Kummer's congruences for classical numbers

Abstract

In this thesis, we study a special kind of congruences, called Kummer's congruences, which is here classified into three types, namely, Kummer's congruences of a zeroth kind, Kummer's congruences of a first kind and Kummer's congruences of a second kind. Some interesting aspects concerning congruences of each type and their complexity including some relationships among those three types are studied. A crucial identity for Bernoulli numbers employed to prove their Kummer's congruential properties is generalized.