An Open Door to Number Theory

297 Pages·2018·2.454 MB·English

by  Duff Campbell

AMS / MAA TEXTBOOKS VOL 39
An Open Door to
Number Theory
Duff Campbell
An Open Door to
Number Theory

AMS / MAA TEXTBOOKS
VOL 39
An Open Door to
Number Theory
Duff Campbell
Providence, Rhode Island
Committee on Books
Jennifer J. Quinn, Chair
MAA Textbooks Editorial Board
Stanley E. Seltzer, Editor
Bela Bajnok William Robert Green John Lorch
Matthias Beck Charles R. Hampton Virginia A. Noonburg
Otto Bretscher Jacqueline A. Jensen-Vallin Jeffrey L. Stuart
Heather Ann Dye Suzanne Lynne Larson Ruth Vanderpool
2010 Mathematics Subject Classification. Primary 11-01,
11A05, 11A07, 11A15, 11A41, 11A51, 11A55.
For additional informationand updates on this book, visit
www.ams.org/bookpages/text-39
The cover photograph is courtesy of Kristin McCullough/Moonlight Photography.
All illustrationsin this book were made by the author using Mathematica software.
Library of Congress Cataloging-in-Publication Data
Names: Campbell,Duff,1959–author.
Title: Anopendoortonumbertheory/DuffCampbell.
Description: Providence, Rhode Island: MAA Press, an imprint of the American Mathematical
Society,[2018]|Series: AMS/MAAtextbooks;volume39|Includesbibliographicalreferences
andindex.
Identifiers: LCCN2017055802|ISBN9781470443481(alk. paper)
Subjects: LCSH: Number theory–Textbooks. | AMS: Number theory – Instructional exposition
(textbooks, tutorial papers, etc.). msc| Number theory – Elementary number theory – Mul-
tiplicative structure; Euclidean algorithm; greatest common divisors. msc | Number theory
– Elementary number theory – Congruences; primitive roots; residue systems. msc | Num-
ber theory – Elementary number theory – Power residues, reciprocity. msc | Number theory
– Elementary number theory – Primes. msc | Number theory – Elementary number theory
– Factorization; primality. msc | Number theory – Elementary number theory – Continued
fractions. msc
Classification: LCCQA241.C27252018|DDC512.7/2–dc23
LCrecordavailableathttps://lccn.loc.gov/2017055802
Colorgraphicpolicy. Anygraphicscreatedincolorwillberenderedingrayscalefortheprinted
versionunlesscolorprintingisauthorizedbythePublisher. Ingeneral,colorgraphicswillappear
incolorintheonlineversion.
Copying and reprinting. Individual readersofthispublication,andnonprofit librariesacting
for them, are permitted to make fair use of the material, such as to copy select pages for use
in teaching or research. Permission is granted to quote brief passages from this publication in
reviews,providedthecustomaryacknowledgmentofthesourceisgiven.
Republication,systematiccopying,ormultiplereproductionofanymaterialinthispublication
ispermittedonlyunderlicensefromtheAmericanMathematicalSociety. Requestsforpermission
toreuseportionsofAMSpublicationcontentarehandledbytheCopyrightClearanceCenter. For
moreinformation,pleasevisitwww.ams.org/publications/pubpermissions.
Sendreq[e-mail].
(cid:2)c 2018bytheAmericanMathematicalSociety. Allrightsreserved.
TheAmericanMathematicalSocietyretainsallrights
exceptthosegrantedtotheUnitedStatesGovernment.
PrintedintheUnitedStatesofAmerica.
(cid:2)∞ Thepaperusedinthisbookisacid-freeandfallswithintheguidelines
establishedtoensurepermanenceanddurability.
VisittheAMShomepageathttp://www.ams.org/
10987654321 232221201918
To my grandfather, LeRoy Archer Campbell

Contents
1 The Integers, Z 1
1 Number systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Rings and fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
3 Some fundamental facts about Z and N . . . . . . . . . . . . . . . . 7
4 Proofs by induction. . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
5 The binomial theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 18
6 The fundamental theorem of arithmetic (foreshadowing) . . . . . . . 26
7 Divisibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
8 Greatest common divisors . . . . . . . . . . . . . . . . . . . . . . . . 31
9 The Euclidean algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 33
10 The amazing array . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
11 Convergents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
12 The amazing super-array . . . . . . . . . . . . . . . . . . . . . . . . 49
13 The modified division algorithm. . . . . . . . . . . . . . . . . . . . . 56
14 Why does the amazing array work? . . . . . . . . . . . . . . . . . . . 58
15 Primes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
16 The proof of the fundamental theorem of arithmetic . . . . . . . . . 64
17 Unique factorization in other rings . . . . . . . . . . . . . . . . . . . 68
2 Modular Arithmetic in Z/mZ 71
18 The integers mod m, Z/mZ . . . . . . . . . . . . . . . . . . . . . . . 71
vii
viii Contents
19 Congruences. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
20 Units and zero-divisors in Z/mZ . . . . . . . . . . . . . . . . . . . . 81
21 Cancellation law in Z/mZ . . . . . . . . . . . . . . . . . . . . . . . . 85
22 Solving linear equations in Z/mZ . . . . . . . . . . . . . . . . . . . . 87
23 Solving polynomial equations in Z/mZ . . . . . . . . . . . . . . . . . 88
24 Solving systems of linear equations in Z/mZ. . . . . . . . . . . . . . 95
25 Lifting roots in Z/pnZ . . . . . . . . . . . . . . . . . . . . . . . . . . 103
26 Wilson’s theorem and its converse . . . . . . . . . . . . . . . . . . . 108
27 Calculating ϕ(n) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
28 Euler’s and Fermat’s theorems . . . . . . . . . . . . . . . . . . . . . 115
29 The order of an integer modulo m . . . . . . . . . . . . . . . . . . . 118
30 Divisibility tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
√
3 Quadratic Extensions of the Integers, Z[ d] 127
31 Divisibility in Z[i]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
32 The Euclidean algorithm in Z[i] . . . . . . . . . . . . . . . . . . . . . 130
33 Unique factorization in Z[i] . . . . . . . . . . . . . . . . . . . . . . . 135
√
34 The structure of Z[ 2] . . . . . . . . . . . . . . . . . . . . . . . . . . 138
√
35 The Euclidean algorithm in Z[ d] . . . . . . . . . . . . . . . . . . . 140
36 Factoring in Z[i] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
37 The primes in Z[i] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
4 An Interlude of Analytic Number Theory 153
38 The distribution of primes in Z . . . . . . . . . . . . . . . . . . . . . 153
5 Quadratic Residues 157
39 Perfect squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
40 Quadratic residues . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
41 Calculating the Legendre symbol (hard way) . . . . . . . . . . . . . 167
Contents ix
(cid:2) (cid:3)
√
42 The arithmetic of Z[ −2] and the Legendre symbol −2 . . . . . . 169
p
43 Gauss’s lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
44 Calculating the Legendre symbol (easier way) . . . . . . . . . . . . . 174
√
45 The arithmetic of Z[ −3] . . . . . . . . . . . . . . . . . . . . . . . . 180
46 The arithmetic of Z[ρ] . . . . . . . . . . . . . . . . . . . . . . . . . . 182
47 Calculating the Legendre symbol (easiest way) . . . . . . . . . . . . 193
48 The Jacobi symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
6 Further Topics 203
49 When Z/nZ has a primitive root . . . . . . . . . . . . . . . . . . . . 203
50 Minkowski’s theorem (geometry in the aid of algebra) . . . . . . . . 208
Appendix A Tables 223
Appendix B Projects 233
Bibliography 279
Index 281