  The purpose of this memoir is to discuss two very interesting properties of
integer sequences. One is the law of apparition and the other is the law of
repetition. Both have been extensively studied by mathematicians such as Ward,
Lucas, Lehmer, Hall, etc. However, due to the lack of a proper survey in this
area, many results have been rediscovered many decades later. This along with
the necessity of the usefulness of such theory calls for a survey on this
topic.
