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http://dx.doi.org/10.1090/stml/045

Higher Arithmeti c An Algorithmic Introduction to Numbe r Theory

STUDENT MATHEMATICAL LIBRARY Volume 45

Higher Arithmeti c An Algorithmic Introduction to Numbe r Theory

Harold M . Edwards

ilAMS AMERICAN MATHEMATICA L SOCIET Y

Providence, Rhode Islan d

Editorial Boar d Gerald B . Follan d Bra d G . Osgoo d (Chair ) Robin Forma n Michae l S tarb i r d

2000 Mathematics Subject Classification. P r i m a r y 11-01 .

For addi t iona l informatio n an d upda te s o n th i s book , visi t w w w . a m s . o r g / b o o k p a g e s / s t m l - 4 5

Library o f Congres s Cataloging-in-Publicatio n D a t a

Edwards, Harol d M . Higher arithmeti c : a n algorithmi c introductio n t o numbe r theor y / Harol d M .

Edwards. p. cm . — (Studen t mathematica l library , ISS N 1520-912 1 ; v. 45 )

Includes bibliographica l reference s an d index . ISBN 978-0-8218-4439- 7 (alk . paper ) 1. Numbe r theory . I . Title .

QA241 .E39 200 8 512.7—dc22 200706057 8

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Contents

Preface i x

Chapter 1 . Number s 1

Chapter 2 . Th e Proble m AD + B = • 7

Chapter 3 . Congruence s 1 1

Chapter 4 . Doubl e Congruences and the Euclidean Algorithm 1 7

Chapter 5 . Th e Augmente d Euclidea n Algorith m 2 3

Chapter 6 . Simultaneou s Congruence s 2 9

Chapter 7 . Th e Fundamenta l Theore m o f Arithmetic 3 3

Chapter 8 . Exponentiatio n an d Order s 3 7

Chapter 9 . Euler' s </>-Functio n 4 3

Chapter 10 . Findin g th e Orde r o f a mod c 4 5

Chapter 11 . Primalit y Testin g 5 1

VI Higher Arithmeti c

Chapter 12 .

Chapter 13 .

Chapter 14 .

Chapter 15 .

Chapter 16 .

Chapter 17 .

Chapter 18 .

Chapter 19 .

Chapter 20 .

Chapter 21 .

Chapter 22 .

Chapter 23 .

Chapter 24 .

Chapter 25 .

Chapter 26 .

Chapter 27 .

Chapter 28 .

Chapter 29 .

Chapter 30 .

Chapter 31 .

The RS A Ciphe r Syste m

Primitive Root s mo d p

Polynomials

Tables o f Indices mo d p

Brahmagupta's Formul a an d Hypernumber s

Modules o f Hypernumber s

A Canonical Form for Module s o f Hypernumber s

Solution o f AD + B = •

Proof o f the Theore m o f Chapte r 1 9

Euler's Remarkabl e Discover y

Stable Module s

Equivalence o f Module s

Signatures o f Equivalence Classe s

The Mai n Theore m

Modules Tha t Becom e Principa l Whe n Square d

The Possibl e Signature s fo r Certai n Value s o f A

The La w o f Quadrati c Reciprocit y

Proof o f the Mai n Theore m

The Theor y o f Binary Quadrati c Form s

Composition o f Binar y Quadrati c Form s

57

61

67

71

77

81

87

93

99

113

119

123

129

135

137

143

149

153

155

163

Contents vn

Appendix. Cycle s o f Stabl e Module s 16 9

Answers t o Exercise s 17 9

Bibliography 20 7

Index 20 9

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Preface

It i s widel y agree d tha t Car l Friedric h Gauss' s 180 1 boo k Disquisi-tiones Arithmeticae [G ] was the beginning o f modern numbe r theory , the firs t wor k o n th e subjec t tha t wa s systemati c an d comprehen -sive rather tha n a collection o f special problems an d techniques . Th e name "numbe r theory " b y whic h th e subjec t i s know n toda y wa s i n use a t th e time—Gaus s himsel f use d i t (theoria numerorum) i n Arti -cle 5 6 o f th e book—bu t h e chos e t o cal l i t "arithmetic " i n hi s title . He explaine d i n th e first paragrap h o f hi s Prefac e tha t h e di d no t mean arithmeti c i n th e sens e o f everyda y computation s wit h whol e numbers bu t a "highe r arithmetic " tha t comprise d "genera l studie s of specifi c relation s amon g whol e numbers. "

I too prefer "arithmetic " t o "numbe r theory." T o me, number the -ory sounds passive , theoretical , an d disconnecte d fro m reality . Highe r arithmetic sound s active , challenging , an d relate d t o everyda y realit y while aspirin g t o transcen d it .

