A "not well understood" proof, do you know if one knows by now the conceptual background?: http://www.math.u-bordeaux1.fr/~cohen/sum4cub.ps
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asked Jul 24, 2013 at 7:45
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There is a paper by Philippe Revoy that analyses and expands on this result.
Its abstract reads (the paper itself is in French):
In this note, we do a systematic study of first degree identities $\sum_{i=1}^4 P_i(x)^3 = P x + q$, $P_i \in \mathbb{Z}[x]$, occuring in the four cube problem over $\mathbb{Z}$; we try to explain the difficulties to get identities for numbers $18k + 2$ which were found by Demjanenko and we show we can get a lot of similar identities, so that most integers of that residue class are sum of four cubes unless they are divisible by certain prime numbers, possibly an infinity, and we settle the question using second degree identities.
The paper itself seems to be freely available on seals.ch, the link below should take you there directly:
Ph. Revoy, Sur les sommes de quatre cubes, Enseign. Math 29 (1983), 209-220