Computational Methods and Function Theory vol:10 issue:2 pages:585-602

Abstract:

We will describe a method for proving that a given real number is irrational. It amounts to constructing explicit rational approximants to the real number which are "better than possible" should the real number be rational. The rational approximants are obtained by evaluating a Hermite-Padé rational approximant to explicit functions at
a convenient (integer) point. This constructive proof can also be used to prove linear independence over Q of some real numbers and in fact was first used by Hermite to prove the transcendence of e. The method is illustrated by various examples involving the zeta-function.