# Preface... 5 Arithmetic Functions The Euler-Maclaurin Summation Formula Average Orders The Riemann ζ-function...

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### Contents

Preface . . . ix

1 Algebraic Number Theory and Quadratic Fields 1 1.1 Algebraic Number Fields . . . 1

1.2 The Gaussian field. . . 19

1.3 Euclidean Quadratic Fields . . . 34

1.4 Applications of Unique Factorization . . . 50

2 Ideals 59 2.1 The Arithmetic of Ideals in Quadratic Fields. . . 59

2.2 Dedekind Domains . . . 71

2.3 Application to Factoring . . . 93

3 Binary Quadratic Forms 103 3.1 Basics. . . 103

3.2 Composition and the Form Class Group. . . 111

3.3 Applications Via Ambiguity . . . 124

3.4 Genus. . . 135

3.5 Representation . . . 154

3.6 Equivalence Modulo p . . . 161

4 Diophantine Approximation 165 4.1 Algebraic and Transcendental Numbers . . . 165

4.2 Transcendence . . . 177

4.3 Minkowski’s Convex Body Theorem . . . 188

5 Arithmetic Functions 197 5.1 The Euler-Maclaurin Summation Formula . . . 197

5.2 Average Orders . . . 214

5.3 The Riemann ζ-function . . . 224

6 Introduction to p-Adic Analysis 235 6.1 Solving Modulo pn . . . 235

6.2 Introduction to Valuations . . . 239

6.3 Non- Vs. Archimedean Valuations . . . 246

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viii

6.4 Representation of p-Adic Numbers . . . 249

7 Dirichlet: Characters, Density, and Primes in Progression 253 7.1 Dirichlet Characters . . . 253

7.2 Dirichlet’s L-Function and Theorem . . . 258

7.3 Dirichlet Density. . . 269

8 Applications to Diophantine Equations 277 8.1 Lucas-Lehmer Theory . . . 277

8.2 Generalized Ramanujan-Nagell Equations. . . 282

8.3 Bachet’s Equation . . . 288

8.4 The Fermat Equation . . . 292

8.5 Catalan and the ABC-Conjecture . . . 300

9 Elliptic Curves 307 9.1 The Basics. . . 307

9.2 Mazur, Siegel, and Reduction . . . 316

9.3 Applications: Factoring & Primality Testing . . . 323

9.4 Elliptic Curve Cryptography (ECC) . . . 332

10 Modular Forms 337 10.1 The Modular Group . . . 337

10.2 Modular Forms and Functions . . . 342

10.3 Applications to Elliptic Curves. . . 353

10.4 Shimura-Taniyama-Weil & FLT . . . 359

Appendix A: Sieve Methods . . . 374

Bibliography . . . 397

Solutions to Odd-Numbered Exercises . . . 405

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### Preface

This book is designed as a second course in number theory at the senior undergraduate/junior graduate level to follow a course in elementary methods, such as that given in [65], the contents of which the reader is assumed to have knowledge. The material covered in the ten chapters of this book constitute a course outline for one semester.

Chapter One begins with algebraic techniques including specialization to quadratic fields with applications to solutions of the Ramanujan-Nagell equa-tions, factorization of Gaussian integers, Euclidean quadratic fields, and Gauss’ proof of Fermat’s Last Theorem (FLT) forp= 3. Applications of unique fac-torization are given in terms of both Euler’s and Fermat’s solution to Bachet’s equation, concluding with a look at norm-Euclidean quadratic fields.

In Chapter Two ideal theory is covered beginning with quadratic fields, and decomposition into prime ideals therein. Dedekind domains make up the second section, leading into Noetherian domains, and the unique factorization theorem for Dedekind domains. Principal Ideal Domains and Unique Factoriza-tion Domains are compared and contrasted. The secFactoriza-tion ends with the Chinese Remainder Theorem for ideals. The chapter concludes with an application to factoring using Pollard’s cubic integer method, which serves as a preamble for the introduction of the number field sieve presented in Appendix A. Pollard’s method is illustrated via factoring of the seventh Fermat number.

