four cubes of integers. For this, he used the following identities:
The problem therefore only arises for integers congruent to 4 or to −4 modulo 9. One example is
18x±2 case edit
According to Henri Cohen's translation of Demjanenko's paper, these identities
See also edit
Notes and references edit
- ^ Referred to as the "four cube problem" in H. Davenport, The Higher Arithmetic: An Introduction to the Theory of Numbers, Cambridge University Press, 7th edition, 1999, p. 173, 177.
- ^ At least in 1982. See Philippe Revoy, “Sur les sommes de quatre cubes”, L’Enseignement Mathématique, t. 29, 1983, p. 209-220, online here or here, p. 209 on the point in question.
- ^ V.A. Demjanenko, "On sums of four cubes", Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, vol. 54, no. 5, 1966, p. 63-69, available online at the site Math-Net.Ru. For a demonstration in French, see Philippe Revoy, “Sur les sommes de quatre cubes”, L’Enseignement Mathématique, t. 29, 1983, p. 209-220, online here or here.