We have compiled the list of Best Reference Books on Number Theory subject. These books are used by students of top universities, institutes and colleges. Here is the full list of best books on Number Theory along with reviews.
Kindly note that we have put a lot of effort into researching the best books on Number Theory subject and came out with a recommended list of best books. The table below contains the Name of these best books, their authors, publishers and an unbiased review of books on “Number Theory” as well as links to the Amazon website to directly purchase these books. As an Amazon Associate, we earn from qualifying purchases, but this does not impact our reviews, comparisons, and listing of these top books; the table serves as a ready reckoner list of these best books.
List of Number Theory Books with author’s names, publishers, and an unbiased review as well as links to the Amazon website to directly purchase these books.
 Basic Number Theory
 Analytic Number Theory
 Computational Number Theory
 Algebraic Number Theory
 Number Theory and Cryptography
1. Basic Number Theory
1. “Introduction to the Theory of Numbers” by W W Adams and L J Goldstein  
2. “A Concise Introduction to the Theory of Numbers” by A Baker
“A Concise Introduction to the Theory of Numbers” Book Review: The book is readerfriendly text, featuring the rudiments of number theory. The number theory has a long and distinct history and the concepts and problems relating to that subject have been of great help in the foundation of many mathematics. The chapters of this book are comprehensive, easytounderstand, and describe all the major topics related to visibility, arithmetic functions, congruences, quadratic residues, quadratic forms, Diophantine approximation, quadratic fields, and Diophantine equations. Each chapter ends with a bunch of exercises. The book explains the basics of number theory in a concise, simple and direct way. Although much of the text is oldfashioned, it incorporates numerous study guides that will encourage the reader to immerse himself in the vast resources available to the subject. The book is based on Professor Baker’s lectures at the University of Cambridge and is designed for undergraduate mathematics students.


3. “An Introduction to the Theory of Numbers” by I Niven and H S Zuckerman
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“An Introduction to the Theory of Numbers” book review: The book is an updated and revised piece of work featuring latest topics and recent developments of number theory. The chapters of this book are broadly based on visibility, congruences, quadratic reciprocity, quadratic forms, functions of number theory, Diophantine equations, farey fractions, simple continued fractions, and prime and multiplicative number theory. The topics like irrational numbers, algebraic numbers, partition function, and sequences of integers are discussed in detail. The treatment of the binomial theorem and techniques of numerical calculation are highlighted in this text. Several problems are included in this book for selfstudy and selfassessment of readers. The fifth edition is one of the most effective levels in number theory, written by worldrenowned mathematicians. The chapters are selfexplanatory for maximum flexibility. The book included new features as extended behaviour of the binomial theorem, techniques of numerical calculation and a section on a public key cryptography. The book also contains numerous sets of problems.


4. “Basic College Mathematics with Early Integers” by Elayn MartinGay
“Basic College Mathematics with Early Integers” Book Review: The book lays an emphasis on conceptual understanding. For better understanding of the readers, many applications and exercises are included in this piece of writing. This package contains a textbook and access kit for MyMathLab / MyStatLab. The Bittinger Worktext Series has transformed the nature of development education through the introduction of purposeful worktexts that provide mathematical concepts one at a time. The book introduces objectivebased worktexts, thus presenting one math concept at a time. The chapters of this book are wellstructured and readerfriendly. The topics featured in this book are described in detail and explained stepbystep. This approach allowed students to gain the basics of each concept before embarking on a series of concepts. Through this review, the book continues to focus on building success with cognitive comprehension, while also supporting students with advanced applications, tests, and new reviews and resources to help them apply and retain their knowledge. MyMathLab offers a variety of homework tools, tutorials, and tests that make it easy to manage your lessons online.