Although Gauss' s explanation o f what h e means by "highe r arith -metic" i n his Preface i s unclear, a strong indicatio n o f what h e had i n mind come s a t th e en d o f hi s Preface whe n h e mentions th e materia l in hi s Sectio n 7 o n th e constructio n o f regula r polygons . (I n mod -ern terms , Sectio n 7 i s th e Galoi s theor y o f th e algebrai c equatio n xn — 1 = 0. ) H e admit s tha t thi s materia l doe s no t trul y belon g t o arithmetic bu t tha t "it s principle s mus t b e draw n fro m arithmetic. "

IX

X Higher Arithmeti c

What h e mean s b y arithmetic , I believe , i s exact computation, clos e to wha t Leopol d Kronecke r late r calle d "genera l arithmetic." 1

In 21s t centur y terms , Gauss' s subjec t i s "algorithmi c mathe -matics," mathematic s i n whic h th e emphasi s i s o n algorithm s an d computations. Instea d o f set-theoreti c abstraction s an d unrealizabl e constructions, suc h mathematic s deal s wit h specifi c operation s tha t arrive a t concret e answers . Regardles s o f wha t Gaus s migh t hav e meant b y hi s titl e Disquisitiones Arithmeticae, wha t I mea n b y m y title Higher Arithmetic i s a n algorithmi c approac h t o th e number -theoretic topic s i n th e book , mos t o f whic h ar e draw n fro m Gauss' s great work .

Mathematics i s abou t reasoning , bot h inductiv e an d deductive . Computations ar e simpl y ver y articulat e deductiv e arguments . Th e best theoretica l mathematic s i s a n inductiv e proces s b y whic h suc h arguments ar e found , organized , motivated , an d explained . Tha t i s why I think ample computational experience is indispensable to math-ematical education .

In teachin g th e numbe r theor y cours e a t Ne w Yor k Universit y several time s i n recen t years , I hav e foun d tha t student s enjo y an d feel the y profi t fro m doin g computationa l assignments . M y ow n ex -perience i n readin g Gaus s ha s usuall y bee n tha t I don' t understan d what h e i s doin g unti l h e give s a n example , s o I tr y t o ski p t o th e example righ t away . Moreover , o n anothe r level , i n writin g thi s an d previous books , I hav e ofte n foun d tha t creatin g exercise s lead s t o a clearer understandin g o f th e materia l an d a muc h improve d versio n of th e tex t tha t th e exercise s ha d bee n mean t t o illustrate . (Ver y often, th e greates t enlightenmen t cam e whe n writin g answers t o th e exercises. Fo r thi s reason , amon g others , answer s ar e give n fo r mos t of the exercises , beginnin g o n pag e 179. )

Fortunately, numbe r theor y i s a n idea l subjec t fro m th e poin t o f view o f providin g illustrativ e example s o f al l order s o f difficulty . I n this ag e o f computers , student s ca n tackl e problem s wit h rea l com -putational substanc e withou t havin g t o d o a lo t o f tediou s work . I

xSee Essa y 1. 1 o f m y boo k [E3] . Fo r th e relatio n o f genera l arithmeti c t o Galois theory , se e Essa y 2.1 .

Preface XI

have trie d t o provid e a t th e en d o f eac h chapte r enoug h example s and experiment s fo r student s t o try , bu t I' m sur e tha t enterprisin g students an d teacher s wil l b e abl e t o inven t man y more .

What bega n a s a n experimen t i n th e NY U cours e turne d int o a substantia l revisio n o f th e course . Th e experimen t wa s t o see ho w much of number theor y coul d be formulated i n terms of "numbers " i n the mos t primitiv e sense—th e number s 0 , 1 , 2 , . . . use d i n counting . To m y surprise , I foun d tha t no t onl y coul d I avoid negativ e num -bers bu t tha t I didn't miss them . Th e simpl e reaso n fo r thi s i s tha t the basi c question s o f numbe r theor y ca n b e state d i n term s o f con -gruences, an d subtractio n i s alway s possibl e i n congruence s withou t any need fo r negativ e numbers . Negativ e numbers hav e alway s led t o metaphysical conundrums—wh y shoul d a negativ e time s a negativ e be a positive?—which caus e confusing distraction s righ t a t th e outse t when th e meanin g o f "number " i s being mad e precise . I n thi s book , the meaning of "number " derive s simply from th e activit y o f countin g and arithmeti c ca n begi n immediately . Kronecker' s famou s dictum , "God create d th e whole numbers; al l the res t i s human work, " ca n b e amended t o say , "nonnegativ e whol e numbers, " whic h i s ver y likel y what Kronecke r mean t anyway .

