Elementary number theory with programming

229 Pages·2016·1.41 MB·English

by  Marty Lewinter

ELEMENTARY
NUMBER THEORY
WITH PROGRAMMING

ELEMENTARY
NUMBER THEORY
WITH PROGRAMMING
MARTY LEWINTER
JEANINE MEYER
Copyright©2016byJohnWiley&Sons,Inc.Allrightsreserved
PublishedbyJohnWiley&Sons,Inc.,Hoboken,NewJersey
PublishedsimultaneouslyinCanada
Nopartofthispublicationmaybereproduced,storedinaretrievalsystem,ortransmittedinany
formorbyanymeans,electronic,mechanical,photocopying,recording,scanning,orotherwise,
exceptaspermittedunderSection107or108ofthe1976UnitedStatesCopyrightAct,without
eitherthepriorwrittenpermissionofthePublisher,orauthorizationthroughpaymentofthe
appropriateper-copyfeetotheCopyrightClearanceCenter,Inc.,222RosewoodDrive,Danvers,
MA01923,[phone],fax[phone],oronthewebatwww.copyright.com.Requeststo
thePublisherforpermissionshouldbeaddressedtothePermissionsDepartment,JohnWiley&
Sons,Inc.,111RiverStreet,Hoboken,NJ07030,[phone],fax[phone],oronlineat
http://www.wiley.com/go/permissions.
LimitofLiability/DisclaimerofWarranty:Whilethepublisherandauthorhaveusedtheirbest
effortsinpreparingthisbook,theymakenorepresentationsorwarrantieswithrespecttothe
accuracyorcompletenessofthecontentsofthisbookandspecificallydisclaimanyimplied
warrantiesofmerchantabilityorfitnessforaparticularpurpose.Nowarrantymaybecreatedor
extendedbysalesrepresentativesorwrittensalesmaterials.Theadviceandstrategiescontained
hereinmaynotbesuitableforyoursituation.Youshouldconsultwithaprofessionalwhere
appropriate.Neitherthepublishernorauthorshallbeliableforanylossofprofitoranyother
commercialdamages,includingbutnotlimitedtospecial,incidental,consequential,orother
damages.
Forgeneralinformationonourotherproductsandservicesorfortechnicalsupport,please
contactourCustomerCareDepartmentwithintheUnitedStatesat(800)762-2974,outsidethe
UnitedStatesat(317)572-3993orfax(317)572-4002.
Wileyalsopublishesitsbooksinavarietyofelectronicformats.Somecontentthatappearsin
printmaynotbeavailableinelectronicformats.FormoreinformationaboutWileyproducts,
visitourwebsiteatwww.wiley.com.
LibraryofCongressCataloging-in-PublicationData:
Lewinter,Marty,1950–
Elementarynumbertheorywithprogramming/MartyLewinter,JeanineMeyer.
pages cm
Includesindex.
ISBN978-1-119-06276-9(cloth)
1. Numbertheory. 2. Numbertheory–Problems,exercises,etc. 3. Computerprogramming.
I. Meyer,Jeanine. II. Title. III. Title:Numbertheorywithprogramming.
QA241.L58152015
512.7–dc23
2015000699
Setin11/13ptTimesbySPiGlobal,Pondicherry,India
PrintedintheUnitedStatesofAmerica
10 9 8 7 6 5 4 3 2 1
The first author dedicates this book to his son and fellow
mathematician, Anthony Delgado
The second author dedicates this book to her mother,
Esther Minkin, of blessed memory

CONTENTS
Preface xi
Words xiii
Notation in Mathematical Writing and in Programming xv
1 Special Numbers: Triangular, Oblong, Perfect, Deficient,
and Abundant 1
Theprogramsincludeoneforfactoringnumbersandone
totestaconjectureuptoafixedlimit.
Triangular Numbers 1
Oblong Numbers and Squares 3
Deficient, Abundant, and Perfect Numbers 4
Exercises 7
2 Fibonacci Sequence, Primes, and the Pell Equation 13
Theprogramsincludeexamplesthatcountstepstocompare
twodifferentapproaches.
Prime Numbers and Proof by Contradiction 13
Proof by Construction 17
Sums of Two Squares 18
Building a Proof on Prior Assertions 18
Sigma Notation 19
viii CONTENTS
Some Sums 19
Finding Arithmetic Functions 20
Fibonacci Numbers 22
An Infinite Product 26
The Pell Equation 26
Goldbach’s Conjecture 30
Exercises 31
3 Pascal’s Triangle 44
Theprogramsincludeexamplesthatgeneratefactorialusingiteration
andusingrecursionandthusdemonstrateandcompareimportant
techniquesinprogramming.
Factorials 44
The Combinatorial Numbers n Choose k 46
Pascal’s Triangle 48
Binomial Coefficients 50
Exercises 50
4 Divisors and Prime Decomposition 56
TheprogramsincludeonethatusesthealgorithmtoproducetheGCD
ofapairofnumbersandaprogramtoproducetheprimedecomposition
ofanumber.
Divisors 56
Greatest Common Divisor 58
Diophantine Equations 65
Least Common Multiple 67
Prime Decomposition 68
Semiprime Numbers 70
When Is a Number an mth Power? 71
Twin Primes 73
Fermat Primes 73
Odd Primes Are Differences of Squares 74
When Is n a Linear Combination of a and b? 75
Prime Decomposition of n! 76
No Nonconstant Polynomial with Integer Coefficients
Assumes Only Prime Values 77
Exercises 78
CONTENTS ix
5 Modular Arithmetic 85
Oneprogramchecksifamodequationistrue,andanotherdetermines
thesolvabilityofamodequationandthensolvesanequationthatis
solvablebyabrute-forceapproach.
Congruence Classes Mod k 85
Laws of Modular Arithmetic 87
Modular Equations 90
Fermat’s Little Theorem 91
Fermat’s Little Theorem 92
Multiplicative Inverses 92
Wilson’s Theorem 93
Wilson’s Theorem 95
Wilson’s Theorem (2nd Version) 95
Squares and Quadratic Residues 96
Lagrange’s Theorem 98
Lagrange’s Theorem 99
Reduced Pythagorean Triples 100
Chinese Remainder Theorem 102
Chinese Remainder Theorem 103
Exercises 104
6 Number Theoretic Functions 111
Theprogramsincludetwodistinctapproachestocalculating
thetaufunction.
The Tau Function 111
The Sigma Function 114
Multiplicative Functions 115
Perfect Numbers Revisited 115
Mersenne Primes 116
F(n)=Σf(d) Where d is a Divisor of n 117
The Möbius Function 119
The Riemann Zeta Function 121
Exercises 124
7 The Euler Phi Function 134
Theprogramsdemonstratetwoapproachestocalculatingthephifunction.
The Phi Function 134
Euler’s Generalization of Fermat’s Little Theorem 138