5. “The Trachtenberg Speed System of Basic Mathematics” by Rudolph McShane and Jakow Trachtenberg
“The Trachtenberg Speed System of Basic Mathematics” Book Review: The book aims at introducing ‘The Trachtenberg Speed System’, for dealing with basic math and large sums. The book will be useful for children, struggling with mathematics and the teachers. This book provides a course for basic mathematical skills to deal with large sums before making it easier to increase focus and ability with everyday math. The book describes the shorthand of mathematics and requires only the ability to count from one to eleven. It allows anyone to know numbers and calculations that provide greater speed, ease of use in handling numbers, and increasing accuracy. The chapters of this book cover the topics related to table or no tables, multiplication by the direct method, speed multiplication, addition and the right answer, division, squares and square roots, and algebraic description of the method. It uses fundamental mathematical skills and principles for performing various operations. The book illustrates a set of rules that allow every child to perform multiplication, division, insertion, subtraction, and square roots with consistent accuracy and remarkable speed. A good entry into gaining confidence in numbers.


6. “Basic Algebra” by Anthony W Knapp
“Basic Algebra” Book Review: This book systematically develops algebraic concepts and tools that are important to all mathematicians. Chapter on modern algebra treat groups, rings, fields, modules, and Galois groups are also illustrated with various methods of computation. The book includes many examples and hundreds of problems with tips and complete solutions. The book is an excellent blend of fundamental as well as advanced concepts of algebra. The chapters of this book are based on integers, polynomials, and matrices, vector spaces over Q, R, and C, innerproduct spaces, group and group actions, multilinear algebra, advanced group theory, commutative rings, fields and Galois theory, and noncommutative rings. It also covers applications for science and engineering (e.g. Fourier rapid transformation, error correction code, use of the Jordan canonical form in resolving compatible systems of differential equations, and building interest in mathematical physics) appear in a series of problems. It is suitable as a textbook for undergraduate or first year of algebra graduation, which may be supplemented by some material from Advanced Algebra at the graduate level. It requires the student to familiarize himself only with the matrix algebra, to understand the geometry and the reduction of equal proportions, as well as to know the evidence. The applications of algebra in science and engineering are highlighted in this text. For better understanding of the readers, the book consists of several examples and problems along with the hints of their solutions. The book will be an asset for mathematicians and practitioners.


7. “Basic Essentials of Mathematics” by James T Shea
“Basic Essentials of Mathematics” Book Review: This book describes A basic mathematical system that teaches whole numbers, fractions, and decimal skills in Book 1 and percent, measurement, formulas, arithmetic, measurement, and measurement skills in Book 2, each concludes with a skills test.
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8. “Number Theory” by George E Andrews
“Number Theory” Book Review: The book aims at covering basis principles, key concepts, and major aspects of number theory. This book solves the various problems of maintaining the interest of students at both levels by giving a combinational approach to basic number theory. In studying numerical theory with such a view, mathematics majors survive repetition and are given new details, while other students benefit from the subsequent presence of evidence for multiple theories. The book covered introduction for multiplicativitydivisibility included the topics such as the fundamental theorem of arithmetic, combinatorial and computational number theory, congruences, arithmetic functions, primitive roots and prime numbers. Recent chapters offer a good treatment of quadratic congruences, additions (including partition theory), and geometric number theory. The book offers numerous exercises which provides the opportunity to build number tables with or without a computer. Students can then find themes in tables of such numbers, after which the appropriate theories will appear natural and wellmotivated. The book is enriched with several numerical examples. It will be helpful for the students, seeking a strong foundation for dealing with more advanced issues.


