Number Theory and its Applications

Table of contents

1. Cover Page
2. Title Page
3. Dedication
4. Copyright Page
5. Preface
6. Table of Contents
7. 1 Prerequisites
8. 2 Theory of Divisibility
   1. 2.1 Introduction
   2. 2.2 Divisibility
   3. 2.3 Worked out Exercises
      1. 2.4 Greatest Common Divisor
   4. 2.5 Least Common multiple
   5. 2.6 Worked out Exercises
   6. 2.7 Linear Diophantine Equations
   7. 2.8 Worked out Exercises
   8. 2.9 Exercises:
9. 3 Prime Numbers
   1. 3.1 Introduction
   2. 3.2 Primes & Fundamental Theorem of Arithmetic
   3. 3.3 Worked out Exercises
   4. 3.4 Exercises:
10. 4 Theory of Congruences
    1. 4.1 Introduction
    2. 4.2 Congruences
    3. 4.3 Worked out Exercises
    4. 4.4 Linear Congruences
    5. 4.5 Worked out Exercises
    6. 4.6 System of Linear Congruences
    7. 4.7 Worked out Exercises
    8. 4.8 Exercises:
11. 5 Fermat’s Little Theorem
    1. 5.1 Introduction
    2. 5.2 Fermat’s Little Theorem
    3. 5.3 Worked out Exercises
    4. 5.4 Wilson’s Theorem
    5. 5.5 Worked out Exercises
    6. 5.6 Exercises:
12. 6 Arithmetic Functions
    1. 6.1 Introduction
    2. 6.2 The Sum and Number of Divisors
    3. 6.3 Worked out Exercises
    4. 6.4 Mobüs μ-function
    5. 6.5 Worked out Exercises
    6. 6.6 Greatest Integer function
    7. 6.8 Worked out Exercises
    8. 6.9 Exercises:
13. 7 Euler’s Generalization and ϕ-function
    1. 7.1 Introduction
    2. 7.2 Euler’s ϕ – function
    3. 7.3 Worked out Exercises
    4. 7.4 Euler’s Theorem
    5. 7.5 Worked out Exercises
    6. 7.6 Properties of ϕ – function
    7. 7.7 Worked out Exercises
    8. 7.8 Exercises:
14. 8 Primitive Roots
    1. 8.1 Introduction
    2. 8.2 Multiplicative Order
    3. 8.3 Worked out Exercises
    4. 8.4 Primitive Roots for Primes
    5. 8.5 Worked out Exercises
    6. 8.6 Existence of Primitive Roots
    7. 8.7 Worked out Exercises
    8. 8.8 Index Arithmetic
    9. 8.9 Worked out Exercises
    10. 8.10 Exercises:
15. 9 Theory of Quadratic Residues
    1. 9.1 Introduction
    2. 9.2 Quadratic Residues and Nonresidues
    3. 9.3 Worked out Exercises
    4. 9.4 Quadratic Reciprocity Law
16. 10 Integers of Special Forms
    1. 10.1 Introduction
    2. 10.2 Perfect Numbers
    3. 10.3 Worked out Exercises
    4. 10.4 Mersenne Primes
    5. 10.5 Worked out Exercises
    6. 10.6 Fermat Numbers
    7. 10.7 Worked out Exercises
    8. 10.8 Exercises:
17. 11 Continued Fractions
    1. 11.1 Introduction
    2. 11.2 Finite Continued Fractions
    3. 11.3 Worked out Exercises
    4. 11.4 Infinite Continued Fractions
    5. 11.5 Worked out Exercises
    6. 11.6 Periodic Fractions
    7. 11.7 Worked out Exercises
    8. 11.8 Exercises:
18. 12 Few Non-Linear Diophantine Equations
    1. 12.1 Introduction
    2. 12.2 Pythagorean Triples
    3. 12.3 Worked out Exercises
    4. 12.4 Fermat’s Last Theorem
    5. 12.5 Worked out Exercises
    6. 12.6 Exercises:
19. 13 Integers as Sums of Squares
    1. 13.1 Introduction
    2. 13.2 Sum of Two Squares
    3. 13.3 Worked out Exercises
    4. 13.4 Sum of More than Two Squares
    5. 13.5 Worked out Exercises
    6. 13.6 Exercises:
20. 14 Certain Applications on Number Theory
    1. 14.1 Fibonacci Numbers
    2. 14.2 Worked out Exercises
       1. 14.3 Pseudo-random Numbers
    3. 14.4 Worked out Exercises
    4. 14.5 Cryptology
    5. 14.6 Worked out Exercises
    6. 14.7 Exercises:
21. Bibliography
22. Index