Fundamental Perceptions in Contemporary Number Theory


J. Kannan, M.Sc., M.Phil., PhD – Assistant Professor, Department of Mathematics, Ayya Nadar Janaki Ammal College (Autonomous), Sivakasi, Tamil Nadu, India
Manju Somanath, M.Sc., M.Phil., PhD – Assistant Professor, PG and Research Department of Mathematics, National College (Autonomous), Trichy, Tamil Nadu, India

Series: Computational Mathematics and Analysis
BISAC: MAT022000 MAT000000; MAT002000

The current state and future directions of numerous facets of contemporary number theory are examined in this book from a unified standpoint. The theoretical foundations of contemporary theories are unveiled as a consequence of simple challenges. Additionally, this book makes an effort to present the contents as simply as possible. It is primarily intended for novice mathematicians who have tried reading other works but have struggled to comprehend them due to their complexity.

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Table of Contents


About the Authors


1. Divisibility
1.1 Preliminaries
1.2 Division Algorithm
1.3 GCD, LCM and Euclidean Algorithm
1.4 The Fundamental Theorem of Arithmetic

2. Classical Functions of Number Theory
2.1 Arithmetic Functions
2.2 Some Classical Arithmetic Functions
2.2.1 The Mobius Function
2.2.2 The Euler Totient Function
2.2.3 The Sum and Number of Divisors
2.3 Greatest Integer Function

3. Theory of Congruences
3.1 Basic Properties of Congruences
3.2 Divisibility Tests
3.3 Theory of Residues
3.4 Linear Congruences
3.5 Congruences of Higher Degree
3.6 Fermat- Little Theorem and its Applications

4. Primitive Roots and Indices
4.1 Order of an Integer
4.2 Primitive Roots
4.3 Primitive Root Theorem
4.4 Theory of Indices

5. Quadratic Reciprocity
5.1 Quadratic Residues and Non Residues
5.2 Legendre Symbol and its Properties
5.3 Jacobi Symbol and its Properties
5.4 Quadratic Reciprocity Law

6. Special Numbers
6.1 Perfect Numbers
6.2 Mersenne Numbers
6.3 Amicable Numbers
6.4 Fermat Numbers
6.5 Pell Numbers

7. Waring’s Problem
7.1 Sum of Two Squares
7.2 Difference of Two Squares
7.3 Sum of Three Squares
7.4 Sum of Four Squares
7.5 Waring’s Problem



Author’s ORCID iD

J. Kannan0000-0001-6197-2119
Manju Somanath

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