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Book
1 online resource (375 pages) : illustrations.
  • Preface
  • Table of illustrations
  • Abbreviations, basic references and notations
  • Protohistory
  • Fermat and his Correspondents
  • Euler
  • An Age of Transition: Lagrange and Legendre
  • Additional bibliography and references
  • Index nominum
  • Index rerum.
Number Theory or arithmetic, as some prefer to call it, is the oldest, purest, liveliest, most elementary yet sophisticated field of mathematics. It is no coincidence that the fundamental science of numbers has come to be known as the "Queen of Mathematics." Indeed some of the most complex conventions of the mathematical mind have evolved from the study of basic problems of number theory. Andr Weil, one of the outstanding contributors to number theory, has written an historical exposition of this subject; his study examines texts that span roughly thirty-six centuries of arithmetical work from an Old Babylonian tablet, datable to the time of Hammurapi to Legendres Essai sur la Thorie des Nombres (1798). Motivated by a desire to present the substance of his field to the educated reader, Weil employs an historical approach in the analysis of problems and evolving methods of number theory and their significance within mathematics. In the course of his study Weil accompanies the reader into the workshops of four major authors of modern number theory (Fermat, Euler, Lagrange and Legendre) and there he conducts a detailed and critical examination of their work. Enriched by a broad coverage of intellectual history, Number Theory represents a major contribution to the understanding of our cultural heritage. ----- A very unusual book combining thorough philological exactness, keen observation, apt comments of the essential points, picturesque fantasy, enthusiastic love of the subject, and brilliant literary style: a romantic novel of documents. It is both number theory and its history in an inseparable oneness, helping us understand the very roots and the first big stage of progress of this discipline. The author, one of the most prominent number theoristschose to give us a broad perspective of the birth of modern number theory.--Periodica Mathematica Hungaria The volume under review ... a discursive, expository, leisurely peek over the shoulders of several great authors in number theoryis perhaps unique in the enthusiasm it has inspired. --Mathematical Reviews.
Book
1 online resource (xv, 408 pages).
The book includes several survey articles on prime numbers, divisor problems, and Diophantine equations, as well as research papers on various aspects of analytic number theory such as additive problems, Diophantine approximations and the theory of zeta and L-function. Audience: Researchers and graduate students interested in recent development of number theory.
Book
x, 222 p. : ill. ; 23 cm.
SAL3 (off-campus storage)
Book
376 p. : ill. ; 25 cm.
SAL3 (off-campus storage)
Book
xiv, 247 pages ; 23 cm.
Textbook, with answers to some exercises.
(source: Nielsen Book Data)9781786344717 20180319
Science Library (Li and Ma)
Book
xix, 191 pages ; 24 cm.
  • On Modular Relations (T Arai, K Chakraborty and S Kanemitsu)-- Figurate Primes and Hilbert's 8th Problem (T-X Cai, Y Zhang and Z-G Shen)-- Statistical Distribution of Roots of a Polynomial Modulo Prime Powers (Y Kitaoka)-- A Survey on the Theory of Universality for Zeta and L-Functions (K Matsumoto)-- Complex Multiplication in the Sense of Abel (K Miyake)-- Problems on Combinatorial Properties of Primes (Z-W Sun)--.
  • (source: Nielsen Book Data)9789814644921 20160618
Based on the successful 7th China-Japan seminar on number theory conducted in Kyushu University, this volume is a compilation of survey and semi-survey type of papers by the participants of the seminar. The topics covered range from traditional analytic number theory to elliptic curves and universality. This volume contains new developments in the field of number theory from recent years and it provides suitable problems for possible new research at a level which is not unattainable. Timely surveys will be beneficial to a new generation of researchers as a source of information and these provide a glimpse at the state-of-the-art affairs in the fields of their research interests.
(source: Nielsen Book Data)9789814644921 20160618
Science Library (Li and Ma)

7. Journal of Numbers [2014 - ]

Journal/Periodical
1 online resource
Book
xiv, 603 p. ; 24 cm.