A central theme o f the book i s the problem I denote b y the equa -tion AD + B = • , th e proble m o f finding , fo r tw o give n number s A and B, al l number s x fo r whic h Ax 2 + B i s a square . A s Chapte r 2 explains , version s o f thi s proble m ar e a t leas t a s ol d a s Pythago -ras, althoug h tw o millennia late r th e Disquisitiones Arithmeticae stil l dealt wit h it . A simpl e algorith m fo r th e complet e solutio n i s give n in Chapte r 19 .

Work o n problem s o f the for m A\D + B = • le d Leonhar d Eule r to th e discover y o f wha t I cal l "Euler' s law, " th e statemen t tha t th e answer t o th e questio n "I s A a squar e mo d p? " fo r a prim e numbe r p depends onl y o n th e valu e o f p mod 4A. Thi s statement , o f whic h the la w o f quadrati c reciprocit y i s a byproduct , i s completely prove d in Chapte r 29 .

When Erns t Eduar d Kumme r first introduce d hi s theory o f "idea l complex numbers " i n 1846 , 4 5 year s afte r th e publicatio n o f Disqui-sitiones Arithmeticae, Gaus s sai d tha t h e had worke d ou t somethin g

Xll Higher Arithmeti c

resembling Rummer' s theor y fo r hi s "privat e use " whe n h e was writ -ing abou t th e compositio n o f binar y quadrati c form s i n Sectio n 5 of Disquisitiones Arithmeticae, bu t tha t h e lef t i t ou t o f th e boo k be -cause h e ha d no t bee n abl e t o pu t i t o n firm ground. 2 Althoug h th e proof o f quadrati c reciprocit y give n i n thi s boo k wa s originall y in -spired b y Gauss' s proo f usin g th e compositio n o f forms , i t i s state d in term s close r t o Rummer' s idea l numbers . Specifically :

If, i n additio n t o usin g ordinar y number s 0 , 1 , 2 , . . . , one com -putes wit h a symbo l \J~A whos e squar e i s a fixed numbe r A , on e ha s an arithmetic— I hav e dubbe d i t th e arithmeti c o f "hypernumbers " for tha t A —in whic h th e natura l generalizatio n o f doin g computa -tions mo d n fo r som e numbe r n i s t o d o computation s mo d [a , b] for some pair o f hypernumber s a an d b. (Wit h ordinar y numbers , th e Euclidean algorith m serve s to reduce the numbe r o f numbers i n a se t that describe s a modulus to just one , but wit h hypernumbers tw o may be needed , a s i s show n i n Chapte r 18. ) Wit h natura l definition s o f multiplication an d equivalenc e o f suc h "module s o f hypernumbers, " the computations neede d to solve AD + B = • an d to prove quadrati c reciprocity ca n b e explaine d ver y simply . I n thi s way , Gauss' s diffi -cult compositio n o f forms i s avoided bu t th e essenc e o f hi s method i s preserved.

The las t tw o chapter s relat e th e method s o f the boo k t o Gauss' s binary quadrati c form s s o student s intereste d i n readin g furthe r i n the Disquisitiones Arithmeticae —or student s interested in binary qua -dratic forms—wil l b e abl e t o mak e th e transition .

Finally, a n appendi x give s a table o f the cycle s o f stable module s of hypernumbers fo r al l numbers A < 11 1 that ar e not squares , whic h will be usefu l fo r students , a s they wer e for me , i n understandin g th e general theor y an d i n workin g ou t examples .

2 See [E4].

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Bibliography

[D] L. E . Dickson , History of the Theory of Numbers, Carnegi e Insti -tute, Washington , 1920 , Chelsea reprint , 1971 .

[El] H . M. Edwards, Riemann's Zeta Function, Academi c Press , Ne w York, 1974 , Dover reprint , 2001 .

[E2] H . M . Edwards , FermaVs Last Theorem, Springer-Verlag , Ne w York, 1974 .

[E3] H . M . Edwards , Essays in Constructive Mathematics, Springer -Verlag, Ne w York , 2005 .

[E4] H. M. Edwards, Composition of Binary Quadratic Forms and the Foundations of Mathematics, articl e in The Shaping of Arithmetic, C . Goldstein e t al. , eds. , Springer-Verlag , Berlin , Heidelberg , Ne w York , 2007.

[G] C . F . Gauss , Disquisitiones Arithmeticae, Braunschweig , 1801 . (Reprinted a s vol. 1 of Gauss's Collected Works (Gesammelte Werke) and availabl e i n man y edition s i n whic h i t i s translate d int o man y languages.)

