  We perform measurements of the third moment of atom number fluctuations in
small slices of a very elongated weakly interacting degenerate Bose gas. We
find a positive skewness of the atom number distribution in the ideal gas
regime and a reduced skewness compatible with zero in the quasi-condensate
regime. For our parameters, the third moment is a thermodynamic quantity whose
measurement constitutes a sensitive test of the equation of state and our
results are in agreement with a modified Yang-Yang thermodynamic prediction.
Moreover, we show that the measured skewness reveals the presence of true three
body correlations in the system.
