Number Theory: An Introduction via the Density of Primes

423 Pages·2016·3.54 MB·English

by  Benjamin Fine

Benjamin Fine
Gerhard Rosenberger
Number
Theory
An Introduction via the Density
of Primes
Second Edition

Benjamin Fine Gerhard Rosenberger
(cid:129)
Number Theory
An Introduction via the Density of Primes
Second Edition
Benjamin Fine Gerhard Rosenberger
Department ofMathematics UniversitätHamburg
FairfieldUniversity Hamburg
Fairfield,CT Germany
USA
ISBN978-3-319-43873-3 ISBN978-3-319-43875-7 (eBook)
DOI 10.1007/978-3-319-43875-7
LibraryofCongressControlNumber:2016947201
Mathematics Subject Classification (2010): 11A01, 11A03, 11M01, 11R04, 11Z05, 11T71, 11H01,
20A01,20G01,14G01,08A01
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Preface to the Second Edition
Wewereverypleasedwiththeresponsetothefirsteditionofthisbookandwewere
very happy to do a second edition. In this second edition, we cleaned up various
typospointedoutbyreadersandaddedsomenewmaterialsuggestedbythem.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.
Wehaveaddedanewchapter,Chapter7,onp-adicnumbers,p-adicarithmetic,
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 5 on primality
testing.
We have added material in Chapter 4 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 Benjamin Fine
Hamburg, Germany Gerhard Rosenberger
v
Preface to the First Edition
Numbertheoryisfascinating.Resultsaboutnumbersoftenappearmagical,bothin
their statements and in the elegance of their proofs. Nowhere is this more evident
thaninresultsaboutthesetofprimenumbers.ThePrimeNumberTheorem,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.
Theprimenumbersformthecornerstoneofthetheoryofnumbers.Many,ifnot
most,resultsinnumbertheoryproceed byconsideringthecase ofprimesandthen
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
numbertheory based onthe central theme of thesequence 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
materialaswellaspresentinganoverviewofthewholediscipline.Alltheessential
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 algo-
rithm 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 under-
graduates 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 PrefacetotheFirstEdition
Germany. The prerequisites are a knowledge of Calculus and Multivariable
CalculusandsomeLinearAlgebra.ThenecessaryideasfromAbstractAlgebraand
ComplexAnalysisareintroducedinthebook.Therearemanyinterestingexercises
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.
Therearetwobasicfactsconcerningthesequenceofprimesthatarefocusedon
inthisbookandfromwhichmuchofthetheoryofnumbersisintroduced.Thefirst
factisthatthereareinfinitelymanyprimes.Thisfactwasofcourseknownsinceat
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,awideareaofmodernnumbertheoryiscovered.Thisincludesthefactthat
theprimesarenumerousenoughsothatthereareinfinitelymanyinanyarithmetic
progression anþb 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
sequenceofprimesthinsoutisformalizedinthePrimeNumberTheorem,whichas
alreadymentioned,manypeopleconsiderthemostsurprisingresultinmathematics.
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 fac-
torizationtoalgebraicnumberrings.Weusetheapproachoflookingatprimesand
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,includingtherecentlydevelopedAKSalgorithmandthenprovideabasic
introduction to cryptography.
Thereareseveralwaysthatthisbookcanbeusedforcourses.Chapter1together
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 do 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 4 and selections from 5 and 6, while a year-long course
focusingonalgebraicnumbertheorycanbefashionedfromChapters1,2,3,and6
andselectionsfrom4and5.Therearealsopossibilities forusingthebookforone
semesterintroductorycoursesinanalyticnumbertheory,centeringonChapter4,or
for a one semester introductory course in algebraic number theory, centering on
Chapter 6. Some suggested courses:
PrefacetotheFirstEdition 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 2
(as needed), Sections 3.1, 3.2, 3.3, 3.4, 3.5, Chapter 4
One-SemesterCourseFocusingonAlgebraicNumberTheory:Chapter1,Chapter2
(as needed), Chapter 6
We would like to thank the many people who have read through other prelim-
inary versions of these notes and made suggestions. Included among these people
are Kati Bencsath and Al Thaler, as well as themany 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 4 comes.
Benjamin Fine
Gerhard Rosenberger
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 .. .... .... .... ..... .... 10
