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SELECTEDTOPICS IN NUMBER THEORY In thesacredmemoryof BabuKishan ChandandRev'n'Devi to whomIwasmore thanason Hansraj Gupta SELECTED TOPICS IN NUMBER THEORY Hansraj Gupta HonoraryProfessor, Pan/ab University, Chandigarh, India ABACUS PRESS First publishedin 1980 by ABACUSPRESS Abacus House, Speldhurst Road, Tunbridge Wells, Kent, TN4 OHU, England ©ABACUS PRESS 1980 All rightsreserved. No part ofthis publication may be reproduced, storedin aretrieval system, ortransmittedin any form orby any means, electronic, mechanical, photocopying, recording orotherwise, without the priorpermission ofAbacusPress BritishLibraryCataloguinginPublicationData Gupta, H Selected topicsin numbertheory. 1. Numbers, Theory of I. Title 512’.7 QA241 ISBN 0-85626‘-177-7 Typesetin GreatBritain by PintailStudiosLtd Ringwood, Hampshire andprintedandboundby Tien Wah Press (Pte.) Ltd,Singapore TO THE READER To the Reader Bookslivelonger than their authors. This iswhatprovidedame withtheurge to write orrather compile thisbookfrommy own publishedandunpublishedworkto a largeextent. I amentirely responsible forthe choice oftopics discussed inthebook. It isvery difficult forme to say ifmy choicehasbeen ahappy one,but I amvery hopefulthat you willnot dislike it. ThoughI receivedmy educationthroughthemediumofthe Englishlanguage mostly, Englishis notmymother-tongue. It isquitepossible that inplacesyoumay findme ratherobscure,but alittle effort onyourpartwill, I am sure,make my meaning clear. In severalplaces I havenot given all the stepsinthe proofofa theorem.Thiswillgiveyouagoodexerciseinunderstandingthesubjectifyoufillthe ‘blanks’,so to speak, givingreferencesto the theorems andresultsused. At the end ofthebook Ihave given alist ofbooksandpapers whichwill help you incon- tinuingyour study ofthe subject. Ofthe twelve chapters inthe bookasmany as four(Chapters 7—10) dealwithpartitions: therefore the papersconcernedwiththis topicare listedseparately. Thepresentationis on thewhole expository and some of my friendswhohave looked throughparts ofthe workintypescriptbefore publica- tionhavebeenkind enoughto say agoodword about it. Ofallbranches ofmathematics, numbertheory is the one thathasnot only provided food for seriousthoughtbut also createdenthusiasm for the subject amongyoungandold alike. I mentionedto a smallboy(aschool lad)thatifhe wouldtake any fourconsecutive numbersgreaterthan 11, at least one ofthem wouldbe divisibleby aprime greaterthan 11. He immediately started checkingmy statementmentally. I know ofaverygood appliedmathematicianwho started takingkeeninterestinnumbertheory, orratherthe properties ofnatural numbers, towardsthe end ofhislife. Ifyou like to enthuseyourown son or daughter about the study ofmathematics, givehim orhersome enticing results fromnumbers. For example: 452=2025 and 45=20+25; 992=9801 and 99=98+01 Anyamount ofsuchmaterialisavailable inAlbert H. Beiler’s ‘Recreationsinthe Theory ofNumbers’ (Dover, 1964). It isnot without sense thatnumber theoryhas beencalled ‘TheQueen ofMathematics’. For alargepart ofmylife, Iwasworkingina small township away fromwhat 5 6 TO THE READER M N « wasthen the only University in the Panjab. Therewasnolibraryworth the name in ..,‘.. oraround the place. Iwas, therefore,rediscoveringformyselfmany known results. Forexample, I didnotknowthat symmetric functionsrepresenting the sums ofthe products ofthe first n natural numberstakenrat atime, were Stirling’s Numbers ofthe firstkind. Theworkpresented in Chapters 5 and 6 was done mostly during the years 1931—1935 and formed part ofmy thesisforthe Ph.D. degree. I