9. “An Introduction to the Theory of Numbers” by G H Hardy and Edward M Wright
“An Introduction to the Theory of Numbers” Book Review: The context here is wrapped up with simple words and with a detailed understanding of all the key topics. The text retains the style and clarity of previous editions making it for undergraduates in mathematics as well as a reference for number theorists. This book is the primary and classical textbook in basic number theory. The book is an updated and revised piece of work, featuring latest topics and recent developments in number theory. It is a wellstructured, selfcontained, and readerfriendly book. The new edition of the book is revised and gives guidance to students through the various milestones and developments in number theory. The book updates covers a chapter by J.H. Silverman on one of the most important developments in number theory. The text here is clear and covers all the important topics like The Series of Primes, Farey Series and a Theorem of Minkowski, Irrational Numbers, Congruences and Residues, General Properties of Congruences, Congruences to Composite Moduli, The Representation of Numbers by Decimals, Continued Fractions, Fermat’s Theorem and its Consequences, Approximation of Irrationals by Rationals, The Fundamental Theorem of Arithmetic in k(l), k(i), and k(p), Some Diophantine Equations, Quadratic Fields, The Arithmetical Functions ø(n), µ(n), *d(n), *s(n), r(n), Generating Functions of Arithmetical Functions, The Order of Magnitude of Arithmetical Functions, Partitions, Kronecker’s Theorem, Geometry of Numbers, Elliptic Curves, etc. It also elaborates modular elliptic curves and their role in the proof of Fermat’s theorem. The book also included the suggestions for further study for the most active reader. The text retains the style and clarity of previous versions making it ideal for students studying mathematics from first year onwards and a valuable reference for all numerical theorists. Each chapter concluded with a series of notes featuring major developments in number theory. The book will be a good resource for undergraduate students of mathematics and the number theorists.


10. “An Adventurer’s Guide to Number Theory” by Richard Friedberg
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“An Adventurer’s Guide to Number Theory” Book Review: The book provides a fair introduction to number theory. This book offers an intelligent introduction, based on the history of number theory, which speaks of number and numerical properties as abstract concepts. The book is designed for students with an understanding of mathematics and the beginnings of algebra. It introduces the early discoveries of number theory, including the work of Pythagoras, Euclid, Diophantus, Fermat, Euler, Lagrange and Gauss. This book encourages students to think about imaginary, playful numbers as they look at subjects such as primes and divisions, quadratic forms and arithmetic and quadratic reciprocity and related theorems. It also included a number of unusual things to challenge and motivate readers: some of the early problems in Diophantus’ Arithmetica, the evidence of Fermat’s Last Theorem 3 and 4, and two testimonies of The Wilson’s Theorem. The book will be valuable for the students and professionals of mathematics.


11. “Number Theory and Its Applications (Developments in Mathematics)” by imusti
“Number Theory and Its Applications 1999th Edition” Book Review: This book covers topics ranging from several aspects of number theory to diophantine approximation. Number theory and its applications have also been discussed.


2. Analytic Number Theory
1. “Algebraic Number Theory” by S Lang
“Algebraic Number Theory” Book Review: This book provides an exposition of the classical basic algebraic and analytic number theory and algebraic numbers. It is useful for future research in algebraic number theory as a theoretically wellcontained book. This book offers an excellent overview of the key theories and outlook of the topic. The book is up to date with the latest development in the field. This book treats cyclotomic fields as being analogues for number fields of the constant field extensions of algebraic geometry. This book is a direct global approach to number fields. This book into topics like Algebraic Integers, Completions, The Different and Discriminant, Cyclotomic Fields, Parallelotopes, The Ideal Function, Adele and Adeles, Elementary Properties of the Zeta Function and Lseries, Norm Index Computations, The Artin Symbol, Reciprocity Law, and Class Field Theory, Hecke’s Proof, Functional Equation, Tate’s Thesis, Density of Primes and Tauberian Theorem, The BrauerSiegel Theorem, Explicit Formulas, etc.


2. “A Course in Arithmetic” by J P Serre
“A Course in Arithmetic” Book Review: This book is resourceful and is easy to understand as it is in simple language and is purely algebraic. The book serves as a classification of quadratic forms over the field of rational numbers. The book covers topics such as Finite Fields, pAdic Fields, Hilbert Symbol, Quadratic Forms over Q p and over Q, Integral Quadratic Forms with Discriminant ± 1, The Theorem on Arithmetic Progressions, Modular Forms etc.


