Number theory an introduction via the density of primes second edition

Benjamin Fine Gerhard Rosenberger Number Theory An Introduction via the Density of Primes Second Edition Benjamin Fine Gerhard Rosenberger • Number Theory An Introduction via the Density of Primes Second Edition Gerhard Rosenberger Universität Hamburg Hamburg Germany Benjamin Fine Department of Mathematics Fairfield University Fairfield, CT USA ISBN 978-3-319-43873-3 DOI 10.1007/978-3-319-43875-7 ISBN 978-3-319-43875-7 (eBook) Library of Congress Control Number: 2016947201 Mathematics Subject Classification (2010): 11A01, 11A03, 11M01, 11R04, 11Z05, 11T71, 11H01, 20A01, 20G01, 14G01, 08A01 © Springer International Publishing AG 2007, 2016 This work is subject to copyright All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed The use of general descriptive names, registered names, trademarks, service marks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made Printed on acid-free paper This book is published under the trade name Birkhäuser, www.birkhauser-science.com The registered company is Springer International Publishing AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Preface to the Second Edition We were very pleased with the response to the first edition of this book and we were very happy to a second edition In this second edition, we cleaned up various typos pointed out by readers and added some new material suggested by them We have also included important new results that have appeared since the first edition came out These results include results on the gaps between primes and the twin primes conjecture We have added a new chapter, Chapter 7, on p-adic numbers, p-adic arithmetic, and the use of Hensel’s Lemma This can be included in a year-long course We have extended the material on elliptic curves in Chapter on primality testing We have added material in Chapter on multiple-valued zeta functions As before, we would like to thank the many people who read or used the first edition and made suggestions We would also especially like to thank Anja Moldenhauer and Anja Rosenberger who helped tremendously with editing and LATEX and made some invaluable suggestions about the contents Fairfield, USA Hamburg, Germany Benjamin Fine Gerhard Rosenberger v Preface to the First Edition Number theory is fascinating Results about numbers often appear magical, both in their statements and in the elegance of their proofs Nowhere is this more evident than in results about the set of prime numbers The Prime Number Theorem, which gives the asymptotic density of the prime numbers, is often cited as the most surprising result in all of mathematics It certainly is the result which 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 by 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 theory and mathematics in general is needed to learn and truly understand the prime numbers The 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 the fundamental theorem of arithmetic, theory of congruences, quadratic reciprocity, arithmetic functions, and the distribution of primes In addition, there are firm 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 a complete explanation of the new AKS algorithm that shows that primality testing is of polynomial time In algebraic number theory, there is a complete presentation of primes and prime factorizations in algebraic number fields The book grew out of notes from several courses given for advanced undergraduates in the United States and for teachers in Germany The material on the Prime Number Theorem grew out of seminars also given both at the University of Dortmund and at Fairfield University The intended audience is upper level undergraduates and beginning graduate students The notes upon which the book was