  • Numbers.- Numbers.- Induction.- Euclid's Algorithm.- Unique Factorization.- Congruence.- Congruence classes and rings.- Congruence Classes.- Rings and Fields.- Matrices and Codes.- Congruences and Groups.- Fermat's and Euler's Theorems.- Applications of Euler's Theorem.- Groups.- The Chinese Remainder Theorem.- Polynomials.- Polynomials.- Unique Factorization.- The Fundamental Theorem of Algebra.- Polynomials in ?[x].- Congruences and the Chinese Remainder Theorem.- Fast Polynomial Multiplication.- Primitive Roots.- Cyclic Groups and Cryptography.- Carmichael Numbers.- Quadratic Reciprocity.- Quadratic Applications.- Finite Fields.- Congruence Classes Modulo a Polynomial.- Homomorphisms and Finite Fields.- BCH Codes.- Factoring Polynomials.- Factoring in ?[x].- Irreducible Polynomials.
  • (source: Nielsen Book Data)9780387745275 20160605
This book is an informal and readable introduction to higher algebra at the post-calculus level. The concepts of ring and field are introduced through study of the familiar examples of the integers and polynomials. The new examples and theory are built in a well-motivated fashion and made relevant by many applications - to cryptography, coding, integration, history of mathematics, and especially to elementary and computational number theory. The later chapters include expositions of Rabiin's probabilistic primality test, quadratic reciprocity, and the classification of finite fields. Over 900 exercises are found throughout the book.
(source: Nielsen Book Data)9780387745275 20160605
dx.doi.org SpringerLink
Book
xviii, 384 p. ; 25 cm.
  • Fundamentals.- Divisibility.- Powers of Integers.- Floor Function and Fractional Part.- Digits of Numbers.- Basic Principles in Number Theory.- Arithmetic Functions.- More on Divisibility.- Diophantine Equations.- Some Special Problems in Number Theory.- Problems Involving Binomial Coefficients.- Miscellaneous Problems.- Solutions to Additional Problems.- Divisibility.- Powers of Integers.- Floor Function and Fractional Part.- Digits of Numbers.- Basic Principles in Number Theory.- Arithmetic Functions.- More on Divisibility.- Diophantine Equations.- Some Special Problems in Number Theory.- Problems Involving Binomial Coefficients.- Miscellaneous Problems.
  • (source: Nielsen Book Data)9780817632458 20160605
This introductory textbook takes a problem-solving approach to number theory, situating each concept within the framework of an example or a problem for solving. Starting with the essentials, the text covers divisibility, unique factorization, modular arithmetic and the Chinese Remainder Theorem, Diophantine equations, binomial coefficients, Fermat and Mersenne primes and other special numbers, and special sequences. Included are sections on mathematical induction and the pigeonhole principle, as well as a discussion of other number systems. By emphasizing examples and applications the authors motivate and engage readers.
(source: Nielsen Book Data)9780817632458 20160605
dx.doi.org SpringerLink
Book
xiii, 543 p. : ill. ; 24 cm.
  • 1. Introduction. 2. Divisibility and Primes. 3. Modular Arithmetic. 4. Fundamental Theorems of Modular Arithmetic. 5. Cryptography. 6. Primality Testing and Factoring. 7. Primitive Roots. 8. Applications. 9. Quadratic Congruences. 10. Applications. 11. Continued Fractions. 12. Factoring Methods. 13. Diophantine Approximations. 14. Diophantine Equations. 15. Arithmetical Functions and Dirichlet Series. 16. Distribution of Primes. 17. Quadratic Reciprocity Law 18. Binary Quadratic Forms. 19. Elliptic Curves. Appendix A: Mathematical Induction. Appendix B: Binomial Theorem. Appendix C: Algorithmic Complexity and O-notation. Answers and Hints. Index of Notation. Index.
  • (source: Nielsen Book Data)9780138018122 20160528
Appropriate for most courses in Number Theory. This book effectively integrates computing algorithms into the number theory curriculum using a heuristic approach and strong emphasis on proofs. Its in-depth coverage of modern applications considers the latest trends and topics, such as elliptic curves-a subject that has seen a rise in popularity due to its use in the proof of Fermat's Last Theorem.
(source: Nielsen Book Data)9780138018122 20160528
SAL3 (off-campus storage)
Book
xv, 323 pages ; 29 cm
  • Seeing arithmeticFoundations: The Euclidean algorithmPrime factorizationRational and constructible numbersGaussian and Eisenstein integersModular arithmetic: The modular worldsModular dynamicsAssembling the modular worldsQuadratic residuesQuadratic forms: The topographDefinite formsIndefinite formsIndex of theoremsIndex of termsIndex of namesBibliography.
