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Title Details:
Number Theory and Applications
Authors: Antoniadis, Ioannis
Kontogeorgis, Aristeidis
Reviewer: Theochari Apostolidou, Theodora
Subject: MATHEMATICS AND COMPUTER SCIENCE > MATHEMATICS > NUMBER THEORY
Keywords:
Prime Numbers
Diophantine Equations
Congruences
Quadratic Residues
Law Of Quadratic Reciprocity
Primitive Roots
Indexes
Primality Testing
Factorization
Continued Fractions
Pell Equation
Quadratic Forms
Quadratic Number Fields
Coding Theory
Integral Basis
Discriminant
Fundamental Unit
Decomposition Law
Class Number
Pseudoprimes
Carmichael Numbers
Jacobi Symbol
L-series
Legendre Symbol
Fibonacci And Lucas Numbers
Cryptography
Description:
Abstract:
This book is an introduction to number theory. Emphasis has been given to the historical development of the ideas and to applications. The book covers the teaching needs of all Mathematics departments in Greece. Until the ninth chapter only elementary tools are needed.
Last chapter requires some knowledge of algebra and Galois theory.

Number theory for centuries was considered as part of "pure mathematics". The last 35 years many applications of number theory to cryptography and coding theory were discovered, and some of these applications are explained.

The book is divided into two parts. The first part is devoted to the arithmetic of natural numbers and the second part to the arithmetic of irrational quantities.
Table of Contents:
Arithmetic of Rational number
Divisibility and primes
1.1 Integers
1.2 Divisibility
1.3 Prime numbers
1.4 Bertrand's principle
1.5 GCD and LCM
1.6 Euclid's algorithm
1.7 Main theorem of arithmetic
Diophantine equations
2.1 Introduction
2.2 Linear Diophantine equations
2.3 Pythagorean triples
3 Named Integers
3.1 Friends
3.2 Perfect numbers
3.3 Factorization and cryptography
4 Congruences
4.1 Introducion
4.2 Phi function
4.3 Systems of congruences
4.4 Applications
4.5 Raising in power and square root
4.6 Cryptography
4.7 Higher congruences
4.8 Factorization
4.9 Algorithms
5 Quadratic residues
5.1 Quadratic congruences
5.2 Quadratic reciprocity law
5.3 Composites
5.4 n-residues, roots, indices
II Irrational numbers
6 Fibonacci numbers
6.1 Fibonacci numbers
6.2 Lucas Numbrs
6.4 Lucas sequences
7 Continued fractions
7.1 Continued fractions of rationals
7.2 Convergents
7.3 Linear equations
7.4 Continued freactions of reals
7.5 Best aproximation
7.6 Congruent numbers
7.7 Periodic continued fractions
7.8 Factorization
7.9 Continued fraction of e
7.10 Historical notes
8 Η Pell's equation
8.1 Introduction
8.2 Pell's equation
8.3 Generalized Pell's equation
8.4 Historical notes
9 Quadratic forms
9.1 Introduction
9.2 Equivalent forms
9.3 Representation of integes
9.4 Number of representations
9.5 Historical notes
10 Quadratic number fields
10.1 Arithmetic of Gaussian Integers
10.2 Algebraic integers
10.3 Basis and discriminant
10.4 The group of units
10.5 Decomposition law
10.6 Ideals and class number
10.7 Applications
Linguistic Editors: Kalliaras, Dimitris
Type: Undergraduate textbook
Creation Date: 2015
Item Details:
ISBN 978-618-82124-5-9
License: http://creativecommons.org/licenses/by-nc-nd/3.0/gr
Handle http://hdl.handle.net/11419/107
Bibliographic Reference: Antoniadis, I., & Kontogeorgis, A. (2015). Number Theory and Applications [Undergraduate textbook]. Kallipos, Open Academic Editions. http://hdl.handle.net/11419/107
Language: Greek
Consists of: 1. Divisibility and prime numbers
2. Diopantine Equations
3. Special Integers, cryptography and coding theory
4. Congruences
5. Quadratic residues , roots, indices and applications
6. Fibonacci numbers
7. Pell's equation
8. Quadratic forms
9. Quadratic number fields
10. Continuous Fractions
11. Number Theory and Applications: Annex A
12. Number Theory and Applications: Annex B
Publication Origin: Kallipos, Open Academic Editions