3. “Introduction to Analytic Number Theory” by T Apostol
“Introduction to Analytic Number Theory” Book Review: This book is the crystallization of the topic to undergraduates irrespective of they have or don’t have any knowledge of the topic earlier. The book starts with the most basic properties of the natural integers. The text in the book is resourceful with enormous information which is well presented and easy to read. The book has a good selection of topics which are easy to understand and are nicely structured with exercises at the end of each chapter. This book covers topics like The Fundamental Theorem of Arithmetic, Arithmetical Functions and Dirichlet Multiplication, Averages of Arithmetical Functions, Some Elementary Theorems on the Distribution of Prime Numbers, Congruences, Finite Abelian Groups and Their Characters, Dirichlet’s Theorem on Primes in Arithmetic Progression, Periodic Arithmetical Functions and Gauss Sums, Quadratic Residues and the Quadratic Reciprocity Law, Primitive Roots, Dirichlet Series and Euler Products, Analytic Proof of the Prime Number Theorem, etc.


4. “Analytic Number Theory” by Henryk Iwaniec and Emmanuel Kowalski
“Analytic Number Theory” Book Review: This book teaches the reader to use the subject to establish results. This book has a vast diversity of concepts and methods. The aim of the book is to show the scope of the theory, both in classical and modern directions, and to exhibit its wealth and prospects, beautiful theorems, and powerful techniques. This book is for graduate students. The book balances the clarity, completeness, and generality. This book has exercises and examples which is intended to improve reader understanding and provide additional information. The reader must have a basic idea and covers topics like Arithmetic functions, Elementary theory of prime numbers, Classical analytic theory of Lfunctions, Elementary sieve methods, Bilinear forms and the large sieve, Exponential sums, The Dirichlet polynomials, Zerodensity estimates, Sums over finite fields, Holomorphic modular forms, Spectral theory of automorphic forms, Sums of Kloosterman sums, Primes in arithmetic progressions, The least prime in an arithmetic progression, The Goldbach problem, Effective bounds for the class number, etc.


5. “A Primer of Analytic Number Theory: From Pythagoras to Riemann” by Jeffrey Stopple
“A Primer of Analytic Number Theory: From Pythagoras to Riemann” Book Review: This is an easy to read book that incorporates all the topics essential for detailed knowledge of the subject. The book is aimed to enhance the analytical skills of the reader in course of studying ancient questions on polygonal numbers, perfect numbers and amicable pairs. This book shows how the importance of Riemann zeta function and the significance of the Riemann Hypothesis. The book here develops the basic idea of elementary number theory. The book is included with a series of exercises for better understanding and advanced ideas for representation of the topic which is suitable for undergraduates. The book covers topics like Sums and Differences, Products and Divisibility, Order of Magnitude, Averages, Primes, Basel Problem, Euler’s Product, The Riemann Zeta Function, Stirling’s Formula, Explicit Formula, Pell’s Equation, Elliptic Curves, Analytic Theory of Algebraic Numbers, etc.


6. “Analytic Number Theory: Exploring the Anatomy of Integers” by Jeanmarie De Koninck and Florian Luca
“Analytic Number Theory: Exploring the Anatomy of Integers” Book Review: This book presents a clear introduction to the subject analytic number theory with a very unique focus on the anatomy of integers, that is, on the study of the multiplicative structure of the integers. The book has a collection of problems at the end of each chapter that have been chosen carefully. The book has answers to solved numericals at the end of each chapter which makes it appropriate for readers who want to test their understanding of the topic. The book covers topic like Prime Numbers and Their Properties, The Riemann Zeta Function, Setting the Stage for the Proof of the Prime Number Theorem, The Proof of the Prime Number Theorem, The Global and Local Behavior of Arithmetic Functions, The Fascinating Euler Function, Smooth Numbers, The HardyRamanujan and Landau Theorems, Sieve Methods, Prime Numbers in Arithmetic Progression, Characters and the Dirichlet Theorem, Selected Applications of Primes in Arithmetic Progression, The Index of Composition of an Integer, Preliminary Notions along with solutions to every problem of the chapters at the end.