based were used effectively in such courses in both the United States and vii viii Preface to the First Edition Germany The prerequisites are a knowledge of Calculus and Multivariable Calculus and some Linear Algebra The necessary ideas from Abstract Algebra and Complex Analysis are introduced in the book There are many interesting exercises ranging from simple to quite difficult Solutions and hints are provided to selected exercises We have written the book in what we feel is a user-friendly style with many discussions of the history of various topics It is our opinion that it is also ideal for self-study There are two basic facts concerning the sequence of primes that are focused on in this book and from which much of the theory of numbers is introduced The first fact is that there are infinitely many primes This fact was of course known since at least the time of Euclid However, there are a great many proofs of this result not related to Euclid’s original proof By considering and presenting many of these proofs, a wide area of modern number theory is covered This includes the fact that the primes are numerous enough so that there are innitely many in any arithmetic progression an ỵ b with a; b relatively prime (Dirichlet’s Theorem) The proof of Dirichlet’s Theorem allows us to first introduce analytic methods In distinction to there being infinitely many primes, the density of primes thins out We first encounter this in the startling (but easily proved) result that there are arbitrarily large gaps in the sequence of primes The exact nature of how the sequence of primes thins out is formalized in the Prime Number Theorem, which as already mentioned, many people consider the most surprising result in mathematics Presenting the proof and the ideas surrounding the proof of the Prime Number Theorem allows us to introduce and discuss a large portion of analytic number theory Algebraic Number Theory arose originally as an attempt to extend unique factorization to algebraic number rings We use the approach of looking at primes and prime factorizations to present a fairy comprehensive introduction to algebraic number theory Finally, modern cryptography is intimately tied to number theory Especially crucial in this connection is primality testing We discuss various primality testing methods, including the recently developed AKS algorithm and then provide a basic introduction to cryptography There are several ways that this book can be used for courses Chapter together with selections from the remaining chapters can be used for a one-semester course in number theory for undergraduates or beginning graduate students The only prerequisites are a basic knowledge of mathematical proofs (induction, etc.) and some knowledge of Calculus All the rest is self-contained, although we use algebraic methods so that some knowledge of basic abstract algebra would be beneficial A year-long course focusing on analytic methods can be done from Chapters 1, 2, 3, and and selections from and 6, while a year-long course focusing on algebraic number theory can be fashioned from Chapters 1, 2, 3, and and selections from and There are also possibilities for using the book for one semester introductory courses in analytic number theory, centering on Chapter 4, or for a one semester introductory course in algebraic number theory, centering on Chapter Some suggested courses: Preface to the First Edition ix Basic Introductory One Semester Number Theory Course: Chapter 1, Chapter 2, Sections 3.1, 4.1, 4.2, 5.1, 5.3, 5.4, 6.1 Year-Long Course Focusing on Analytic Number Theory: Chapter 1, Chapter 2, Chapter 3, Chapter 4, Sections 5.1, 5.3, 5.4, 6.1 Year-Long Course Focusing on Algebraic Number Theory: Chapter 1, Chapter 2, Chapter 3, Chapter 6, Sections 4.1, 4.2, 5.1, 