2.3 The Fundamental Theorem of Arithmetic... .... .... ..... .... 16
2.4 Congruences and Modular Arithmetic . .... .... .... ..... .... 22
2.4.1 Basic Theory of Congruences.. .... .... .... ..... .... 22
2.4.2 The Ring of Integers Mod N... .... .... .... ..... .... 23
2.4.3 Units and the Euler Phi Function ... .... .... ..... .... 27
2.4.4 Fermat’s Little Theorem and the Order of an Element .... 32
2.4.5 On Cyclic Groups... .... .... .... .... .... ..... .... 36
2.5 The Solution of Polynomial Congruences Modulo m.. ..... .... 39
2.5.1 Linear Congruences and the Chinese
Remainder Theorem . .... .... .... .... .... ..... .... 39
2.5.2 Higher Degree Congruences ... .... .... .... ..... .... 45
2.6 Quadratic Reciprocity. ..... .... .... .... .... .... ..... .... 48
2.7 Exercises .. .... .... ..... .... .... .... .... .... ..... .... 55
3 The Infinitude of Primes.. ..... .... .... .... .... .... ..... .... 59
3.1 The Infinitude of Primes.... .... .... .... .... .... ..... .... 59
3.1.1 Some Direct Proofs and Variations.. .... .... ..... .... 59
3.1.2 Some Analytic Proofs and Variations.... .... ..... .... 62
3.1.3 The Fermat and Mersenne Numbers. .... .... ..... .... 66
3.1.4 The Fibonacci Numbers and the Golden Section .... .... 71
3.1.5 Some Simple Cases of Dirichlet’s Theorem ... ..... .... 84
3.1.6 A Topological Proof and a Proof Using Codes. ..... .... 89
3.2 Sums of Squares. .... ..... .... .... .... .... .... ..... .... 92
3.2.1 Pythagorean Triples.. .... .... .... .... .... ..... .... 93
3.2.2 Fermat’s Two-Square Theorem. .... .... .... ..... .... 96
xi
xii Contents
3.2.3 The Modular Group . .... .... .... .... .... ..... .... 100
3.2.4 Lagrange’s Four Square Theorem... .... .... ..... .... 107
3.2.5 The Infinitude of Primes Through Continued Fractions.... 110
3.3 Dirichlet’s Theorem .. ..... .... .... .... .... .... ..... .... 112
3.4 Twin Prime Conjecture and Related Ideas .. .... .... ..... .... 131
3.5 Primes Between x and 2x... .... .... .... .... .... ..... .... 132
3.6 Arithmetic Functions and the Möbius Inversion Formula.... .... 133
3.7 Exercises .. .... .... ..... .... .... .... .... .... ..... .... 138
4 The Density of Primes.... ..... .... .... .... .... .... ..... .... 143
4.1 The Prime Number Theorem—Estimates and History . ..... .... 143
4.2 Chebyshev’s Estimate and Some Consequences.. .... ..... .... 147
4.3 Equivalent Formulations of the Prime Number Theorem .... .... 159
4.4 The Riemann Zeta Function and the Riemann Hypothesis... .... 169
4.4.1 The Real Zeta Function of Euler.... .... .... ..... .... 170
4.4.2 Analytic Functions and Analytic Continuation . ..... .... 175
4.4.3 The Riemann Zeta Function ... .... .... .... ..... .... 179
4.5 The Prime Number Theorem .... .... .... .... .... ..... .... 186
4.6 The Elementary Proof. ..... .... .... .... .... .... ..... .... 193
4.7 Multiple Zeta Values . ..... .... .... .... .... .... ..... .... 198
4.8 Some Extensions and Comments . .... .... .... .... ..... .... 206
4.9 Exercises .. .... .... ..... .... .... .... .... .... ..... .... 213
5 Primality Testing—An Overview.... .... .... .... .... ..... .... 219
5.1 Primality Testing and Factorization ... .... .... .... ..... .... 219
5.2 Sieving Methods. .... ..... .... .... .... .... .... ..... .... 220
5.2.1 Brun’s Sieve and Brun’s Theorem .. .... .... ..... .... 226
5.3 Primality Testing and Prime Records.. .... .... .... ..... .... 236
5.3.1 Pseudo-Primes and Probabilistic Testing.. .... ..... .... 241
5.3.2 The Lucas–Lehmer Test and Prime Records... ..... .... 249
5.3.3 Some Additional Primality Tests.... .... .... ..... .... 255
5.3.4 Elliptic Curve Methods... .... .... .... .... ..... .... 257
5.4 Cryptography and Primes... .... .... .... .... .... ..... .... 263
5.4.1 Some Number Theoretic Cryptosystems.. .... ..... .... 267
5.5 Public Key Cryptography and the RSA Algorithm.... ..... .... 270
5.6 Elliptic Curve Cryptography. .... .... .... .... .... ..... .... 273
5.7 The AKS Algorithm.. ..... .... .... .... .... .... ..... .... 276
5.8 Exercises .. .... .... ..... .... .... .... .... .... ..... .... 282
6 Primes and Algebraic Number Theory ... .... .... .... ..... .... 285
6.1 Algebraic Number Theory .. .... .... .... .... .... ..... .... 285
6.2 Unique Factorization Domains... .... .... .... .... ..... .... 287
6.2.1 Euclidean Domains and the Gaussian Integers . ..... .... 293
6.2.2 Principal Ideal Domains .. .... .... .... .... ..... .... 301
6.2.3 Prime and Maximal Ideals .... .... .... .... ..... .... 304