  • (source: Nielsen Book Data)9781470434939 20171009
An Illustrated Theory of Numbers gives a comprehensive introduction to number theory, with complete proofs, worked examples, and exercises. Its exposition reflects the most recent scholarship in mathematics and its history. Almost 500 sharp illustrations accompany elegant proofs, from prime decomposition through quadratic reciprocity. Geometric and dynamical arguments provide new insights, and allow for a rigorous approach with less algebraic manipulation. The final chapters contain an extended treatment of binary quadratic forms, using Conway's topograph to solve quadratic Diophantine equations (e.g. Pell's equation) and to study reduction and the finiteness of class numbers.Data visualizations introduce the reader to open questions and cutting-edge results in analytic number theory such as the Riemann hypothesis, boundedness of prime gaps, and the class number 1 problem. Accompanying each chapter, historical notes curate primary sources and secondary scholarship to trace the development of number theory within and outside the Western tradition.Requiring only high school algebra and geometry, this text is recommended for a first course in elementary number theory, and to all mathematicians seeking a fresh perspective on an ancient subject.
(source: Nielsen Book Data)9781470434939 20171009
Science Library (Li and Ma)
Book
x, 240 pages ; 24 cm.
  • Part I. Elementary Number Theory: 1. Prelude-- 2. Arithmetic functions and integer points-- 3. Congruences-- 4. Quadratic reciprocity and Fourier series-- 5. Sums of squares-- Part II. Fourier Analysis and Geometric Discrepancy: 6. Uniform distribution and completeness of the trigonometric system-- 7. Discrepancy and trigonometric approximation-- 8. Integer points and Poisson summation formula-- 9. Integer points and exponential sums-- 10. Geometric discrepancy and decay of Fourier transforms-- 11. Discrepancy in high dimension and Bessel functions-- References-- Index.
  • (source: Nielsen Book Data)9781107619852 20160616
The study of geometric discrepancy, which provides a framework for quantifying the quality of a distribution of a finite set of points, has experienced significant growth in recent decades. This book provides a self-contained course in number theory, Fourier analysis and geometric discrepancy theory, and the relations between them, at the advanced undergraduate or beginning graduate level. It starts as a traditional course in elementary number theory, and introduces the reader to subsequent material on uniform distribution of infinite sequences, and discrepancy of finite sequences. Both modern and classical aspects of the theory are discussed, such as Weyl's criterion, Benford's law, the Koksma-Hlawka inequality, lattice point problems, and irregularities of distribution for convex bodies. Fourier analysis also features prominently, for which the theory is developed in parallel, including topics such as convergence of Fourier series, one-sided trigonometric approximation, the Poisson summation formula, exponential sums, decay of Fourier transforms, and Bessel functions.
(source: Nielsen Book Data)9781107619852 20160616
Science Library (Li and Ma)
Book
xv, 256 pages : illustrations ; 25 cm.
  • * On the greatest prime factor of some divisibility sequences by A. Akbary and S. Yazdani* A number field extension of a question of Milnor by T. Chatterjee, S. Gun, and P. Rath* Mixing rates of random walks with little backtracking by S. M. Cioaba and P. Xu* Additive and multiplicative functions with similar global behavior by J.-M. De Koninck and N. Doyon* Multidimensional sequences uniformly distributed modulo 1 created from normal numbers by J.-M. De Koninck and I. Katai* The index of $a$ modulo $p$ by A. T. Felix* Determining optimal test functions for bounding the average rank in families of $L$-functions by J. Freeman and S. J. Miller* Familles d'equations de Thue associees a un sous-groupe de rang 1 d'unites totalement reelles d'un corps de nombres by C. Levesque and M. Waldschmidt* Cyclicity of quotients of non-CM elliptic curves modulo primes by G. Meleleo* On the Euler Kronecker constant of a cyclotomic field, II by M. Mourtada and V. K. Murty* The generalized Dedekind determinant by M. R. Murty and K. Sinha* A remark on elliptic curves with a given number of points over finite fields by J. Parks* Recovering cusp forms on GL(2) from symmetric cubes by D. Ramakrishnan* Arithmetic nature of some infinite series and integrals by N. Saradha and D. Sharma* Points on varieties over finite fields in small boxes by I. E. Shparlinski* Bounds for the Lang-Trotter conjectures by D. Zywina.
