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2021 | Book

Preliminaries Read chapter Read first chapter

Integer Sequences

Divisibility, Lucas and Lehmer Sequences

Authors: Masum Billal, Samin Riasat

Publisher: Springer Singapore

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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About this book

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

Frontmatter
Chapter 1. Preliminaries
Abstract
In this chapter, we discuss some topics from algebra that are prerequisites to the theory we will develop. First, we discuss groups, rings, fields, vector spaces, and matrices very briefly in Sects. 1.1 and 1.2.
Masum Billal, Samin Riasat
Chapter 2. Linear Recurrent Sequences
Abstract
In this chapter, we discuss linear recurrent sequences over a field. We give results on when such a sequence is periodic and obtain an upper bound on the length of the period, as well as show how to produce this bound. Finally, we discuss the theory developed by Morgan Ward on the periodicity of such sequences with the help of the double modulus. We will see a lot of results that are of fundamental importance in this theory.
Masum Billal, Samin Riasat
Chapter 3. Divisibility Sequences
Abstract
Consider the classic problem that the product of n consecutive integers is divisible by n!. The proof of this fact is the basis of our study on this topic. A beginner usually tries to prove this with some basic modular arithmetic, for example, at least one of the n consecutive integers is divisible by n since each of them leaves a different remainder upon division by n. Similarly, at least one of those integers is divisible by i for \(1\le i\le n\). However, this does not prove that the product of all \(1\le i\le n\) divides n! as well, although the least common multiple of them \({{\,\mathrm{lcm}\,}}(1,2,\ldots ,n)\) does.
Masum Billal, Samin Riasat
Chapter 4. Lucas Sequences
Abstract
This chapter discusses generalizations on Lucas sequence. We will establish some results regarding general Lucas sequences and find out when a Lucas sequence is divisible. The usual Fibonacci sequence \(1,1,2,3,5,8,\ldots \) is a special case of Lucas sequence. Therefore, pretty much every theorem discussed in this chapter along with the results in Chapter 3 are applicable to Fibonacci numbers.
Masum Billal, Samin Riasat
Chapter 5. Lehmer Sequences
Abstract
In this chapter, we discuss a generalization of Lucas sequences due to D. H. Lehmer.
Masum Billal, Samin Riasat
Chapter 6. On Primitive Divisors
Abstract
This chapter is dedicated to investigating special divisors of the sequences we have discussed so far. More specifically, we are mostly interested in primitive divisors of such sequences. We will first see a chronological history of this literature and then go through each one of them sequentially. We will also show some results regarding the size of primitive divisors and the number of distinct primitive divisors.
Masum Billal, Samin Riasat
Chapter 7. Exercises
Abstract
This book is not fully committed to Olympiad purposes. However, a lot of the theory in this topic can be used for Olympiad purposes and that has been the case for a while. So we will discuss some problems and see how to solve them.
Masum Billal, Samin Riasat
Backmatter
Metadata
Title
Integer Sequences
Authors
Masum Billal
Samin Riasat
Copyright Year
2021
Publisher
Springer Singapore
Electronic ISBN
978-981-16-0570-3
Print ISBN
978-981-16-0569-7
DOI
https://doi.org/10.1007/978-981-16-0570-3

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