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El. knyga: Introduction to Number Theory

(Truman State University, Kirksville, Missouri, USA), (Truman State University, Kirksville, Missouri, USA)
  • Formatas: 426 pages
  • Serija: Textbooks in Mathematics
  • Išleidimo metai: 18-Nov-2015
  • Leidėjas: Chapman & Hall/CRC
  • ISBN-13: 9781498717526
Kitos knygos pagal šią temą:
Introduction to Number TheoryDidesnis paveikslėlis
  • Formatas: 426 pages
  • Serija: Textbooks in Mathematics
  • Išleidimo metai: 18-Nov-2015
  • Leidėjas: Chapman & Hall/CRC
  • ISBN-13: 9781498717526
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Introduction to Number Theory is a classroom-tested, student-friendly text that covers a diverse array of number theory topics, from the ancient Euclidean algorithm for finding the greatest common divisor of two integers to recent developments such as cryptography, the theory of elliptic curves, and the negative solution of Hilberts tenth problem. The authors illustrate the connections between number theory and other areas of mathematics, including algebra, analysis, and combinatorics. They also describe applications of number theory to real-world problems, such as congruences in the ISBN system, modular arithmetic and Eulers theorem in RSA encryption, and quadratic residues in the construction of tournaments.

Ideal for a one- or two-semester undergraduate-level course, this Second Edition:





Features a more flexible structure that offers a greater range of options for course design Adds new sections on the representations of integers and the Chinese remainder theorem Expands exercise sets to encompass a wider variety of problems, many of which relate number theory to fields outside of mathematics (e.g., music) Provides calculations for computational experimentation using SageMath, a free open-source mathematics software system, as well as Mathematica® and Maple, online via a robust, author-maintained website Includes a solutions manual with qualifying course adoption

By tackling both fundamental and advanced subjectsand using worked examples, numerous exercises, and popular software packages to ensure a practical understandingIntroduction to Number Theory, Second Edition instills a solid foundation of number theory knowledge.

Recenzijos

Praise for the Previous Edition

"The authors succeed in presenting the topics of number theory in a very easy and natural way, and the presence of interesting anecdotes, applications, and recent problems alongside the obvious mathematical rigor makes the book even more appealing. a valid and flexible textbook for any undergraduate number theory course." International Association for Cryptologic Research Book Reviews, May 2011

" a welcome addition to the stable of elementary number theory works for all good undergraduate libraries." J. McCleary, Vassar College, Poughkeepsie, New York, USA, from CHOICE, Vol. 46, No. 1, August 2009

" a reader-friendly text. provides all of the tools to achieve a solid foundation in number theory." LEnseignement Mathématique, Vol. 54, No. 2, 2008

The theory of numbers is a core subject of mathematics. The authors have written a solid update to the first edition (CH, Aug'09, 46-6857) of this classic topic. There is no shortage of introductions to number theory, and this book does not offer significantly different information. Nonetheless, the authors manage to give the subject a fresh, new feel. The writing style is simple, clear, and easy to follow for standard readers. The book contains all the essential topics of a first-semester course and enough advanced topics to fill a second. In particular, it includes several modern aspects of number theory, which are often ignored in other texts, such as the use of factoring in computer security, searching for large prime numbers, and connections to other branches of mathematics. Each section contains supplementary homework exercises of various difficulties, a crucial ingredient of any good textbook. Finally, much emphasis is placed on calculating with computers, a staple of modern number theory. Overall, this title should be considered by any student or professor seeking an excellent text on the subject.