7. “Analytic Number Theory: An Introductory Course” by Paul Trevier Bateman and Harold G Diamond
“Analytic Number Theory: An Introductory Course” Book Review: This gives a nice introduction to the topic and is enough for a graduate course.The book is well written and is resourceful. This is an introductory book on the topic and the reader is expected to have a basic idea on real analysis, complex analysis, number theory and abstract algebra.There are various exercises throughout the entire book and at end of the each chapter developments of each particular subject or theorem are given together with references. The book focuses on a collection of powerful methods of analysis that yield deep number theoretical estimates. The book covers topics like Calculus of Arithmetic Functions, Summatory Functions, The Distribution of Prime Numbers, An Elementary Proof of the P.N.T., Dirichlet Series and Mellin Transforms, Inversion Formulas, The Riemann Zeta Function, Primes in Arithmetic Progressions, Applications of Characters, Oscillation Theorems, Sieves, Application of Sieves, etc along with appendices.


8. “Geometric and Analytic Number Theory” by Edmund Hlawka and Johannes Schoißengeier
“Geometric and Analytic Number Theory” Book Review: The book presents the information in a unique way that the reader can not only know how to solve it but also why to solve it. This book includes important findings, examples & exercises. The aim of the book is to present number theory and to introduce basic results from the areas of the geometry of numbers, diophantine approximation, prime number theory, and the asymptotic calculation of number theoretic functions to a beginner. The prerequisite of the book is the reader must have knowledge analytic geometry, differential and integral calculus, together with the elements of complex variable theory, The book covers topics like The Dirichlet Approximation Theorem, The Kronecker Approximation Theorem, Geometry of Numbers, Number Theoretic Functions, The Prime Number Theorem, Characters of Groups of Residues, The Algorithm of Lenstra, Lenstra and Lovász, etc.


9. “Abstract Analytic Number Theory” by John Knopfmacher
“Abstract Analytic Number Theory” Book Review: This book is well written. This book applies classical analytic number theory to a wide variety of mathematical subjects in an arithmetical way. The book deals with the arithmetical semigroups and algebraic enumeration problems, and focuses on arithmetical semigroups with analytical properties of classical type, and gives a deep dive of analytical properties of other arithmetical systems. The book carefully treats fundamental concepts and theorems. The book covers topics like Arithmetical Asymptotic Enumeration, Functions, Enumeration Problems, Arithmetical Semigroups, Arithmetical Semigroups with Analytical Properties of Classical Type and Further Statistical Properties of Arithmetical Functions, The Abstract Prime Number Theorem, Fourier Analysis of Arithmetical Functions, Additive Arithmetical Semigroups, Arithmetical Formations, etc along with appendices and bibliography.


3. Computational Number Theory
1. “Computational Number Theory (Discrete Mathematics and Its Applications)” by Abhijit Das
“Computational Number Theory (Discrete Mathematics and Its Applications) 1st Edition” Book Review: This book is designed for advanced undergraduate and beginning graduate students in engineering. It is also helpful for researchers new to the field. This book presents the major topics of compactional number theory. Main chapters included are arithmetic of integers and polynomials, elliptic curves, primality testing. Practitioners of cryptography in industry can also benefit from this book. The book provides a complete coverage of numbertheoretic algorithms. Coverage of all important applications in the area of engineering along with explaining all computational aspects has been done. Elliptic curves, primality testing etc are also discussed. Theoretic tools are used in cryptography and cryptanalysis are explored. Examples and exercises are also provided. All the topics are mentioned in a detailed manner with plenty of examples. This book is suitable for graduate students of engineering.
Other topics mentioned are cryptography, cryptanalysis, public key cryptography. 

