Number Theory: An Introduction via the Distribution of Primes

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Springer Science & Business Media, Jun 4, 2007 - Mathematics - 342 pages
Number theory is fascinating. Results about numbers often appear magical, both in theirstatementsandintheeleganceoftheirproofs. Nowhereisthismoreevidentthan inresultsaboutthesetofprimenumbers. Theprimenumbertheorem,whichgivesthe asymptotic density of the prime numbers, is often cited as the most surprising result in all of mathematics. It certainly is the result that is hardest to justify intuitively. The prime numbers form the cornerstone of the theory of numbers. Many, if not most, results in number theory proceed by considering the case of primes and then pasting the result together for all integers using the fundamental theorem of arithmetic. The purpose of this book is to give an introduction and overview of number theory based on the central theme of the sequence of primes. The richness of this somewhat unique approach becomes clear once one realizes how much number theoryandmathematicsingeneralareneededinordertolearnandtrulyunderstandthe prime numbers. Our approach provides a solid background in the standard material as well as presenting an overview of the whole discipline. All the essential topics are covered: fundamental theorem of arithmetic, theory of congruences, quadratic reciprocity, arithmetic functions, the distribution of primes. In addition, there are ?rm introductions to analytic number theory, primality testing and cryptography, and algebraic number theory as well as many interesting side topics. Full treatments and proofs are given to both Dirichlet’s theorem and the prime number theorem. There is acompleteexplanationofthenewAKSalgorithm,whichshowsthatprimalitytesting is of polynomial time. In algebraic number theory there is a complete presentation of primes and prime factorizations in algebraic number ?elds.
 

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I was disappointed by this book overall, but especially the proofs. Many were jumpy and hard to follow, not clearly explaining what was going on. Some left out important truth conditions, and several had fatal typos that made the math impossible to follow. The whole text seemed like it had been rushed to the printer without thorough editing or review. The 2nd edition will hopefully be better. 

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good phi

Contents

Introduction and Historical Remarks
1
Basic Number Theory
6
22 Divisibility Primes and Composites
11
23 The Fundamental Theorem of Arithmetic
16
24 Congruences and Modular Arithmetic
21
241 Basic Theory of Congruences
22
242 The Ring of Integers Modulo n
23
243 Units and the Euler Phi Function
26
47 Some Extensions and Comments
185
Primality Testing An Overview
197
52 Sieving Methods
198
521 Bruns Sieve and Bruns Theorem
204
53 Primality Testing and Prime Records
212
531 Pseudoprimes and Probabilistic Testing
218
532 The LucasLehmer Test and Prime Records
225
533 Some Additional Primality Tests
231

244 Fermats Little Theorem and the Order of an Element
31
245 On Cyclic Groups
34
25 The Solution of Polynomial Congruences Modulo m
37
252 HigherDegree Congruences
42
26 Quadratic Reciprocity
45
The Infinitude of Primes
55
312 Some Analytic Proofs and Variations
58
313 The Fermat and Mersenne Numbers
61
314 The Fibonacci Numbers and the Golden Section
65
315 Some Simple Cases of Dirichlets Theorem
78
316 A Topological Proof and a Proof Using Codes
83
32 Sums of Squares
86
321 Pythagorean Triples
87
322 Fermats TwoSquare Theorem
89
323 The Modular Group
94
324 Lagranges FourSquare Theorem
100
325 The Infinitude of Primes Through Continued Fractions
102
33 Dirichlets Theorem
104
34 Twin Prime Conjecture and Related Ideas
121
35 Primes Between x and 2x
122
36 Arithmetic Functions and the Möbius Inversion Formula
123
The Density of Primes
133
42 Chebychevs Estimate and Some Consequences
136
43 Equivalent Formulations of the Prime Number Theorem
149
44 The Riemann Zeta Function and the Riemann Hypothesis
157
441 The Real Zeta Function of Euler
158
442 Analytic Functions and Analytic Continuation
163
443 The Riemann Zeta Function
166
45 The Prime Number Theorem
173
46 The Elementary Proof
180
54 Cryptography and Primes
234
541 Some NumberTheoretic Cryptosystems
237
542 Public Key Cryptography and the RSA Algorithm
240
55 The AKS Algorithm
243
Primes and Algebraic Number Theory
252
62 Unique Factorization Domains
255
621 Euclidean Domains and the Gaussian Integers
261
622 Principal Ideal Domains
268
623 Prime and Maximal Ideals
272
63 Algebraic Number Fields
275
631 Algebraic Extensions of Q
282
632 Algebraic and Transcendental Numbers
284
633 Symmetric Polynomials
287
634 Discriminant and Norm
290
64 Algebraic Integers
294
641 The Ring of Algebraic Integers
296
642 Integral Bases
297
643 Quadratic Fields and Quadratic Integers
300
644 The Transcendence of e and π
303
Minkowski Theory
306
646 Dirichlets Unit Theorem
308
65 The Theory of Ideals
311
651 Unique Factorization of Ideals
313
652 An Application of Unique Factorization
319
653 The Ideal Class Group
321
654 Norms of Ideals
323
655 Class Number
326
Bibliography and Cited References
333
Index
337
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Page 10 - A then k + 1 e A. It can be concluded that in this case, A = N. This generalization provides the basis for proof by mathematical induction. The corresponding theorem is as follows: If for a given statement about positive integers, the statement is true for 1 and if it is true for n = k, then it is true for n = k + 1 , then the statement is true for all positive integers.