Nuggets of Number Theory: A Visual Approach

165 Pages·2018·4.78 MB·English

by  Roger B. Nelsen

AMS / MAA CLASSROOM RESOURCE MATERIALS VOL 55
Nuggets of Number Theory
A Visual Approach
Roger B. Nelsen
Nuggets of Number Theory
A Visual Approach

AMS / MAA CLASSROOM RESOURCE MATERIALS
VOL 55
Nuggets of Number Theory
A Visual Approach
Roger B. Nelsen
Providence, Rhode Island
Classroom Resource Materials Editorial Board
Susan G. Staples, Editor
Bret J. Benesh Paul R. Klingsberg
Christina Eubanks-Turner Tamara J. Lakins
Christoper Hallstrom Brian Lins
Cynthia J. Huffman Mary Eugenia Morley
Brian Paul Katz Darryl Yong
Haseeb A. Kazi
2010 Mathematics Subject Classification. Primary 11-01;
Secondary 11A07, 11B39, 11D04.
For additional information and updates on this book, visit
www.ams.org/bookpages/clrm-55
Library of Congress Cataloging-in-Publication Data
Names: Nelsen,RogerB.,author.
Title: Nuggetsofnumbertheory: avisualapproach/RogerB.Nelsen.
Description: Providence,RhodeIsland: MAAPress,animprintoftheAmericanMathematical
Society, [2018] | Series: Classroom resource materials ; volume 55 | Includes bibliographical
referencesandindex.
Identifiers: LCCN2018000043|ISBN9781470443986(alk. paper)
Subjects: LCSH: Number theory–Study and teaching. | Mathematics–Study and teaching.
| AMS: Number theory – Instructional exposition (textbooks, tutorial papers, etc.). msc |
Numbertheory–Elementarynumbertheory–Congruences;primitiveroots;residuesystems.
msc | Number theory – Sequences and sets – Fibonacci and Lucas numbers and polynomials
andgeneralizations. msc|Numbertheory–Diophantineequations–Linearequations. msc
Classification: LCCQA241.N4352018|DDC512.7–dc23
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Contents
Preface ix
Chapter1. FigurateNumbers 1
1.1. Polygonalnumbers 1
1.2. Triangularnumberidentities 7
1.3. Oblongnumbersandthein(cid:976)initudeofprimes 13
1.4. Pentagonalandother(cid:976)iguratenumbers 14
1.5. Politenumbers 16
1.6. Three-dimensional(cid:976)iguratenumbers 18
1.7. Exercises 21
Chapter2. Congruence 25
2.1. Congruenceresultsfortriangularnumbers 25
2.2. Congruenceresultsforother(cid:976)iguratenumbers 27
2.3. Fermat’slittletheorem 30
2.4. Wilson’stheorem 32
2.5. Exercises 33
Chapter3. DiophantineEquations 35
3.1. Trianglesandsquares 36
3.2. LinearDiophantineequations 38
3.3. LinearcongruencesandtheChineseremaindertheorem 41
3.4. ThePellequation𝑥(cid:2870) −2𝑦(cid:2870) = 1 44
3.5. ThePellequation𝑥(cid:2870) −3𝑦(cid:2870) = 1 46
3.6. ThePellequations𝑥(cid:2870) −𝑑𝑦(cid:2870) = 1 49
3.7. Exercises 52
Chapter4. PythagoreanTriples 55
4.1. Euclid’sformula 56
4.2. Pythagoreantriplesandmeansofoddsquares 57
4.3. Thecarpetstheorem 58
4.4. Pythagoreantriplesandthefactorsofevensquares 59
4.5. AlmostisoscelesprimitivePythagoreantriples 61
4.6. APythagoreantripletree 64
4.7. PrimitivePythagoreantripleswithsquaresides 67
v
vi CONTENTS
4.8. Pythagoreanprimesandtriangularnumbers 67
4.9. Divisibilityproperties 69
4.10. Pythagoreantriangles 70
4.11. Pythagoreanruns 73
4.12. Sumsoftwosquares 73
4.13. PythagoreanquadruplesandPythagoreanboxes 76
4.14. Exercises 79
Chapter5. IrrationalNumbers 83
5.1. Theirrationalityof√2 83
5.2. Rationalapproximationsto√2: Pellequations 88
5.3. Rationalapproximationsto√2: AlmostisoscelesPPTs 89
5.4. Theirrationalityof√3and√5 90
5.5. Theirrationalityof√𝑑 fornon-square𝑑 93
5.6. Thegoldenratioandthegoldenrectangle 94
5.7. Thegoldenratioandtheregularpentagon 96
5.8. Periodiccontinuedfractions 98
5.9. Exercises 101
Chapter6. FibonacciandLucasNumbers 103
6.1. TheFibonaccisequenceinartandnature 104
6.2. Fibonacciparallelograms,triangles,andtrapezoids 106
6.3. Fibonaccirectanglesandsquares 107
6.4. DiagonalsumsinPascal’striangle 113
6.5. Lucasnumbers 115
6.6. ThePellequations𝑥(cid:2870) −5𝑦(cid:2870) = ±4andBinet’sformula 117
6.7. Exercises 121
Chapter7. PerfectNumbers 123
7.1. Euclid’sformula 123
7.2. Evenperfectnumbersandgeometricprogressions 125
7.3. Evenperfectnumbersandtriangularnumbers 126
7.4. Evenperfectnumbersmodulo9 128
7.5. Evenperfectnumbersendin6or28 128
7.6. Evenperfectnumbersmodulo7 130
7.7. Evenperfectnumbersandsumsofoddcubes 131
7.8. Oddperfectnumbers 131
7.9. Exercises 133
SolutionstotheExercises 135
Chapter1 135
Chapter2 137
Chapter3 138
CONTENTS vii
Chapter4 141
Chapter5 142
Chapter6 144
Chapter7 145
Bibliography 149
Index 151