2. “Elementary Number Theory: Primes, Congruences, and Secrets: A Computational Approach (Undergraduate Texts in Mathematics)” by William Stein
“Elementary Number Theory: Primes, Congruences, and Secrets: A Computational Approach (Undergraduate Texts in Mathematics) 2009th Edition” Book Review: This book is designed for undergraduate students. It aims to bring readers closer to active research that can resolve the congruent number problem. Deeper understanding of prime numbers is also provided. The book contains problems solved using Sage mathematical software. Exercises and examples are present throughout the text. The book assumes basic knowledge regarding reading and writing mathematical proofs.


3. “Computational Algebra and Number Theory (Mathematics and Its Applications)” by Wieb Bosma and Alf van der Poorten
“Computational Algebra and Number Theory (Mathematics and Its Applications (325)” Book Review: This book highlights the interaction between computers and maths. It explores the depth of the combination of these two fields. Examples from the field are drawn for better understanding. Different theories like theory, graphs etc have been explained in detail. The main emphasis of the book is on emphasising on the basic principles and commonly used methods.


4. “C++ Toolbox for Verified Computing I: Basic Numerical Problems Theory, Algorithms, and Programs” by Ulrich Kulisch and Matthias Hocks
“C++ Toolbox for Verified Computing I: Basic Numerical Problems Theory, Algorithms, and Programs” Book Review: This book provides a number of C++ programmes for solving numeric problems. Results are also verified using these programs. The text uses the CXSC library. The boom does not assume a great amount of knowledge about C++ programming.


5. “Cryptography and Computational Number Theory (Progress in Computer Science and Applied Logic)” by Kwok Y Lam and Igor Shparlinski  
6. “Advanced Topics in Computational Number Theory (Graduate Texts in Mathematics)” by Henri Cohen
“Advanced Topics in Computational Number Theory (Graduate Texts in Mathematics (193)” Book Review: This book can be used by those doing a sixmonth or yearlong course in computational number theory. It explores different topics in computational number theories. Chapters move from the basics to complex topics.


7. “Numerical Methods for Scientists and Engineers (Dover Books on Mathematics)” by Richard W Hamming
“Numerical Methods for Scientists and Engineers (Dover Books on Mathematics)” Book Review: This book is designed for undergraduate and graduate students of mathematics, science and engineering. It can also be used by professionals and researchers. would be of interest to mathematicians and computer scientists. Numerical methods and its importance in computers are explained. Five important concepts that aim at producing meaningful numbers are also highlighted.


8. “The Mathematical Theory of Communication” by Claude E Shannon and Warren Weaver  
9. “The Mathematical Theory of Finite Element Methods (Texts in Applied Mathematics)” by Susanne Brenner and Ridgway Scott
“The Mathematical Theory of Finite Element Methods (Texts in Applied Mathematics Book 15) 3rd Edition” Book Review: This book is designed for mathematicians as well as engineers and physical scientists. The book explores the basic mathematical theories of finite methods. The book highlights the most common tools used by researchers in this field. The book provides basic knowledge about theories, functional & numeral analysis and approximation. Additional exercises are provided with all chapters for better practice.


4. Algebraic Number Theory
1. “A Classical Introduction to Modern Number Theory” by K Ireland and M Rosen
“A Classical Introduction to Modern Number Theory” Book Review: The book bridges the gap between elementary number theory and the systematic study of advanced topics. It gives a classic introduction to Modern Number Theory and requires the prior knowledge of basic abstract algebra as a prerequisite. The book also includes extensive bibliography and many challenging exercises to explore more knowledge about number theory. It covers the wide range of significant results with basic elementary proofs. The second edition of the book has been revised and contains two new chapters that provide ample evidence of MordellWeil’s belief in elliptic curves.


2. “Number Fields” by D A Marcus
“Number Fields” Book Review: This textbook provides the basics of algebraic number theory in a straightforward manner. The book doesn’t require prior knowledge of abstract algebra. It presents different proofs in a way that it highlights key concepts of the chapter. It provides many exercises as well as appendices to summarize the required background of an algebra. It also maintains the perfect balance between both computational and theoretical practice. The newly edition book is a highly regarded textbook for students to learn number theory.