5.3, 5.4 One-Semester Course Focusing on Analytic Number Theory: Chapter 1, Chapter (as needed), Sections 3.1, 3.2, 3.3, 3.4, 3.5, Chapter One-Semester Course Focusing on Algebraic Number Theory: Chapter 1, Chapter (as needed), Chapter We would like to thank the many people who have read through other preliminary versions of these notes and made suggestions Included among these people are Kati Bencsath and Al Thaler, as well as the many students who have taken the courses In particular, we would like to thank Peter Ackermann, who read through the whole manuscript both proofreading and making mathematical suggestions Peter was also heavily involved in the seminars on the Prime Number Theorem from which much of the material in Chapter comes Benjamin Fine Gerhard Rosenberger Contents Introduction and Historical Remarks Basic Number Theory 2.1 The Ring of Integers 2.2 Divisibility, Primes, and Composites 2.3 The Fundamental Theorem of Arithmetic 2.4 Congruences and Modular Arithmetic 2.4.1 Basic Theory of Congruences 2.4.2 The Ring of Integers Mod N 2.4.3 Units and the Euler Phi Function 2.4.4 Fermat’s Little Theorem and the Order of an Element 2.4.5 On Cyclic Groups 2.5 The Solution of Polynomial Congruences Modulo m 2.5.1 Linear Congruences and the Chinese Remainder Theorem 2.5.2 Higher Degree Congruences 2.6 Quadratic Reciprocity 2.7 Exercises 7 10 16 22 22 23 27 32 36 39 39 45 48 55 The Infinitude of Primes 3.1 The Infinitude of Primes 3.1.1 Some Direct Proofs and Variations 3.1.2 Some Analytic Proofs and Variations 3.1.3 The Fermat and Mersenne Numbers 3.1.4 The Fibonacci Numbers and the Golden Section 3.1.5 Some Simple Cases of Dirichlet’s Theorem 3.1.6 A Topological Proof and a Proof Using Codes 3.2 Sums of Squares 3.2.1 Pythagorean Triples 3.2.2 Fermat’s Two-Square Theorem 59 59 59 62 66 71 84 89 92 93 96 xi xii Contents 3.3 3.4 3.5 3.6 3.7 The 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 3.2.3 The Modular Group 3.2.4 Lagrange’s Four Square Theorem 3.2.5 The Infinitude of Primes Through Continued Fractions Dirichlet’s Theorem Twin Prime Conjecture and Related Ideas Primes Between x and 2x Arithmetic Functions and the Möbius Inversion Formula Exercises 100 107 110 112 131 132 133 138 Density of Primes The Prime Number Theorem—Estimates and History Chebyshev’s Estimate and Some Consequences Equivalent Formulations of the Prime Number Theorem The Riemann Zeta Function and the Riemann Hypothesis 4.4.1 The Real Zeta Function of Euler 4.4.2 Analytic Functions and Analytic Continuation 4.4.3 The Riemann Zeta Function The Prime Number Theorem The Elementary Proof Multiple Zeta Values Some Extensions and Comments Exercises 143 143 147 159 169 170 175 179 186 193 198 206 213 Primality Testing—An Overview 5.1 Primality Testing and Factorization 5.2 Sieving Methods 5.2.1 Brun’s Sieve and Brun’s Theorem 5.3 Primality Testing and Prime Records 5.3.1 Pseudo-Primes and Probabilistic Testing 5.3.2 The Lucas–Lehmer Test and Prime Records 5.3.3 Some Additional Primality Tests 5.3.4 Elliptic Curve Methods 5.4 Cryptography and Primes 5.4.1 Some Number Theoretic Cryptosystems 5.5 Public Key Cryptography and the RSA Algorithm 5.6 Elliptic Curve Cryptography 5.7 The AKS Algorithm 5.8 Exercises 219 219 220 226 236 241 249 255 257 263 267 270 273 276 282 Primes and Algebraic Number Theory 6.1 Algebraic Number Theory 6.2 Unique Factorization Domains 6.2.1 Euclidean Domains and the Gaussian Integers 6.2.2 Principal Ideal Domains 6.2.3 Prime and Maximal Ideals 285 285 287 293 301 304 7.8 Hensel’s Lemma and Applications 399 Theorem 7.8.1 (Hensel’s Lemma) Let f (x) = c0 + c1 x + · · · + cn x n be a polynomial in Z p [x] (coefficients are p-adic integers) Let f (x) be the formal derivative of f (x) Suppose a ∈ Z p with f (a ) ≡ mod p and f (a ) ≡ mod p Then, there exists a unique p-adic integer a such that f (a) = and a ≡ a mod p As preparation for the proof of Hensel’s lemma we recall Newton’s method for solving a non-linear equation f (x) = over the reals where f (x) is a differentiable real-valued function We start with an initial guess x0 This initial guess must be sufficiently close to a solution for this method to work but we will ignore this here and refer to [A] for the technical requirements Given x0 we form the tangent line to the curve y = f (x) at the point (x0 , f (x0 )) This has the equation y − f (x0 ) = f (x0 )(x − x0 ) Let x1 be where the tangent line crosses the x-axis, that is where y = We then have f (x0 ) − f (x0 ) = f (x0 )(x1 − x0 ) =⇒ x1 = x0 − f (x0 ) assuming that f (x0 ) = This provides the initial step in an iteration scheme Consider the tangent line at (x1 , f (x1 )) and obtain x2 = x1 − and in general xn+1 = xn − f (x1 ) assuming f (x1 ) = f (x1 ) f (xn ) assuming f (xn ) = f (xn ) Under appropriate conditions (see [A]) this iteration scheme will converge to a solution of f (x) = How close the initial guess must be to a solution for the method to converge depends on the function f (x) (see [A]) This method can be applied to polynomial equations P(x) = over the reals The proof of Hensel’s lemma in the p-adic field Q p utilizes a p-adic version of Newton’s technique Proof Let f (x) be an p-adic integral polynomial, that is, f (x) has p-adic coefficients, and let a be as in the statement of Hensel’s lemma We will prove the existence of a solution a by inductively constructing its canonical p-adic expansion a = d0 + d1 p + · · · + dk p k + · · · where di are p-adic digits to be determined Let ak be the k-th convergent for a, ak = d0 + d1 + · · · + dk p k The Fields Q p of p-Adic Numbers: Hensel’s Lemma 400 We will use an induction and a p-adic version of Newton’s method to show that we can find p-adic digits so that f (ak ) ≡ mod p k+1 and ak ≡ a mod p Then as ak → a we have a as the desired solution Let a have the canonical p-adic expansion a = b0 + b1 p + · · · + bk p k + · · · Take a0 = d0 = b0 Then a0 ≡ a mod p and f (a0 ) ≡ mod p This establishes the lowest level of an induction Now, suppose we have ak−1 satisfying f (ak−1 ) ≡ mod p k and ak−1 ≡ a mod p Now let ak = ak−1 + dk p k where dk is a p-adic digit to be determined Then n f (ak ) = f (ak−1 + dk p k ) = ci (ak−1 + dk p k )i i=0 Then n f (ak ) = c0 + i ci (ak−1 + i(aki−1 dk p k + terms in powers higher than p k+1 )) i=1 This implies that f (ak ) = f (ak−1 ) + dk p k f (ak−1 ) By the inductive hypothesis we have f (ak−1 ) ≡ mod p k and hence there is a p-adic digit ek with f (ak ) = ek p k + dk p k f (ak−1 ) To obtain the appropriate digit dk we must then have ek + dk f (ak−1 ) ≡ mod p Since ak−1 ≡ a mod p we have f (ak−1 ) ≡ f (a ) ≡ mod p Therefore, the digit dk can be found by ek mod p dk = − f (ak−1 ) and hence f (ak ) ≡ mod p Notice that approximating the p-adic digits uses essentially the same iteration scheme as Newton’s method over the reals 7.8 Hensel’s Lemma and Applications 401 Now consider a = d0 + d1 p + · · · + dk p k + · · · Since f (a) ≡ f (ak ) mod p k+1 for all k we must have f (a) = Now assume that ak−1 has the desired properties and consider ak Let dk be a p-adic digit to be determined and consider ak = ak−1 + dk p k The uniqueness of a follows from the uniqueness of the sequence of convergents ak The proof of Hensel’s lemma provides an algorithm for constructing the solution to an equation f (x) = with f (x) ∈ Z p [x] This algorithm is analogous to Newton’s Method for solving real polynomial equations Suppose a is a solution to f (x) ≡ mod p Then follow the procedure outlined in the proof Take d0 the first p-adic digit of a and let a0 = a Let ak = ak−1 + dk p k