--A. Misseldine, Southern Utah University, Choice magazine 2016

Preface xi
1 Introduction
1(12)
1.1 What is number theory?
1(3)
1.2 The natural numbers
4(3)
1.3 Mathematical induction
7(5)
1.4 Notes
12(1)
The Peano axioms
12(1)
2 Divisibility
13(10)
2.1 Basic definitions and properties
13(2)
2.2 The division algorithm
15(2)
2.3 Representations of integers
17(6)
3 Greatest Common Divisor
23(22)
3.1 Greatest common divisor
23(5)
3.2 The Euclidean algorithm
28(4)
3.3 Linear Diophantine equations
32(9)
3.4 Notes
41(4)
Euclid
41(1)
The number of steps in the Euclidean algorithm
41(2)
Geometric interpretation of the equation ax + by = c
43(2)
4 Primes
45(16)
4.1 The sieve of Eratosthenes
45(4)
4.2 The fundamental theorem of arithmetic
49(5)
4.3 Distribution of prime numbers
54(4)
4.4 Notes
58(3)
Eratosthenes
58(1)
Nonunique factorization and Fermat's Last Theorem
58(3)
5 Congruences
61(18)
5.1 Residue classes
61(4)
5.2 Linear congruences
65(5)
5.3 Application: Check digits and the ISBN-10 system
70(3)
5.4 The Chinese remainder theorem
73(6)
6 Special Congruences
79(14)
6.1 Fermat's theorem
80(4)
6.2 Euler's theorem
84(5)
6.3 Wilson's theorem
89(3)
6.4 Notes
92(1)
Leonhard Euler
92(1)
7 Primitive Roots
93(18)
7.1 Order of an element mod n
93(3)
7.2 Existence of primitive roots
96(4)
7.3 Primitive roots modulo composites
100(3)
7.4 Application: Construction of the regular 17-gon
103(5)
7.5 Notes
108(3)
Groups
108(1)
Straightedge and compass constructions
109(2)
8 Cryptography
111(20)
8.1 Monoalphabetic substitution ciphers
111(5)
8.2 The Pohlig--Hellman cipher
116(5)
8.3 The Massey--Omura exchange
121(4)
8.4 The RSA algorithm
125(4)
8.5 Notes
129(2)
Computing powers mod p
129(1)
RSA cryptography
130(1)
9 Quadratic Residues
131(18)
9.1 Quadratic congruences
131(2)
9.2 Quadratic residues and nonresidues
133(4)
9.3 Quadratic reciprocity
137(7)
9.4 The Jacobi symbol
144(4)
9.5 Notes
148(1)
Carl Friedrich Gauss
148(1)
10 Applications of Quadratic Residues
149(12)
10.1 Application: Construction of tournaments
149(5)
10.2 Consecutive quadratic residues and nonresidues
154(2)
10.3 Application: Hadamard matrices
156(5)
11 Sums of Squares
161(22)
11.1 Pythagorean triples
161(4)
11.2 Gaussian integers
165(6)
11.3 Factorization of Gaussian integers
171(5)
11.4 Lagrange's four squares theorem
176(5)
11.5 Notes
181(2)
Diophantus
181(2)
12 Further Topics in Diophantine Equations
183(20)
12.1 The case n = 4 in Fermat's Last Theorem
183(3)
12.2 Pell's equation
186(8)
12.3 The abc conjecture
194(5)
12.4 Notes
199(4)
Pierre de Fermat
199(1)
The p-adic numbers
200(3)
13 Continued Fractions
203(32)
13.1 Finite continued fractions
204(8)
13.2 Infinite continued fractions
212(7)
13.3 Rational approximation of real numbers
219(13)
13.4 Notes
232(3)
Continued fraction expansion of e
232(1)
Continued fraction expansion of tan x
232(1)
Srinivasa Ramanujan
233(2)
14 Continued Fraction Expansions of Quadratic Irrationals
235(28)
14.1 Periodic continued fractions
235(13)
14.2 Continued fraction factorization
248(6)
14.3 Continued fraction solution of Pell's equation
254(6)
14.4 Notes
260(3)
Three squares and triangular numbers
260(2)
History of Pell's equation
262(1)
15 Arithmetic Functions
263(40)
15.1 Perfect numbers
263(7)
15.2 The group of arithmetic functions
270(8)
15.3 Mobius inversion
278(5)
15.4 Application: Cyclotomic polynomials
283(4)
15.5 Partitions of an integer
287(12)
15.6 Notes
299(4)
The lore of perfect numbers
299(1)
Pioneers of integer partitions
300(3)
16 Large Primes
303(16)
16.1 Fermat numbers
303(6)
16.2 Mersenne numbers
309(3)
16.3 Prime certificates
312(3)
16.4 Finding large primes
315(4)
17 Analytic Number Theory
319(36)
17.1 Sum of reciprocals of primes
319(3)
17.2 Orders of growth of functions
322(2)
17.3 Chebyshev's theorem
324(7)
17.4 Bertrand's postulate
331(4)
17.5 The prime number theorem
335(6)
17.6 The zeta function and the Riemann hypothesis
341(5)
17.7 Dirichlet's theorem
346(6)
17.8 Notes
352(3)
Paul Erdos
352(3)
18 Elliptic Curves
355(42)
18.1 Cubic curves
355(3)
18.2 Intersections of lines and curves
358(9)
18.3 The group law and addition formulas
367(5)
18.4 Sums of two cubes
372(3)
18.5 Elliptic curves mod p
375(3)
18.6 Encryption via elliptic curves
378(3)
18.7 Elliptic curve method of factorization
381(5)
18.8 Fermat's Last Theorem
386(4)
18.9 Notes
390(7)
Projective space
390(3)
Associativity of the group law
393(4)
A Web Resources 397(4)
B Notation 401(4)
References 405(4)
Index 409
Martin Erickson (1963-2013) received his Ph.D in mathematics in 1987 from the University of Michigan, Ann Arbor, USA, studying with Thomas Frederick Storer. He joined the faculty in the Mathematics Department of Truman State University, Kirksville, Missouri, USA, when he was twenty-four years old, and remained there for the rest of his life. Professor Erickson authored and coauthored several mathematics books, including the first edition of Introduction to Number Theory (CRC Press, 2007), Pearls of Discrete Mathematics (CRC Press, 2010), and A Student's Guide to the Study, Practice, and Tools of Modern Mathematics (CRC Press, 2010).

Anthony Vazzana received his Ph.D in mathematics in 1998 from the University of Michigan, Ann Arbor, USA. He joined the faculty in the Mathematics Department of Truman State University, Kirksville, Missouri, USA, in 1998. In 2000, he was awarded the Governor's Award for Excellence in Teaching and was selected as the Educator of the Year. In 2002, he was named the Missouri Professor of the Year by the Carnegie Foundation for the Advancement of Teaching and the Council for Advancement and Support of Education.

David Garth received his Ph.D in mathematics in 2000 from Kansas State University, Manhattan, USA. He joined the faculty in the Mathematics Department of Truman State University, Kirksville, Missouri, USA, in 2000. In 2005, he was awarded the Golden Apple Award from Truman State University's Theta Kappa chapter of the Order of Omega. His areas of research include analytic and algebraic number theory, especially Pisot numbers and their generalizations, and Diophantine approximation.
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