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Introduction to Number Theory [Kietas viršelis]

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(Truman State University, Kirksville, Missouri, USA), (Truman State University, Kirksville, Missouri, USA), (Truman State University, Kirksville, Missouri, USA)
  • Formatas: Hardback, 536 pages, aukštis x plotis: 234x156 mm, weight: 860 g, 300 equations; 9 Tables, black and white; 26 Illustrations, black and white, Contains 47 hardbacks
  • Serija: Textbooks in Mathematics
  • Išleidimo metai: 01-Oct-2007
  • Leidėjas: Chapman & Hall/CRC
  • ISBN-10: 1584889373
  • ISBN-13: 9781584889373
Kitos knygos pagal šią temą:
Introduction to Number TheoryDidesnis paveikslėlis
  • Formatas: Hardback, 536 pages, aukštis x plotis: 234x156 mm, weight: 860 g, 300 equations; 9 Tables, black and white; 26 Illustrations, black and white, Contains 47 hardbacks
  • Serija: Textbooks in Mathematics
  • Išleidimo metai: 01-Oct-2007
  • Leidėjas: Chapman & Hall/CRC
  • ISBN-10: 1584889373
  • ISBN-13: 9781584889373
Kitos knygos pagal šią temą:
Pastovi nuoroda: https://www.kriso.lt/db/9781584889373.html
Raktažodžiai:
One of the oldest branches of mathematics, number theory is a vast field devoted to studying the properties of whole numbers. Offering a flexible format for a one- or two-semester course, Introduction to Number Theory uses worked examples, numerous exercises, and two popular software packages to describe a diverse array of number theory topics. This classroom-tested, student-friendly text covers a wide range of subjects, from the ancient Euclidean algorithm for finding the greatest common divisor of two integers to recent developments that include 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. The book interweaves the theoretical development of the material with Mathematica® and Maple calculations while giving brief tutorials on the software in the appendices.

Highlighting both fundamental and advanced topics, this introduction provides all of the tools to achieve a solid foundation in number theory.

Recenzijos

Introduction to Number Theory is a well-written book on this important branch of mathematics. 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. I would certainly recommend it to a vast audience, and it is to be considered a valid and flexible textbook for any undergraduate number theory course. IACR Book Reviews, May 2011



Erickson and Vazzana provide a solid book, comprising 12 chapters, for courses in this area All in all, a welcome addition to the stable of elementary number theory works for all good undergraduate libraries. J. McCleary, Vassar College, CHOICE, Vol. 46, No. 1, August 2009



reader-friendly text 'Highlighting both fundamental and advanced topics, this introduction provides all of the tools to achieve a solid foundation in number theory.' LEnseignement Mathématique, Vol. 54, No. 2, 2008

I Core Topics 1
1 Introduction
3
1.1 What is number theory?
3
1.2 The natural numbers
6
1.3 Mathematical induction
9
1.4 Notes
14
The Peano Axioms
14
2 Divisibility and Primes
15
2.1 Basic definitions and properties
15
2.2 The division algorithm
17
2.3 Greatest common divisor
21
2.4 The Euclidean algorithm
26
2.5 Linear Diophantine equations
33
2.6 Primes and the Fundamental Theorem of Arithmetic
41
2.7 Notes
50
Euclid
50
The number of steps in the Euclidean algorithm
50
Nonunique factorization
53
3 Congruences
57
3.1 Residue classes
57
3.2 Linear congruences
61
3.3 Application: Check digits and the ISBN system
66
3.4 Fermat's theorem and Euler's theorem
69
3.5 The Chinese remainder theorem
75
3.6 Wilson's theorem
80
3.7 Order of an element mod n
83
3.8 Existence of primitive roots
86
3.9 Application: Construction of the regular 17-gon
92
3.10 Notes
98
Leonhard Euler
98
Groups
98
Straightedge and compass constructions
100
4 Cryptography
101
4.1 Monoalphabetic substitution ciphers
101
4.2 The Pohlig—Hellman cipher
107
4.3 The Massey—Omura exchange
114
4.4 The RSA algorithm
120
4.5 Notes
125
Computing powers mod p
125
RSA cryptography
127
5 Quadratic Residues
129
5.1 Quadratic congruences
129
5.2 Quadratic residues and nonresidues
131
5.3 Quadratic reciprocity
136
5.4 The Jacobi symbol
143
5.5 Application: Construction of tournaments
147
5.6 Consecutive quadratic residues and nonresidues
151
5.7 Application: Hadamard matrices
155
5.8 Notes
157
Carl Friedrich Gauss
157
II Further Topics 159
6 Arithmetic Functions
161
6.1 Perfect numbers
161
6.2 The group of arithmetic functions
169
6.3 Mains inversion
177
6.4 Application: Cyclotoinic polynomials
182
6.5 Partitions of an integer
186
6.6 Notes
201
The lore of perfect numbers
201
Pioneers of integer partitions
202
7 Large Primes
205
7.1 Prime listing, primality testing, and prime factorization
205
7.2 Fermat numbers
217
7.3 Mersenne numbers
226
7.4 Prime certificates
232
7.5 Finding large primes
236
7.6 Notes
243
Eratosthenes
243
8 Continued Fractions
245
8.1 Finite continued fractions
246
8.2 Infinite continued fractions
258
8.3 Rational approximation of real numbers
267
8.4 Periodic continued fractions
279
8.5 Continued fraction factorization
292
8.6 Notes
299
Continued fraction expansion of e
299
Continued fraction expansion of tan x
300
Srinivasa Ramanujan
300
9 Diophantine Equations
303
9.1 Linear equations
303
9.2 Pythagorean triples
306
9.3 Gaussian integers
309
9.4 Sums of squares
321
9.5 The case n = 4 in Fermat's Last Theorem
326
9.6 Pell's equation
329
9.7 Continued fraction solution of Pell's equation
338
9.8 The abc conjecture
344
9.9 Notes
349
Diophantus
349
Pierre de Fermat
349
Three squares and triangular numbers
350
History of Pell's equation
352
The p-adic numbers
352
III Advanced Topics 357
10 Analytic Number Theory
359
10.1 Sum of reciprocals of primes
359
10.2 Orders of growth of functions
362
10.3 Chebyshev's theorem
364
10.4 Bertrand's Postulate
371
10.5 The Prime Number Theorem
375
10.6 The zeta function and the Riemann hypothesis
381
10.7 Dirichlet's theorem
386
10.8 Notes
393
Paul Eras
393
11 Elliptic Curves
395
11.1 Cubic curves
396
11.2 Intersections of lines and curves
398
11.3 The group law and addition formulas
407
11.4 Sums of two cubes
413
11.5 Elliptic curves mod p
417
11.6 Encryption via elliptic curves
421
11.7 Elliptic curve method of factorization
426
11.8 Fermat's Last Theorem
433
11.9 Notes
439
Projective space
439
Associativity of the group law
441
Elliptic curve calculations
444
12 Logic and Number Theory
453
12.1 Solvable and unsolvable equations
453
12.2 Diophantine equations and Diophantine sets
455
12.3 Positive values of polynomials
461
12.4 Logic background
466
12.5 The negative solution of Hilbert's Tenth Problem
475
12.6 Diophantine representation of the set of primes
486
12.7 Notes
489
Julia Bowman Robinson
489
A Mathematica Basics 491
B Maple Basics 499
C Web Resources 503
D Notation 507
References 511
Index 515


Truman State University, Kirksville, Missouri, USA Truman State University, Kirksville, Missouri, USA