k−1 and iteratively find the digits dk by dk = f−a for k ≥ (ak−1 ) Theorem 7.8.2 A polynomial with rational integer coefficients (in Z[x]) has a root in Z p if and only if it has an integer root modulo p k for any k ≥ Proof Suppose that f (x) ∈ Z[x] and suppose that f (a) = where a ∈ Z p Then from the proof of Hensel’s lemma there exists a sequence of integers (ak ) with ak ≡ a mod p k Since f (ak ) ≡ f (a) mod p k and f (a) = we must have an integer solution mod p k for each k Conversely, suppose that for each k there is an integer ak with f (ak ) ≡ mod p k We have seen that the p-adic integers are complete so the sequence ak has a convergent subsequence (ak ) Suppose that the limit of this subsequence is a A polynomial is a continuous function on any normed field (see exercises) and hence f (a) = lim f (ak ) However f (ak ) ≡ mod p k for all k and therefore f (a) ≡ mod p k for all k and hence f (a) = Corollary 7.8.1 If a polynomial F(x) with integer coefficients has no roots modulo p then it has no roots Hensel’s lemma can be used to describe the roots of unity in Q p Theorem 7.8.3 For any prime p and (m, p) = there exists a primitive m-th root of unity in Q p if and only if m|( p − 1) In this case every m-th root of unity is also a ( p − 1)-th root of unity The set of ( p − 1)-th roots of unity forms a cyclic subgroup of U (Z p ) of order p − 402 The Fields Q p of p-Adic Numbers: Hensel’s Lemma Proof If m|( p − 1) then p − = km and hence every m-th root of unity in Q p is also a ( p − 1)-th root of unity Consider the polynomial f (x) = x p−1 − Then its formal derivative is f (x) = ( p − 1)x p−2 Now let a be a rational integer with ≤ a ≤ p − Then from Fermat’s theorem, we have f (a) = and further f (a) = since | f (a)| p = Therefore Hensel’s lemma implies that there are exactly p − solutions to f (x) = and they are all ( p − 1)-th roots of unity Conversely suppose that a ∈ Q p with a m = then |a| p = and a is a p-adic integer Let · · · a1 a0 = a then a ≡ a0 mod p and hence a0m = Since a0 is a rational integer this implies that m|( p − 1) The set of ( p − 1)-th roots of unity in Q p is then a finite subgroup of a field and as we saw in Theorem 2.4.13 this must be cyclic As we have seen in this book, quadratic residues modulo a prime are important in several different areas of number theory In fact determining quadratic residues was crucial in the Rabin encryption system The final result of this section ties quadratic residues modulo a prime p to square roots in the p-adic integers Lemma 7.8.1 A rational integer a not divisible by p has a square root in Z p ( p = 2) if and only if a is a quadratic residue modulo p Proof Let a ∈ Z with (a, p) = Consider the polynomial P(x) = x − a in Z p [x] Suppose that a is a quadratic residue mod p Then there exists a with a ∈ {1, 2, , p − 1} and a ≡ a 20 mod p Further P (x) = 2x and P (a ) = 2a = mod p since (a, p) = Therefore by Hensel’s lemma P(x) has a solution in Z p Conversely suppose that a is not a quadratic residue Then P(x) ≡ mod p and hence P(x) ≡ mod p k for any k It follows that P(x) can have no solution in Z p 7.8.1 The Non-isomorphism of the p-Adic Fields Since each p-adic field is non-archimedean we have seen from the characterization of R that for any prime p the p-adic field Q p is not isomorphic to R In the next theorem we use the results on square roots in Q p to provide another proof of this and to show that p-adic fields for different primes are non-isomorphic Theorem 7.8.4 The p-adic field Q p is not isomorphic to R for any prime p Further, if p1 and p2 are distinct primes then the corresponding p-adic fields are nonisomorphic Proof Let p be a prime and suppose that f : R → Q p is an isomorphism Then p has a square root in R and hence by the isomorphism f ( p) has a square