http://dx.doi.org/10.1090/stml/045 This page intentionally left blank Higher Arithmeti c An Algorithmic Introduction to Number Theory STUDENT MATHEMATICAL LIBRARY Volume 45 Higher Arithmetic An Algorithmic Introduction to Number 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 Starbir d 2000 Mathematics Subject Classification. Primar y 11-01 . For additiona l informatio n an d update s o n thi s book , visi t www.ams.org/bookpages/stml-45 Library o f Congres s Cataloging-in-Publicatio n Dat a Edwards, Harol d M . Higher arithmetic : an algorithmic introduction t o number theory / 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. Number theory . I . Title . QA241 .E39 200 8 512.7—dc22 200706057 8 Copying an d reprinting . Individua l reader s o f thi s publication , an d nonprofi t libraries actin g fo r them , ar e permitte d t o mak e fai r us e o f th e material , suc h a s t o copy a chapte r fo r us e i n teachin g o r research . Permissio n i s grante d t o quot e brie f passages fro m thi s publicatio n i n reviews , provide d th e customar y acknowledgmen t o f the sourc e i s given . Republication, systemati c copying , o r multiple reproduction o f any materia l i n this publication i s permitted onl y unde r licens e fro m th e America n Mathematica l Society . Requests fo r suc h permissio n shoul d b e addresse d t o th e Acquisition s Department , American Mathematica l Society , 20 1 Charles Street , Providence , Rhod e Islan d 02904 - 2294, USA . Request s ca n als o b e mad e b y e-mail t o [email protected] . © 200 8 by the America n Mathematica l Society . Al l rights reserved . The America n Mathematica l Societ y retain s al l right s except thos e grante d t o th e Unite d State s Government . Printed i n the Unite d State s o f America . @ Th e pape r use d i n this boo k i s acid-free an d fall s withi n th e guideline s established t o ensur e permanenc e an d durability . Visit th e AM S hom e pag e a t http://www.ams.org / 10 9 8 7 6 5 4 3 2 1 1 3 1 2 1 1 1 0 09 0 8 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 Fundamental 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 the Orde r o f a mod c 4 5 Chapter 11 . Primalit y Testin g 5 1 VI Higher Arithmeti c Chapter 12 . The RS A Ciphe r Syste m 57 Chapter 13 . Primitive Root s mo d p 61 Chapter 14 . Polynomials 67 Chapter 15 . Tables o f Indices mo d p 71 Chapter 16 . Brahmagupta's Formul a an d Hypernumber s 77 Chapter 17 . Modules o f Hypernumber s 81 Chapter 18 . A Canonical Form fo r Modules o f Hypernumbers 87 Chapter 19 . Solution o f AD + B = • 93 Chapter 20 . Proof o f the Theore m o f Chapter 1 9 99 Chapter 21. Euler's Remarkabl e Discover y 113 Chapter 22 . Stable Module s 119 Chapter 23 . Equivalence o f Module s 123 Chapter 24 . Signatures o f Equivalence Classe s 129 Chapter 25 . The Mai n Theore m 135 Chapter 26 . Modules That Becom e Principal Whe n Square d 137 Chapter 27 . The Possibl e Signature s fo r Certai n Value s o f A 143 Chapter 28 . The La w o f Quadratic Reciprocit y 149 Chapter 29 . Proof o f the Mai n Theore m 153 Chapter 30 . The Theor y o f Binary Quadrati c Form s 155 Chapter 31. Composition o f Binary Quadrati c Form s 163 Contents vn Appendix. Cycle s o f Stable Module s 16 9 Answers to Exercise s 17 9 Bibliography 20 7 Index 20 9 This page intentionally left blank Preface It i s widel y agree d tha t Car l Friedric h Gauss' s 180 1 book Disquisi- tiones Arithmeticae [G ] was the beginning o f modern number theory , the firs t wor k o n th e subjec t tha t wa s systemati c an d comprehen - sive rather than a collection o f special problems an d techniques . Th e name "numbe r theory " b y whic h the subjec t i s known toda y wa s i n use at th e time—Gauss himsel f use d i t (theoria numerorum) i n Arti- cle 5 6 o f the 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 whole numbers. " I too prefer "arithmetic " to "numbe r theory." T o me, number the- ory sounds passive, theoretical, and disconnected fro m reality . Highe r arithmetic sound s active , challenging, an d relate d to everyday realit y while aspirin g to transcend 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 indication o f what h e had i n mind come s at th e end o f his Preface whe n h e mentions the materia l in hi s Sectio n 7 on 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 what 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-theoretic 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 the 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 writing thi s an d previous books , I have ofte n foun d tha t creatin g exercise s lead s to a clearer understandin g o f the 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 .