Chapter Three is devoted to binary quadratic forms, starting with the ba-sics on equivalence, discriminants, reduction, and class number. In the next section, composition is covered and linked to ideal theory. The form and ideal class groups are compared and contrasted, including an explicit formula for the relationship between the form class number and both the narrow and wide ideal class numbers. A proof of the finiteness of the ideal class number is achieved via the form class number, rather than the usual method of using Minkowski’s Convex Body Theorem, which we cover in§4.3. Section Three investigates the notion of ambiguous forms and ideals and the relationship between their classes. We show how this applies to representations of integers as a sum of two squares and to Markov triples. In the fourth section, genus is introduced and the as-signed values of generic characters is developed via Jacobi symbols. This is then applied to the principal genus, via a coset interpretation, using Dirichlet’s Theorem on Primes in Arithmetic Progression, the proof of which is given in Chapter Seven. This is a valuable vehicle for demonstrating the fact that two forms are in the same genus exactly when their cosets are equal. We tie the above together with the fact that the genus group is essentially the group of ambiguous forms. The fifth section uses the above to investigate representation problems. We begin with the algebraic interpretation of prime power repre-sentation as binary quadratic forms using the ideal class number. Numerous applications to representations of primes in the formp=a2+Db2are provided.

The chapter ends with representations modulo a prime.

Chapter Four develops Diophantine approximation techniques, starting with ix

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x Advanced Number Theory with Applications

Roth’s celebrated result. We prove Liouville’s Theorem, leading into an anal-ysis of enumerable sets, including a proof that the set of all algebraic numbers is enumerable, followed by the countablity of the rational numbers and the un-countability of the reals. Indeed, it follows from this that almost all reals are transcendental. The first section is completed with a proof of the fact that the n-th root of a rational integer is an algebraic integer of degree n, when that integer is not a certain power. Transcendence is covered in the second section with proofs that Liouville numbers, e, and πare all transcendental. Next the Lindemann-Weierstrass Theorem is established, allowing the statement of the more general Schanuel conjecture. The discussion is rounded out by a look at some renowned constants including that of Gel´fond, Gel´fond-Schneider, Proulet-Thue-Morse, Euler, Ap´ery, and Catalan. Section Three introduces the geometry of numbers and its techniques with a goal of proving Minkowski’s Convex Body Theorem that ends the chapter.

In Chapter Five, we extend the knowledge of arithmetic functions gained in a first course, by proving the Euler-Maclaurin summation formula, for which we introduce Bernoulli numbers, Bernoulli polynomials, and Fourier series. With this we are able to apply the formula to obtain Wallis’ formula, Stirling’s con-stant, Stirling’s formula, and perhaps the slickest of applications, namely the accurate approximation of the Euler-Mascheroni constant. Average orders is the topic of the second section starting with a proof of Hermite’s formula. This puts us into a position where we can derive the average order of the number of divisors function, the sum of divisors function, and Euler’s totient φ(m). The third section concentrates upon the Riemannζ-function. We apply the Euler-Maclaurin summation formula to obtain a formula for ζ(s). Then we discuss the Prime Number Theorem (PNT), Merten’s Theorem, and various arithmetic function equivalences to the PNT. Then the Riemann hypothesis (RH) and its equivalent formulations are considered, after which we develop techniques to provide a rather straightforward proof of the functional equation forζ(s) as a closing feature of the chapter.

In the sixth chapter, we introducep-adic analysis, commencing with solving modulopn for successively higher powers of a primep. Hensel’s Lemma is the featured result of the first section. The second section introduces valuations, including the p-adic versions. Then Cauchy sequences come into play giving rise top-adic fields and domains. We have tools to prove that equivalent powers are valuations, which ends the section. We compare Archimedean and non-Archimedean valuations in the third section, featuring a proof of Ostrowski’s Theorem. In the last section, we apply what we have learned to representation of p-adic numbers. This involves the proof that every rational number has a representation as a periodic power series in a given prime p to close out the chapter.

Chapter Seven delves into Dirichlet, his characters, L-functions, and their zeros related to the RH. We see the implications of his theorem for primes in arithmetic progression, proved in the second section. In the third section we introduce Dirichlet density and applications such as Beatty’s theorem. The chapter ends with Dirichlet density on primes in arithmetic progression modulo

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Preface xi

mwhich have density 1/φ(m).