root in Q p However, p is not a quadratic residue mod p and therefore p has no square root in Q p providing a contradiction If p1 = p2 then there are p12−1 quadratic residues mod p1 and p22−1 quadratic residues mod p2 It follows that if p2 > p1 there must exist an integer a which is 7.8 Hensel’s Lemma and Applications 403 a quadratic residue mod p2 but not mod p1 Use this integer a and then follow the same proof as above We leave the details to the exercises As a final application of both Hensel’s lemma and the utility of the p-adic fields in general we mention without proof the local-global principle of Hasse The rational numbers Q are called a global field while its Completions, the real numbers R and the p-adic fields Q p are called local fields Any relationship among a set of rational numbers which is true globally, that is in Q is also true locally, that is in R and all the p-adic fields Q p Hasse’s Global-Local Principle provides a partial converse for equations involving quadratic forms with integer coefficients: j xi x j + i, j bi xi + c = i If such an equation has solutions in R and in Q p for every prime p, then it has a rational solution in Q In other words, a quadratic equation with integer coefficients has a global solution, that is in Q if and only if it has solutions in all the local fields, that is in R and in Q p for all p 7.9 Exercises 7.1 Find the p-adic norm and p-adic expansion in Q7 of: (a) 15 (b) −1 (c) −3 (d) 13 7.2 Describe in detail, analogously as for R, the Cauchy completion of the rational numbers Q equipped with the p-adic norm for a prime p 7.3 Fill in the details of the proof of Theorem 7.8.4, that is if p1 = p2 then the p-adic fields Q p1 and Q p2 are not isomorphic 7.4 Let p be a prime number and Z p the p-adic integers Show that Z p / p n Z p is isomorphic to Z / p n Z for any n > 7.5 Let p be a prime number and Z p the p-adic integers Show that the additive group of Z p is torsion-free 7.6 Use the algorithm in the proof of Hensel’s Lemma to find a solution (if there exists one) of the polynomial equations: (a) x − 3x + 2x + = in Q7 (b) x − in Q11 7.7 Complete the proof that a p-adic expansion for x is periodic if and only if x is rational 7.8 Show that if x ∈ Q p and x ≡ mod p k for all k ≥ then x = 404 The Fields Q p of p-Adic Numbers: Hensel’s Lemma 7.9 Let f (x) ∈ Q p [x] that is a polynomial with p-adic coefficients Show that f (x) is a continuous function of Q p 7.10 Complete the 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Algebraically independent, 201 Analytic continuation, 177 Analytic function, 176 Analytic number theory, 2, 143 Apery’s constant, 173 Archimedean, 381, 384 Arithmetic function, 133 Ascending chain condition, 302 Associates, 21, 287 Asymptotically equal, 154 Attack, 264 Authentication, 270 B Basis, 309 Beale cipher, 266 Bernoulli numbers, 173, 199 Bertrand’s theorem, 132 Big O notation, 153 Binary expansion, 373 Brun’s constant, 228 © Springer International Publishing AG 2016 B Fine and G Rosenberger, Number Theory, DOI 10.1007/978-3-319-43875-7 C Caesar code, 265 Canonical p-adic expansion, 390 Carmichael number, 243 Cauchy completion, 382 Cauchy completion procedure, 373 Cauchy integral formula, 176 Cauchy sequence, 374 Cauchy’s Theorem, 176 Chebyshev functions, 160 Cheyshev’s estimate, 147 Chinese Remainder Theorem, 39 Ciphertext message, 263 Class number, 364 Classical cryptography, 264 Common divisor, 11 Commutative ring, Commutative ring with identity, Complete lattice, 342 Complete metric space, 374, 376 Complete residue system, 23 Complex integers, 286 Complext integral, 176 Composite, 11 Congruence, 22 Conjugates, 317 Continued fractions, 110 Convergent sequence, 374 Coprime, 12 Coset of an ideal, 306 Cousin primes, 132, 229 Cramer’s conjecture, 211 