3. “Algebraic Number Theory” by Jürgen Neukirch and Norbert Schappacher
“Algebraic Number Theory” Book Review: The present book is mainly focused on resolving an anomaly in the textbook content. It provides a comprehensive introduction to algebraic number theory for the beginner. It illustrates the basic concepts from the viewpoint of Arakelov theory. The content of class field theory is very nicely elaborate. The book also includes hints for further study and various examples to gain more knowledge. The concluding chapter on zetafunctions and Lseries is the most important advantage of the new edition of the bok. The book is a very simple, systematic and theoretically balanced textbook on algebraic number field theory available.


4. “Algebraic Number Theory” by John William Scott Cassels and Albrecht Frhlich
“Algebraic Number Theory” Book Review: This book is essential reading for future algebraic number theorists as beginners. It contains the lecture notes from an instructional conference of 1965. That event introduced class field theory as a standard tool of mathematics. The book is a standard textbook for taught courses in algebraic number theory. The second edition of the book includes a valuable list of errata compiled by mathematicians which help students to clear concepts more thoroughly.


5. “Algebraic Number Theory” by Frazer Jarvis
“Algebraic Number Theory” Book Review: This book is essential reading for future algebraic number theorists as beginners. It contains the lecture notes from an instructional conference of 1965. That event introduced class field theory as a standard tool of mathematics. The book is a standard textbook for taught courses in algebraic number theory. The second edition of the book includes a valuable list of errata compiled by mathematicians which help students to clear concepts more thoroughly.


6. “Algebraic Number Theory and Fermat’s Last Theorem” by Ian Stewart and David Tall
“Algebraic Number Theory and Fermat’s Last Theorem” Book Review: This book introduces fundamental basics of algebraic numbers. It also illustrates one of the most interesting stories in mathematical history – the quest for a proof of Fermat’s Last theorem. It helps to motivate a general study of the theory of algebraic numbers from a point of view. The new edition of the book gives uptodate information on unique prime factorization for real quadratic number fields, mainly Harper’s proof that Z(√14) is Euclidean. It revises and explores one chapter into two, covering basic ideas about modular function and highlights the new ideas of Frey, Wiles and others. The book improves and updates the index, figures, bibliography and further reading list to gain more knowledge. It also explains how basic ideas from the concept of algebraic numbers can be used to solve problems from a numerical perspective.


7. “Algebraic Number Theory” by Richard A Mollin
“Algebraic Number Theory” Book Review: The new edition of the book concentrates on integral domains, ideals, and unique factorization in the first chapter, field extensions in the second chapter, class groups in the third chapter. Applications are now collected in chapter four and at the end of chapter five, where quantitative testing is highlighted as a KroneckerWeber thought application. In the fifth chapter, the sections on ideal decomposition in number fields are distributed. The final chapter goes on to elaborate the reciprocity laws. The book includes minibiographies of remarkable mathematicians, easy alignment, a complete index, and a lot of exercise. The book is ideal for a firstsemester course, this accessible, selfcontained offers a comprehensive, indepth knowledge of many applications.


8. “Algebraic Number Theory and Fermat’s Last Theorem” by Ian Stewart and David Tall  
9. “A Course in Algebraic Number Theory” by Robert B Ash  
5. Number Theory and Cryptography
1. “A Course in Number Theory and cryptography” by Neal Koblitz
“A Course in Number Theory and cryptography” Book Review: This book provides the introduction to modern and ancient arithmetic topics. Major chapters included are elementary number theory, finite fields and quadratic residues, cryptography, public key. Other topics mentioned are primality and factoring, elliptic curves. Exercises and problems are provided for student’s practice. mathematics and engineering students can use this book.