[J] C . G . J . Jacobi , Canon Arithmeticus, Berlin , Typi s Academicis , 1839.

207

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Index

Acharya, Bhaskara , 93 , 19 7 algebraic integer , 10 9 algebraic numbe r theory , 10 9 algorithm

augmented Euclidean , 24-2 6 comparison, 9 3 division wit h remainder , 4 , 5 Euclidean, 19 , 18 2 exponentiation, 3 7 factorization, 53 , 5 4 Miller's test , 53 , 18 9 multiplication, 2 , 3 reduction, 10 4

Archimedes, 8 , 9 augmented Euclidea n algorithm ,

23-26, 59 , 6 4

binary quadrati c form , xii , 155-16 0 Brahmagupta, 77 , 93 , 9 7 Brahmagupta's formula , 77-79 , 16 3

canonical form , 87-9 1 Chinese remainde r theorem , 3 0 class group , 12 7 comparison algorithm , 9 3 composite number , 3 3 composition o f forms , 163-16 7 congruence, 11-1 3 congruence o f hypernumbers , 81-8 3 conjugate o f a module , 10 2

content o f a module , 10 2 counting, 1- 5 cube roo t mo d p , 7 3

Dirichlet, G . Lejeune , 11 5 discriminant o f a form , 15 6 Disquisitiones Arithmeticae , ix-xii ,

11, 62 , 16 3 division, 4 division b y a mo d 6 , 2 7 division wit h remainder , 4 , 6 8 double congruence , 1 8

equality o f modules , 8 1 equivalence o f forms , 15 7 equivalence o f modules , 123 , 12 4 Euclidean algorithm , 18 , 24 , 8 2 Euler's criterion , 11 4 Euler's generalizatio n o f Fermat' s

theorem, 4 8 Euler's law , 116 , 129 , 136 , 15 4 Euler, Leonhard , xi , 43 , 114-11 6 exponentiation, 37-3 9

Farey series , 2 7 Fermat's theorem , 48 , 51 , 63 form, see binar y quadrati c for m fundamental theore m o f arithmetic ,

33, 3 4 fundamental unit , 108 , 10 9

209

210 Index

Gauss, Car l Friedrich , ix-xii , 11 , 62, 116 , 154 , 156 , 157 , 163-16 7

greatest commo n divisor , 19 , 2 0

hyperinteger, 16 1 hypernumber, xii , 7 9

indeterminates, 6 7 index o f a numbe r mo d p, 71 , 72 invertible module , 12 6

Jacobi, C . G . J. , 71-7 4

Kronecker, Leopold , x , x i Kummer, Erns t Eduard , xi , xi i

Law o f Quadrati c Reciprocity , see quadratic reciprocity , la w o f

Main Theorem , 135 , 136 , 153 , 15 4 matrix computations , 9 7 Mersenne primes , 4 9 Miller's test , 53-5 5 module, 8 1 modulus, 1 2 monic polynomial , 6 8 multiplication o f modules , 83 , 84,

163

Nicomachus, 3 1 norm o f a module , 102 , 110 , 12 6 number, 1

one-to-one, 14 4 onto, 14 4 orbit, 46 , 4 7 order o f a mo d c , 3 9

</>-function, 4 3 Pell's equation , 98 , 111 , 11 2 permutation, 45-4 7

pivotal module , 139-14 2 pivotal o f typ e 1 , 13 9 pivotal o f typ e 2 , 13 9 Plato, 8 polynomial, 6 7 prime number , 3 3 primitive module , 102 , 12 6 primitive roo t mo d p , 6 1 primitive solution , 9 4 principal cycle , 12 6 principal module , 123 , 16 2 Pythagoras, 7

quadratic characte r o f A mo d p (CP(A)), 113 , 11 4

quadratic numbe r field, 10 9 quadratic reciprocity , la w of , 116 ,

117, 149 , 150 , 154 , 20 1

reciprocal o f a mo d b, 2 7 reduction algorithm , 10 4 relatively prime , 2 0 RSA system , 57-6 0

signature o f a module , 130-133 , 143-145

signature relativ e t o A o f a number, 13 5

simultaneous congruences , 29-3 1 square roo t mo d p, 7 3 squarefree number , 12 9 stable module , 119-121 , 169-17 7 successor o f a module , 11 9 sum o f tw o squares , 16 2 Supplementary La w o f Quadrati c

