Successful proofs of irrationality of some fundamental constants are extremely rare and count only a few examples. All these examples are based on the properties of divisibility of numerators and denominators of the obtained rational approximations as well as on the Dirichlet criterion. This criterion also implicitly underlies other less known criteria of irrationality such as, for example, the Brun and Lambert-Legendre criteria. Here we give two irrationality criteria, which generalize the Brun and Dirichlet criteria, and give some examples of their application.

irrationality criteria, Gaussian integers, asymptotics of sequences

Mathematical problems and theory of numerical methods