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Number theory : an introduction via the distribution of primes

Author: Benjamin Fine; Gerhard Rosenberger
Publisher: Boston : Birkhäuser, ©2007.
Edition/Format:   Print book : EnglishView all editions and formats
Summary:
"This book provides an introduction and overview of number theory based on the distribution and properties of primes. This unique approach provides both a firm background in the standard material as well as an overview of the whole discipline. All the essential topics are covered: fundamental theorem of arithmetic, theory of congruences, quadratic reciprocity, arithmetic functions, and the distribution of  Read more...
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Additional Physical Format: Online version:
Fine, Benjamin, 1948-
Number theory.
Boston : Birkhäuser, ©2007
(OCoLC)681503498
Material Type: Internet resource
Document Type: Book, Internet Resource
All Authors / Contributors: Benjamin Fine; Gerhard Rosenberger
ISBN: 9780817644727 0817644725 9780817645410 0817645411
OCLC Number: 68804135
Notes: Textbook.
Description: xiii, 342 pages : illustrations ; 24 cm
Contents: 1 Introduction and Historical Remarks 1 --
2 Basic Number Theory 7 --
2.1 The Ring of Integers 7 --
2.2 Divisibility, Primes, and Composites 11 --
2.3 The Fundamental Theorem of Arithmetic 16 --
2.4 Congruences and Modular Arithmetic 21 --
2.4.1 Basic Theory of Congruences 22 --
2.4.2 The Ring of Integers Modulo n 23 --
2.4.3 Units and the Euler Phi Function 26 --
2.4.4 Fermat's Little Theorem and the Order of an Element 31 --
2.4.5 On Cyclic Groups 34 --
2.5 The Solution of Polynomial Congruences Modulo m 37 --
2.5.1 Linear Congruences and the Chinese Remainder Theorem 37 --
2.5.2 Higher-Degree Congruences 42 --
2.6 Quadratic Reciprocity 45 --
3 The Infinitude of Primes 55 --
3.1 The Infinitude of Primes 55 --
3.1.1 Some Direct Proofs and Variations 55 --
3.1.2 Some Analytic Proofs and Variations 58 --
3.1.3 The Fermat and Mersenne Numbers 61 --
3.1.4 The Fibonacci Numbers and the Golden Section 65 --
3.1.5 Some Simple Cases of Dirichlet's Theorem 78 --
3.1.6 A Topological Proof and a Proof Using Codes 83 --
3.2 Sums of Squares 86 --
3.2.1 Pythagorean Triples 87 --
3.2.2 Fermat's Two-Square Theorem 89 --
3.2.3 The Modular Group 94 --
3.2.4 Lagrange's Four-Square Theorem 100 --
3.2.5 The Infinitude of Primes Through Continued Fractions 102 --
3.3 Dirichlet's Theorem 104 --
3.4 Twin Prime Conjecture and Related Ideas 121 --
3.5 Primes Between x and 2x 122 --
3.6 Arithmetic Functions and the Mobius Inversion Formula 123 --
4 The Density of Primes 133 --
4.1 The Prime Number Theorem: Estimates and History 133 --
4.2 Chebychev's Estimate and Some Consequences 136 --
4.3 Equivalent Formulations of the Prime Number Theorem 149 --
4.4 The Riemann Zeta Function and the Riemann Hypothesis 157 --
4.4.1 The Real Zeta Function of Euler 158 --
4.4.2 Analytic Functions and Analytic Continuation 163 --
4.4.3 The Riemann Zeta Function 166 --
4.5 The Prime Number Theorem 173 --
4.6 The Elementary Proof 180 --
4.7 Some Extensions and Comments 185 --
5 Primality Testing: An Overview 197 --
5.1 Primality Testing and Factorization 197 --
5.2 Sieving Methods 198 --
5.2.1 Brun's Sieve and Brun's Theorem 204 --
5.3 Primality Testing and Prime Records 212 --
5.3.1 Pseudoprimes and Probabilistic Testing 218 --
5.3.2 The Lucas-Lehmer Test and Prime Records 225 --
5.3.3 Some Additional Primality Tests 231 --
5.4 Cryptography and Primes 234 --
5.4.1 Some Number-Theoretic Cryptosystems 237 --
5.4.2 Public Key Cryptography and the RSA Algorithm 240 --
5.5 The AKS Algorithm 243 --
6 Primes and Algebraic Number Theory 253 --
6.1 Algebraic Number Theory 253 --
6.2 Unique Factorization Domains 255 --
6.2.1 Euclidean Domains and the Gaussian Integers 261 --
6.2.2 Principal Ideal Domains 268 --
6.2.3 Prime and Maximal Ideals 272 --
6.3 Algebraic Number Fields 275 --
6.3.1 Algebraic Extensions of Q 282 --
6.3.2 Algebraic and Transcendental Numbers 284 --
6.3.3 Symmetric Polynomials 287 --
6.3.4 Discriminant and Norm 290 --
6.4 Algebraic Integers 294 --
6.4.1 The Ring of Algebraic Integers 296 --
6.4.2 Integral Bases 297 --
6.4.3 Quadratic Fields and Quadratic Integers 300 --
6.4.4 The Transcendence of e and [pi] 303 --
6.4.5 The Geometry of Numbers: Minkowski Theory 306 --
6.4.6 Dirichlet's Unit Theorem 308 --
6.5 The Theory of Ideals 311 --
6.5.1 Unique Factorization of Ideals 313 --
6.5.2 An Application of Unique Factorization 319 --
6.5.3 The Ideal Class Group 321 --
6.5.4 Norms of Ideals 323 --
6.5.5 Class Number 326.
Responsibility: Benjamin Fine, Gerhard Rosenberger.
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Abstract:

This book provides an introduction and overview of number theory based on the distribution and properties of primes. This unique approach provides both a firm background in the standard material as  Read more...

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This book attempts the rather ambitious task of covering much of elementary, analytic and algebraic number theory in a bit over 300 pages. -MathSciNet

 
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