Chapter Eight comprises applications of the first seven chapters to Dio-phantine equations. We begin with an overview of Lucas-Lehmer theory, prov-ing results promised earlier in the text such as solutions of the generalized Ramanujan-Nagell equations in the second section and Bachet’s equation in the third section. The Fermat equation is the topic of the fourth section with Kum-mer’s proof of FLT for regular primes. The chapter is rounded out with the ABC conjecture and Catalan’s conjecture. We discuss the recent proof of the latter and its generalization, the still open Fermat-Catalan conjecture. More than a half-dozen consequences of the ABC conjecture are displayed and discussed, including the Thue-Siegel-Roth Theorem, Hall’s conjecture, the Erd¨os-Mollin-Walsh conjecture, and the Granville-Langevin conjecture. We demonstrate how these follow from ABC.

Chapter Nine studies elliptic curves, launched by an introduction of the ba-sics, illustrated and presented as a foundation. The second section defines tor-sion points, the Nagell-Lutz Theorem, Mazur’s Theorem, Siegel’s Theorem, and the notion of reduction. This sets the stage for Lenstra’s elliptic curve factoring method and his primality testing method. We also look at the Goldwasser-Killian primality proving algorithm. The chapter closes with a description of the Menezes-Vanstone Elliptic Curve Cryptosystem as an application.

The last chapter is on modular forms. The modular group, and modular forms are introduced as vehicles for much deeper considerations later in the chapter. Spaces and levels of modular forms are used as applications to elliptic curves including j-invariants and the Weierstrass ℘-function. The main text ends with section four that looks, in detail, at the Shimura-Taniyama-Weil con-jecture both in terms of L-functions and modular parametrizations. Modular elliptic curves are introduced as the stepping-stone to the proof of FLT. The tenth chapter ends with Ribet’s Theorem and a one-paragraph proof of FLT emanating from it, called the Frey-Serre-Ribet approach, a fitting conclusion and demonstration of the power of the theory.

An overview, without proofs, of sieve theory is relegated to Appendix A. We begin with a description of the goals of sieve theory and the effects its study has had on such open problems as the twin prime conjecture, the Goldbach conjecture, and Artin’s conjecture, among others. We provide a description of the Eratosthenes sieve from the perspective of the M¨obius function in order to lay the foundation for modern-day sieves. We begin with Brun’s Theorem and his constant, including a discussion of how computation of Brun’s constant led to the discovery of a flaw in the Pentium computer chip. Then we set the groundwork for presentation of Selberg’s sieve by painting the picture of the ba-sic sieve problem in terms of upper and lower limits on certain related functions. Selberg’s sieve has many applications including the Brun-Titchmarsh Theorem, bounds for the twin prime conjecture, and the Goldbach conjecture. Then Lin-nik’s large sieve is developed as a generalization of Brun’s results and illustrated via applications to Artin’s conjecture. Next is the Bombieri-Vinogradov Theo-rem and its applications to the Titchmarsh divisor problem. Then the classic result, Bombieri’s asymptotic sieve, is presented via a hypothesis involving the

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xii Advanced Number Theory with Applications

generalized Mangoldt function. The most striking of the applications of the asymptotic sieve is the Friedlander-Iwaniec Theorem that there are infinitely many primes of the forma2+b4. The aforementioned hypothesis involves the

Elliot-Halberstram conjecture (EHC), so we are naturally led to the recent re-sults by Goldston, Pintz, and Yilidrim on gaps between primes. In particular, their result based upon the validity of the EHC is the satisfying conclusion that limn→∞inf(pn+1−pn) 16, wherepn is the n-th prime. With these results

as an illustration of the power of sieve theory, we turn our attention to the use of sieves in factoring by bringing out the big gun, the number field sieve and illustrate in detail its use in factoring of the ninth Fermat number.

The Bibliography has been set up in such a way that maximum information is imparted. This includes a page reference for each and every citing of a given item, so that no guess work is involved as to where this reference is used. The index has more than 1,500 entries presented for maximum cross-referencing. Similarly, any reference, in text, to a theorem, definition, etc. is coupled with the page number on which it sits. These conventions ensure that the reader will find data with ease. There are nearly 50 mini-biographies of the mathematicians who helped to develop the results presented, in order to give a human face to the number theory and its applications. There are nearly 340 exercises with solutions of the odd-numbered exercises included at the end of the text, and a solutions manual for the even-numbered exercises available to instructors who adopt the text for a course. The website below is designed for the reader to access any updates and the e-mail address below is available for any comments.