Integer Sequences: Divisibility, Lucas and Lehmer Sequences

✍ Scribed by Masum Billal, Samin Riasat

• Publisher: Springer
• Year: 2021
• Tongue: English
• Leaves: 173
• Edition: 1st ed. 2021
• Category: Library

✦ Synopsis

This book discusses special properties of integer sequences from a unique point of view. It generalizes common, well-known properties and connects them with sequences such as divisible sequences, Lucas sequences, Lehmer sequences, periods of sequences, lifting properties, and so on. The book presents theories derived by using elementary means and includes results not usually found in common number theory books. Considering the impact and usefulness of these theorems, the book also aims at being valuable for Olympiad level problem solving as well as regular research. This book will be of interest to students, researchers and faculty members alike.

✦ Table of Contents

Preface
References
Who This Book Is For
Contents
About the Authors
1 Preliminaries
1.1 Groups, Rings, Fields, and Vector Spaces
1.1.1 Groups
1.1.2 Rings and Fields
1.1.3 Vector Spaces
1.2 Matrices
1.3 Polynomials
1.4 Cyclotomic Polynomials
References
2 Linear Recurrent Sequences
2.1 Introduction
2.2 Periodicity
2.3 Fundamental Theorems on Periodicity
2.4 Finding Pre-Period and Least Period
References
3 Divisibility Sequences
3.1 Introduction
3.2 Characterization of Divisibility Sequences
3.3 Binomial Coefficients
References
4 Lucas Sequences
4.1 Introduction
4.2 Divisibility of Second-Order Lucas Sequences
4.3 Lucas Sequence of Arbitrary Order
References
5 Lehmer Sequences
5.1 Introduction
5.2 Divisibility of Lehmer Sequence
5.3 Periodicity of Lehmer Sequence
5.4 Extending Wilson's Theorem
5.5 LCM Sequence of Lehmer Sequences
5.6 Subsequence of Lehmer Sequences
References
6 On Primitive Divisors
6.1 The History of Primitive Divisors
6.2 Primitive Divisors of an-bn
6.3 Primitive Divisors of Real Lucas Sequences
6.4 Primitive Divisors of Real Lehmer Sequences
6.5 Primitive Divisors of Complex Lehmer Sequences
References
7 Exercises
Reference
Appendix Glossary
References
Index