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We investigate whether the (planar, two complex scalar) dilatation operatorof N=4 gauge theory can be, perturbatively and, perhaps, non-perturbatively,described by an integrable long range spin chain with elliptic exchangeinteraction. Such a chain was introduced some time ago by Inozemtsev. In thelimit of sufficiently ``long'' operators a Bethe ansatz exists, which we applyat the perturbative two- and three-loop level. Spectacular agreement is foundwith spinning string predictions of Frolov and Tseytlin for the two-loopenergies of certain large charge operators. However, we then go on to show thatthe agreement between perturbative gauge theory and semi-classical stringtheory begins to break down, in a subtle fashion, at the three-loop level. Thiscorroborates a recently found disagreement between three-loop gauge theory andnear plane-wave string theory results, and quantitatively explains a previouslyobtained puzzling deviation between the string proposal and a numericalextrapolation of finite size three-loop anomalous dimensions. At four loops andbeyond, we find that the Inozemtsev chain exhibits a generic breakdown ofperturbative BMN scaling. However, our proposal is not necessarily limited toperturbation theory, and one would hope that the string theory results can berecovered from the Inozemtsev chain at strong 't Hooft coupling.Comment: 31 pages, no figure; v1: one reference added, minor changes; v2:  slightly extended discussion of rapidity, references adde

PlanarN=4 gauge theory and the Inozemtsev long range spin chain