Contents
Notation vii
Introduction ix
Chapter 1. Definitions and Techniques 1
1.1. Main Definitions and Principal Properties 1
1.2. p-adic Analysis 12
1.3. Linear Forms in Logarithms 15
1.4. Diophantine Approximation and Roth's Theorem 17
1.5. Sums of S-Units 19
Chapter 2. Zeros, Multiplicity and Growth 25
2.1. The Skolem-Mahler-Lech Theorem 25
2.2. Multiplicity of a Linear Recurrence Sequence 26
2.3. Finding the Zeros of Linear Recurrence Sequences 31
2.4. Growth of Linear Recurrence Sequences 31
2.5. Further Equations in Linear Recurrence Sequences 37
Chapter 3. Periodicity 45
3.1. Periodic Structure 45
3.2. Restricted Periods and Artin's Conjecture 49
3.3. Problems Related to Artin's Conjecture 52
3.4. The Collatz Sequence 61
Chapter 4. Operations on Power Series and Linear Recurrence Sequences 65
4.1. Hadamard Operations and their Inverses 65
4.2. Shrinking Recurrence Sequences 71
4.3. Transcendence Theory and Recurrence Sequences 72
Chapter 5. Character Sums and Solutions of Congruences 75
5.1. Bounds for Character Sums 75
5.2. Bounds for other Character Sums 83
5.3. Character Sums in Characteristic Zero 85
5.4. Bounds for the Number of Solutions of Congruences 86
Chapter 6. Arithmetic Structure of Recurrence Sequences 93
6.1. Prime Values of Linear Recurrence Sequences 93
6.2. Prime Divisors of Recurrence Sequences 95
6.3. Primitive Divisors and the Index of Entry 103
6.4. Arithmetic Functions on Linear Recurrence Sequences 109
6.5. Powers in Recurrence Sequences 113
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