Critical line, 183 Critical strip, 183 Cryptanalysis, 264 Cryptography, 219, 263 Cyclic subgroup, 32 409 410 D Decryption, 264 Decryption map, 264 Degree of a polynomial, 288 Degree of an element, 312 Degree of an extension, 309 Dense subset, 376 Deterministic primality test, 236 Diffie–Hellman method, 271 Dimension, 309 Diophantime analysis, Dirichlet character, 100, 113 Dirichlet convolution, 196 Dirichlet series, 100, 119 Dirichlet’s theorem, 59, 112 Dirichlet’s Unit Theorem, 345 Discrete log, 271 Discrete log problem, 271 Discrete valuation, 397 Discrete valuation ring, 397 Discriminant, 326, 335 Divisibility, 10 Division algorithm, 11 Divisor, 10 E ECES, 274 Elementary number theory, Elementary symmetric function, 229 Elementary symmetric polynomial, 322 ElGamal encryption, 273 Elliptic curve, 257 Elliptic curve cryptosystem, 274 Elliptic curve ElGamal cryptosystem, 274 Elliptic curve elliptography, 274 Elliptic curve encryption scheme, 274 Elliptic pseudocurve, 258 Encryption, 264 Encryption map, 264 Entire function, 176 Equivalence relation, 22 Equivalent norms, 383 Euclid’s Lemma, 15 Euclidean algorithm, 13 Euclidean domain, 21, 294 Euclidean norm, 294 Euler Phi Function, 28 Euler products, 64 Euler pseudoprime, 246 Euler zeta function, 63 Euler’s constant, 178 Euler’s sum formula, 204 Index Euler’s Theorem, 33 Extension field, 309 F F-isomorphism, 315 Factor, 10 Factor ring, 306, 307 Fermat numbers, 66 Fermat prime, 66 Fermat’s Big Theorem, Fermat’s Little Theorem, 33 Fermat’s probable prime test, 237 Fermat’s Theorem, 33 Fermat’s Two-Square Theorem, 96 Fibonacci numbers, 71 Field, Field of p-adic numbers, 387 Finite extension, 310 Fourier coefficients, 172 Fourier Convergence Theorem, 172 Fourier inversion theorem, 138 Fourier series, 171 Fractional ideal, 353 Fractional principal ideal, 360 Free module, 353 Free product, 107 Frequency analysis, 264 Fuchsian group, 101 Function elements, 177 Fundamental mesh, 342 Fundamental Theorem for Finitely Generated Abelian Groups, 33 Fundamental Theorem of Arithmetic, 18 Fundamental unit, 338 Fundmaental theorem of arithmetic, G Gamma function, 173 Gauss sum, 51 Gaussian integers, 21, 285, 296 Gaussian primes, 299 Gaussian rationals, 298 GCD, 12 General linear group, 100 General primality testing algorithm, 220 Generalized Riemann hypothesis, 186, 212 Geometric number theory, 345 Global field, 403 Goldbach conjecture, 1, 213 Golden ratio, 71, 72 Golden rectangle, 73 Golden section, 72 Index Golden spiral, 74 Goldwasser–Killian algorithm, 262 Greatest common divisor, 12 Ground field, 309 Group, 27 Group presentation, 105 Group representation, 119 H Hardy–Littlewood conjecture, 228 Hasse’s principle, 403 Hensel’s Lemma, 398 Holomorphic function, 176 Homomorphism, 10 I Ideal, 301 Ideal class group, 359, 360 Ideal class number, 364 Ideal generators, 301 Ideal numbers, 348 Ideals, 286 Imaginary quadratic field, 335 Independent vectors, 309 Inductive property, Industrial grade primes, 237 Integers, 1, Integral basis, 333 Integral domain, Integral ideal, 353 Intermediate field, 309 Irr(α, F), 312 Irreducibe polynomial, 289 Isolated singularity, 178 Isomorphism, 10 J Jacobi symbol, 245 Jacobi theta function, 181 K Keyword, 266 Korselt criterion, 243 L Lattice, 342 Law of quadratic reciprocity, 48 LCM, 12 Leading coefficient, 288 411 Least common multiple, 12 Legendre symbol, 49 Length of MZV, 202 Lenstra’s factorization algorithm, 258 Lindelof hypothesis, 211 Linear combination, 11 Linear congruence, 26 Little o notation, 153 Local field, 403 Local-global principle, 403 Lucas–Lehmer test, 249 Lucas-Lehmer test, 70 M m-ary expansion, 372 Möbius function, 121 Möbius inversion formula, 138 Maximal ideal, 305 Mellin transform, 187 Meneses-Vanstone Cryptosystem, 274 Meromorphic function, 