# Density of Prime Divisors of Linear Recurrences

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*Christian Ballot*

A result due to Hasse says that, on average, 17 out of 24 consecutive primes will divide a number in the sequence \(U_n = 2^n+1\). There are few sequences of integers for which this relative density can be computed exactly. In this work, Ballot links Hasse's method to the concept of the group associated with the set of second-order recurring sequences having the same characteristic polynomial and to the concept of the rank of prime division in a Lucas sequence. This combination of methods and ideas allows the establishment of new density results. Ballot also shows that this synthesis can be generalized to recurring sequences of any order, for which he also obtains new density results. All the results can be shown to be in close agreement with the densities computed using only a small set of primes. This well-written book is fairly elementary in nature and requires only some background in Galois theory and algebraic number theory.

#### Table of Contents

# Table of Contents

## Density of Prime Divisors of Linear Recurrences

- CONTENTS v6 free
- ABSTRACT vii8 free
- INTRODUCTION 110 free
- CHAPTER 1 GENERAL PRELIMINARIES 716
- CHAPTER 2 BACKGROUND MATERIAL 1221
- CHAPTER 3 MORE ABOUT RECURRING SEQUENCES OF ORDER TWO 2433
- CHAPTER 4 A STUDY OF THE CUBIC CASE 3443
- CHAPTER 5 STUDY OF THE GENERAL CASE m ≥ 2 7079
- APPENDIX A - LIST OF THEOREMS 96105
- APPENDIX B - LIST OF SYMBOLS 98107
- REFERENCES 100109