Abstract We present significant numerical evidence, based on the entropy analysis by lumping of the binary expansion of certain values of the Gamma function, that some of these values correspond to incompressible algorithmic information. In particular, the value Γ(1/5) corresponds to a peak of non-compressibility as anticipated on a priori grounds from number-theoretic considerations. Other fundamental constants are similarly considered. This work may be viewed as an invitation for other researchers to apply information theoretic and decision theory techniques in number theory and analysis