178 Mersenne numbers, 67 Mersenne prime, 67 Message units, 264 Metric, 373 Metric space, 376 Miller–Rabin test, 241 Minkowski’s theorem, 345 Modular group, 96 Modular ring, 23 Module, 352 Module basis, 353 Modulo, 22 Monic polynomial, 288 Multiple, 11 Multiple zeta values, 202 Multiplicative group of a field, Multiplicative order, 33 Multiplicity of roots, 293 MZV, 202 N Natural numbers, 1, Newton’s method, 399 Nielsen reduction formula, 203 Noetherian, 349 Non-archimedean, 384 Norm in an algebraic number field, 327 Norm on a field, 382 Normed field, 373, 382 Number theoretic function, 133 412 O One-way function, 270 Order of magnitude, 153 Ordered field, 375 Ordered integral domain, Orthogonality relations, 116 P p-adic expansion, 388 p-adic fields, 371 p-adic integer, 394 p-adic norm, 386 p-adic numbers, 371, 387 p-adic valuation, 386 Pascal’s triangle, 148 Pell’s equation, 337 Perfect number, 69 Permutation cipher, 265 PID, 302 Piece, 323 Plaintest message, 263 Pole, 178 Polyalphabetic cipher, 265 Polynomial, 288 Polynomial congruence, 39 Positive definite form, 92 Primality test, 236 Primality testing, 35, 219 Prime, 11 Prime ideal, 305 Prime number function, 143 Prime number Theorem, Prime number theorem, 1, 59, 143 Prime polynomial, 289 Primitive element, 318 Primitive integral polynomial, 330 Primitive root, 33 Principal ideal, 301 Principal ideal domain, 302 Probabilistic number theory, 212 Probabilistic primality test, 236 Probable primes, 237 Product of ideals, 350 Projective special linear group, 101 Pseudoprime, 242 Public key cryptography, 219, 264 Publickey cryptography, 270 Pythagorean triple, 93 Q Q p , 387 Index Quadratic fields, 335 Quadratic form, 92 Quadratic integers, 336 Quadratic residue, 47 Quotient ring, 24, 307 R Ramification index, 363 Ramified, 339 Rational integers, 286, 298 Rational primes, 298 Real quadratic field, 335 Reduced residue system, 29 Relatively prime, 12 Remainder, 11 Removable singularity, 178 Residue, 22 Residue class, 22 Riemann hypothesis, 145, 183, 207 Riemann zeta function, 63, 144, 179 Ring, Ring with identity, Root of a polynomial, 292 RSA algorithm, 271 S Selberg’s formula, 194 Shift algorithm, 265 Sieve, 220 Sieve of Eratosthenes, 220 Sieving methods, 220 Simple extension, 313 Solovay–Strassen test, 241 Sophie Germain prime, 282 Span, 309 Special linear group, 100 Standard prime decomposition, 19 Stirling’s approximation, 91 Strong pseudoprime, 248 Strong triangle inequality, 384 Submodule, 353 Substitution process, 292 Symmetric functions, 229 Symmetric key cryptography, 264 Symmetric polynomials, 229, 321 T Trace, 328 Transcendence of e and π , 339 Transcendental extension, 311 Transcendental number, 310 Index Triangular numbers, 10 Trivial zeros, 183 Twin primes, 131, 227 Twin primes conjecture, 131, 227 Twin primes constant, 227 U UFD, 21, 287 Ultra-metric, 384 Unique factorization domain, 21, 287 Unit, 21 Unit group, 27 V Vandermonde determinant, 327 413 Vector space, 309 Vigenére code, 265 Von Mangoldt function, 119 W Waring’s problem, 107 Weight of MZV, 202 Well-ordering property, Wilson’s Theorem, 26 Witness, 249 Z Zero divisor, Zero of a polynomial, 292 Zeros of the zeta function, 183 ... few The second major branch is analytic number theory This is the branch of the theory of numbers that studies the integers by using methods of real and complex analysis The final major branch... proof and the ideas surrounding the proof of the Prime Number Theorem allows us to introduce and discuss a large portion of analytic number theory Algebraic Number Theory arose originally as an. .. odd primes Another of the fascinations of number theory is that many results seem almost magical The prime number theorem which describes the asymptotic distribution of the prime numbers has often