Reciprocity, 15 0

table o f indices , 7 3

Titles i n Thi s Serie s

45 Harol d M . Edwards , Highe r arithmetic : A n algorithmi c introductio n t o

number theory , 200 8

44 Yi tzha k Katznelso n an d Yonata n R . Katznelson , A (terse )

introduction t o linea r algebra , 200 8

43 Ilk a Agricol a an d Thoma s Friedrich , Elementar y geometry , 200 8

42 C . E . Silva , Invitatio n t o ergodi c theory , 200 7

41 Gar y L . Mul le n an d Car l M u m m e r t , Finit e fields an d applications ,

2007

40 Deguan g Han , Ker i Kornelson , Davi d Larson , an d Eri c Weber ,

Frames fo r undergraduates , 200 7

39 A le x losevich , A vie w fro m th e top : Analysis , combinatoric s an d numbe r

theory, 200 7

38 B . Fristedt , N . Jain , an d N . Krylov , Filterin g an d prediction : A

primer, 200 7

37 Svet lan a Katok , p-adi c analysi s compare d wit h real , 200 7

36 Mar a D . Neusel , Invarian t theory , 200 7

35 Jor g Bewersdorff , Galoi s theor y fo r beginners : A historica l perspective ,

2006

34 Bruc e C . Berndt , Numbe r theor y i n th e spiri t o f Ramanujan , 200 6

33 Rekh a R . Thomas , Lecture s i n geometri c combinatorics , 200 6

32 Sheldo n Katz , Enumerativ e geometr y an d strin g theory , 200 6

31 Joh n McCleary , A firs t cours e i n topology : Continuit y an d dimension ,

2006

30 Serg e Tabachnikov , Geometr y an d billiards , 200 5

29 Kristophe r Tapp , Matri x group s fo r undergraduates , 200 5

28 Emmanue l Lesigne , Head s o r tails : A n introductio n t o limi t theorem s i n

probability, 200 5

27 Reinhar d Illner , C . Sea n Bohun , Samanth a McCol lum , an d The a

van R o o d e , Mathematica l modelling : A cas e studie s approach , 200 5

26 Rober t Hardt , Editor , Si x theme s o n variation , 200 4

25 S . V . Duzh i n an d B . D . Chebotarevsky , Transformatio n group s fo r

beginners, 200 4

24 Bruc e M . Landma n an d Aaro n Robertson , Ramse y theor y o n th e

integers, 200 4

23 S . K . Lando , Lecture s o n generatin g functions , 200 3

22 Andrea s Arvanitoyeorgos , A n introductio n t o Li e group s an d th e

geometry o f homogeneou s spaces , 200 3

21 W . J . Kaczo r an d M . T . Nowak , Problem s i n mathematica l analysi s

III: Integration , 200 3

20 Klau s Hulek , Elementar y algebrai c geometry , 200 3

19 A . She n an d N . K . Vereshchagin , Computabl e functions , 200 3

18 V . V . Yaschenko , Editor , Cryptography : A n introduction , 200 2

TITLES I N THI S SERIE S

17 A . She n an d N . K . Vereshchagin , Basi c se t theory , 200 2

16 Wolfgan g Kiihnel , Differentia l geometry : curve s - surface s - manifolds , second edition , 200 6

15 Ger d Fischer , Plan e algebrai c curves , 200 1

14 V . A . Vassiliev , Introductio n t o topology , 200 1

13 Frederic k J . Almgren , Jr. , Plateau' s problem : A n invitatio n t o varifol d

geometry, 200 1

12 W . J . Kaczo r an d M . T . Nowak , Problem s i n mathematica l analysi s

II: Continuit y an d differentiation , 200 1

11 Michae l Mesterton-Gibbons , A n introductio n t o game-theoreti c modelling, 200 0

® 10 Joh n Oprea , Th e mathematic s o f soa p films : Exploration s wit h Mapl e , 2000

9 Davi d E . Blair , Inversio n theor y an d conforma l mapping , 200 0

8 Edwar d B . Burger , Explorin g th e numbe r jungle : A journe y int o

diophantine analysis , 200 0

7 Jud y L . Walker , Code s an d curves , 200 0

6 Geral d Tenenbau m an d Miche l Mende s France , Th e prim e number s

and thei r distribution , 200 0

5 Alexande r Mehlmann , Th e game' s afoot ! Gam e theor y i n myt h an d

paradox, 200 0

4 W . J . Kaczo r an d M . T . Nowak , Problem s i n mathematica l analysi s

I: Rea l numbers , sequence s an d series , 200 0

3 Roge r Knobel , A n introductio n t o th e mathematica l theor y o f waves ,

2000

2 Gregor y F . Lawle r an d Leste r N . Coyle , Lecture s o n contemporar y

probability, 199 9

1 Charle s Radin , Mile s o f tiles , 199 9