2. “Elliptic Curves: Number Theory and Cryptography” by Lawrence C Washington
“Elliptic Curves: Number Theory and Cryptography” Book Review: This book describes the fundamental principles of elliptic curves in numerical analysis. The chapters included are the basic theory, torsion points, elliptic curves and the discrete logarithm problem. Other topics included are elliptic curves cryptography, elliptic curves over Q and isogenies. Equations and problems are discussed at length. This book is useful for mathematical and engineering students.


3. “Computational Number Theory and Modern Cryptography” by Song Y Yan
“Computational Number Theory and Modern Cryptography” Book Review: This book discusses the core principles of computational number theory and cryptography. Chapters introduced are primality testing, integer factorization, secretkey cryptography, discrete logarithmbased cryptography. Other topics discussed are quantum computational number theory, quantum resistant cryptography and elliptic curve. Bibliographic notes and references are added for further reading. Equations and diagrams are discussed in depth. This book is suitable for computer engineering and mathematics students.


4. “Number Theory: Structures, Examples, and Problems” by Titu Andreescu and Dorin Andrica
“Number Theory: Structures, Examples, and Problems” Book Review: This book covers the important topics of number theory. Main chapters included are divisibility, power of integers, floor function and fractional part, digits of numbers. Other topics mentioned are arithmetic functions, Diophantine equations, binomial coefficient. Problems and examples are described thoroughly. Additional problems are added to test the knowledge of students. This book is useful for engineering and mathematics students.


5. “Number Theory Cryptography and Its Applications to GNU/Linux Software” by Giovanni A Orlando  
6. “Number Theory for Computing” by M E Hellmann and Song Y Yan
“Number Theory for Computing” Book Review: This book provides the introduction to classic number theory and its applications. Main chapters involved are elementary number theory, algorithmic number theory, primality testing, integer factorization. Other topics discussed are quantum numbertheoretic algorithms, computer systems design, cryptography and information security. Bibliographic notes are added at the end of every chapter for further reading and review purposes. This book is useful for undergraduate students studying computing and information technology, electrical and electronics engineering.


7. “Elementary Number Theory, Cryptography and Codes” by M Welleda Baldoni and Ciro Ciliberto
“Elementary Number Theory, Cryptography and Codes” Book Review: This book discusses the basic methods of algebra and number theory. Major chapters included are roundup on numbers, computational complexity, factoring integers, continued fractions. Other topics involved are congruences, unique factorization domains, finite fields, quadratic residues, primality tests. Multiple choice questions and computational exercises are provided for student’s practice. Programming exercises containing program questions are also included to test students’ knowledge. This book can be used by advanced mathematical and computational engineering students.


8. “Number Theory in Science and Communication” by Manfred Schroeder
“Number Theory in Science and Communication” Book Review: This book contains complete knowledge of number theory and its applications. Major chapters introduced are the natural numbers, primes, the prime distribution, fractions: continued, Egyptian and farey. Other topics mentioned are linear congruences, Diophantine equations, the theorems of format, Wilson and Euler. Equations and graphs are discussed in a detailed manner. A total of 30 chapters are discussed thoroughly in this book. This book is suitable for undergraduate computer, IT engineering students.


9. “Introduction to Modern Number Theory: Fundamental Problems, Ideas and Theories (Encyclopaedia of Mathematical Sciences)” by Yu I Manin and Alexei A Panchishkin
“Introduction to Modern Number Theory: Fundamental Problems, Ideas and Theories (Encyclopaedia of Mathematical Sciences)” Book Review: This book covers the principles of number theory along with uptodate modern problems. Major chapters included are nonAbelian generalizations of class field theory, recursive computability and Diophantine equations. Other topics mentioned are zeta and Lfunctions, Wiles’ proof of Fermat’s last theorem, and relevant techniques coming from a synthesis of various theories. Extensive problems and solutions are mentioned in this book for better understanding. This book is useful for graduate level students of computational engineering and applied mathematics students.


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