Preface
Thetheoryofnumbersisthelastgreatuncivilizedcontinentof
mathematics. Itissplitupintoinnumerablecountries,fertile
enoughinthemselves,butallthemoreorlessindifferenttoone
another’swelfareandwithoutavestigeofacentral,intelligent
government. IfanyyoungAlexanderisweepingforanewworldto
conquer,itliesbeforehim.
EricTempleBell
Theelementarytheoryofnumbersshouldbeoneoftheverybest
subjectsforearlymathematicalinstruction. Itdemandsverylittle
previousknowledge;itssubjectmatteristangibleandfamiliar;the
processesofreasoningwhichitemploysaresimple,general,and
few;anditisuniqueamongthemathematicalsciencesinitsappeal
tonaturalhumancuriosity. Amonth’sintelligentinstructioninthe
theoryofnumbersoughttobetwiceasinstructive,twiceasuseful,
andatleasttentimesasentertainingasthesameamountof
“calculusforengineers.”
GodfreyHaroldHardy
Numbertheoristsarelikelotus-eaters—havingoncetastedofthis
foodtheycannevergiveitup.
LeopoldKronecker
SometimeagoIwaslookingatseveraltextbooksfortheundergrad-
uatenumbertheorycourse. Iwasstruckbyhowfewillustrationswere
includedinmanyofthosetextbooks. Anumber—speci(cid:976)icallyapositive
integer—canrepresentmanythings: thecardinalityofaset;thelength
of a line segment; or the area of a plane region. Such representations
naturallyleadtoavarietyofvisualargumentsfortopicsinelementary
number theory. Since the number theory course usually begins with
propertiesofthepositiveintegers,thetextsshouldhavemorepictures.
Thatobservationbecamethemotivationforthisbook.
Work on this book began when I was invited to give a talk at the
MAA’s MathFest in Albuquerque in August 2005, in a session entitled
“Gems of Number Theory” organized by Arthur Benjamin and Ezra
ix