  • (source: Nielsen Book Data)9781470414573 20160619
M. Ram Murty has had a profound impact on the development of number theory throughout the world. To honor his mathematical legacy, a conference focusing on new research directions in number theory inspired by his most significant achievements was held from October 15-17, 2013, at the Centre de Recherches Mathematiques in Montreal. This proceedings volume is representative of the broad spectrum of topics that were addressed at the conference, such as elliptic curves, function field arithmetic, Galois representations, $L$-functions, modular forms and automorphic forms, sieve methods, and transcendental number theory.
(source: Nielsen Book Data)9781470414573 20160619
Science Library (Li and Ma)
Book
1 online resource (240 pages). Digital: text file; PDF.
The aim of this book is to serve as an introductory text to the theory of linear forms in the logarithms of algebraic numbers, with a special emphasis on a large variety of its applications. We wish to help students and researchers to learn what is hidden inside the blackbox ‚Baker's theory of linear forms in logarithms' (in complex or in $p$-adic logarithms) and how this theory applies to many Diophantine problems, including the effective resolution of Diophantine equations, the $abc$-conjecture, and upper bounds for the irrationality measure of some real numbers. Written for a broad audience, this accessible and self-contained book can be used for graduate courses (some 30 exercises are supplied). Specialists will appreciate the inclusion of over 30 open problems and the rich bibliography of over 450 references.
Book
1 online resource
  • Preface.- Numbers: Problems Involving Integers.- Further Study of Integers.- Diophantine Equations and More.- Pythagorean Triples, Additive Problems, and More.- Homework.
  • (source: Nielsen Book Data)9783319909141 20180910
Through its engaging and unusual problems, this book demonstrates methods of reasoning necessary for learning number theory. Every technique is followed by problems (as well as detailed hints and solutions) that apply theorems immediately, so readers can solve a variety of abstract problems in a systematic, creative manner. New solutions often require the ingenious use of earlier mathematical concepts - not the memorization of formulas and facts. Questions also often permit experimental numeric validation or visual interpretation to encourage the combined use of deductive and intuitive thinking. The first chapter starts with simple topics like even and odd numbers, divisibility, and prime numbers and helps the reader to solve quite complex, Olympiad-type problems right away. It also covers properties of the perfect, amicable, and figurate numbers and introduces congruence. The next chapter begins with the Euclidean algorithm, explores the representations of integer numbers in different bases, and examines continued fractions, quadratic irrationalities, and the Lagrange Theorem. The last section of Chapter Two is an exploration of different methods of proofs. The third chapter is dedicated to solving Diophantine linear and nonlinear equations and includes different methods of solving Fermat's (Pell's) equations. It also covers Fermat's factorization techniques and methods of solving challenging problems involving exponent and factorials. Chapter Four reviews the Pythagorean triple and quadruple and emphasizes their connection with geometry, trigonometry, algebraic geometry, and stereographic projection. A special case of Waring's problem as a representation of a number by the sum of the squares or cubes of other numbers is covered, as well as quadratic residuals, Legendre and Jacobi symbols, and interesting word problems related to the properties of numbers. Appendices provide a historic overview of number theory and its main developments from the ancient cultures in Greece, Babylon, and Egypt to the modern day. Drawing from cases collected by an accomplished female mathematician, Methods in Solving Number Theory Problems is designed as a self-study guide or supplementary textbook for a one-semester course in introductory number theory. It can also be used to prepare for mathematical Olympiads. Elementary algebra, arithmetic and some calculus knowledge are the only prerequisites. Number theory gives precise proofs and theorems of an irreproachable rigor and sharpens analytical thinking, which makes this book perfect for anyone looking to build their mathematical confidence.
(source: Nielsen Book Data)9783319909141 20180910

16. Number fields [2018]

Book
1 online resource (xviii, 203 pages). Digital: text file; PDF.
