  We derive the perturbative five loop anomalous dimension of the Konishi
operator in N = 4 SYM theory from the integrable string sigma model by
evaluating finite size effects using Luscher formulas adapted to multimagnon
states at weak coupling. In addition, we derive the five loop wrapping
contribution for the L=2 single impurity state in the beta deformed theory,
which may be within reach of a direct perturbative computation. The Konishi
expression exhibits two new features - a modification of Asymptotic Bethe
Ansatz quantization and sensitiveness to an infinite set of coefficients of the
BES/BHL dressing phase. The result satisfies nontrivial self-consistency
conditions - simple transcendentality structure and cancellation of mu-term
poles. It may be a testing ground for the proposed AdS/CFT TBA systems.
