Number Theory and Its Applications

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