  • 1: A Special Case of Fermat's Conjecture.- 2: Number Fields and Number Rings.- 3: Prime Decomposition in Number Rings.- 4: Galois Theory Applied to Prime Decomposition.- 5: The Ideal Class Group and the Unit Group.- 6: The Distribution of Ideals in a Number Ring.- 7: The Dedekind Zeta Function and the Class Number Formula.- 8: The Distribution of Primes and an Introduction to Class Field Theory.- Appendix A: Commutative Rings and Ideals.- Appendix B: Galois Theory for Subfields of C.- Appendix C: Finite Fields and Rings.- Appendix D: Two Pages of Primes.- Further Reading.- Index of Theorems.- List of Symbols.
  • (source: Nielsen Book Data)9783319902326 20180910
Requiring no more than a basic knowledge of abstract algebra, this text presents the mathematics of number fields in a straightforward, pedestrian manner. It therefore avoids local methods and presents proofs in a way that highlights the important parts of the arguments. Readers are assumed to be able to fill in the details, which in many places are left as exercises.
(source: Nielsen Book Data)9783319902326 20180910
Book
1 online resource.
  • General Framework.- Number Theoretic Aspects of Regular Sequences.- First-order Logic and Numeration System.- Some Applications of Algebra to Automatic Sequences.- Avoiding or Limiting Regularities in Words.- Coloring Problems for Infinite Words.- Normal Numbers and Computer Science.- Normal Numbers and Symbolic Dynamics.- About the Domino Problem for Subshifts on Groups.- Automation (Semi)Groups: Wang Tilings and Schreier Tries.- Amenability of Groups and G-Sets.- Index.- References.
  • (source: Nielsen Book Data)9783319691510 20180730
This collaborative book presents recent trends on the study of sequences, including combinatorics on words and symbolic dynamics, and new interdisciplinary links to group theory and number theory. Other chapters branch out from those areas into subfields of theoretical computer science, such as complexity theory and theory of automata. The book is built around four general themes: number theory and sequences, word combinatorics, normal numbers, and group theory. Those topics are rounded out by investigations into automatic and regular sequences, tilings and theory of computation, discrete dynamical systems, ergodic theory, numeration systems, automaton semigroups, and amenable groups. This volume is intended for use by graduate students or research mathematicians, as well as computer scientists who are working in automata theory and formal language theory. With its organization around unified themes, it would also be appropriate as a supplemental text for graduate level courses.
(source: Nielsen Book Data)9783319691510 20180730
Book
1 online resource.
Number theory is the branch of mathematics concerned with the counting numbers, 1, 2, 3, ... and their multiples and factors. Of particular importance are odd and even numbers, squares and cubes, and prime numbers. But in spite of their simplicity, you will meet a multitude of topics in this book: magic squares, cryptarithms, finding the day of the week for a given date, constructing regular polygons, pythagorean triples, and many more.In this revised edition, John Watkins and Robin Wilson have updated the text to bring it in line with contemporary developments. They have added new material on Fermat's Last Theorem, the role of computers in number theory, and the use of number theory in cryptography, and have made numerous minor changes in the presentation and layout of the text and the exercises.
(source: Nielsen Book Data)9780883856536 20180219
Book
xi, 279 pages : illustrations (some color) ; 25 cm
  • Forward.- 1. Introduction.- 2. Warming Up: Integers, Sequences, and Experimental Mathematics.- 3. Greatest Prime Factor Sequences.- 4. Conway's Subprime Function and Related Structures with a Touch of Fibonnacci Flavor.- 5. Going All Experimental - More Games and Applications.- Appendix 0.- Visualization.- Appendix 1.- Appendix 2.- Appendix 3.- References.
  • (source: Nielsen Book Data)9783319567617 20180530
With a specific focus on the mathematical life in small undergraduate colleges, this book presents a variety of elementary number theory insights involving sequences largely built from prime numbers and contingent number-theoretic functions. Chapters include new mathematical ideas and open problems, some of which are proved in the text. Vector valued MGPF sequences, extensions of Conway's Subprime Fibonacci sequences, and linear complexity of bit streams derived from GPF sequences are among the topics covered in this book. This book is perfect for the pure-mathematics-minded educator in a small undergraduate college as well as graduate students and advanced undergraduate students looking for a significant high-impact learning experience in mathematics.
(source: Nielsen Book Data)9783319567617 20180530
Science Library (Li and Ma)
Book
1 online resource.
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