hardly made any reference to the work doneby others on the topic. It isvery possible that what Ihave claimedto be newmaybejust a result discoveredby a previous mathematician and only rediscoveredby me. Agreat part ofthe materialpresented inthese two chapterswas publishedin asmall monographby the University of Lucknowin 1938. Ithasbeenincludedherebythekind permission ofthe Univer- sity. Chapter 9 presentsa proofbyProfessorG. Szekeres ofa conjecture which Auluck, Chowla and myselfhad made in 1942. Theversionofhis proofgivenhere waskindlyexaminedbyhim andapproved. The proofhasbeen includedhere with his kindpermission. My thanksto Professor Szekeres. My thanksare due also tonumerousworkersin the field, some ofwhose results Ihave included inthisbook. I amparticularlythankful toProfessor M.V. Subbarao forhismanyusefulsuggestions duringmy stayat Edmonton, where a part ofthis bookwaswrittenin 1969—1970, andto several friends andstudentswho read throughparts ofthebookandhelpedmein clarifying certain points. Ishall thankanyofmy readersheartilyiftheypoint out tome anymistakes or misprintsthat theymay find in the printedwork. Allahabad H. Gupta India August 1979 CONTENTS To the Reader Chapter 1. Preliminary Chapter 2. Primitive Rootsand Indices 37 Chapter 3. Power Residues, Quadratic Reciprocity and Sum ofSquares 70 Chapter4. ClassAlgebraofk-ic Residues 119 Chapter 5. Stirling Numbersand Bernoulli Numbers 137 Chapter 6. Congruence PropertiesofSymmetric Functions 155 Chapter 7. Partition Functions ofVarious Types, Generating 188 Functions, Graphsand Identities Chapter8. Partitions: Inequalitiesand Congruences 232 Chapter9. AnAsymptotic Formula forp(n,k) 255 Chapter 10. Partition ofj—partite Numbers 283 Chapter 11. Some Diophantine Equations 305 Chapter 12. Permutationsofthe First n Natural Numbers 332 MiscellaneousProblems 361 Some InterestingFactsofNumberTheory 364 Bibliography 365 Index 391 CHAPTER 1 PRELIMINARY 1. Introduction We assume that the readerhasalreadytaken acourseinElementaryNumber Theory.Inthis preliminarychapter,wejust give aresume ofthe resultswithwhich the readeris surely familiarbut to whichhe maywant to refernowand thenduring the course ofhisstudy ofthisbook.We do not prove allthe results. 2. Naturalnumbers The numbers 1, 2,3, . . . ,n, . . . ,the so-callednaturalnumbers,canbe placedin three classes: (i)Those whichhavejust one divisor; (ii)Thosethathaveexactly two divisors;and (iii)Those thathave morethan two divisors. Class(i)consistsofonly oneelement namely 1.Natural numberswhich fallinclass (ii)are 2, 3, 5,7,11,13,17,19,... andare calledprimes.We usuallyusep to denote aprime.The rest ofthe natural numbersare placedin class(iii) andare called composite. 3. The fundamental theorem ofarithmetic Itiseasyto prove inductively that everynaturalnumber >1 canbe expressed as a product ofprimes.What is reallyimportant is to realize that, disregardingthe order inwhichthe primes are written,the expressionforanygivenn asaproduct of primesisunique. Thisisthe fundamentaltheoremofarithmetic. Forthe proof,we needthe following LEMMA. Ifthe theorem is trueforacertain m> 1 andpisanyprimedivisor ofm, thenp mustappearin theexpressionformasaproductofprimes. Proof. Ifm =p,there isnothing to prove. Otherwise,let m=p.m1, where m1>1. 10 SELECTED TOPICS IN NUMBER THEORY Let m1=P1P2P3 - - 'pka where thep’s are not necessarilydistinct.Then m=PP1P2P3 - - -Pk isan expression form as aproduct ofprimes. But we have assumed that the funda- mentaltheoremholds form,that is,but forthe orderinwhichthep’sare written, thisexpressionisunique. Hence thisis theonly expressionform asa product of primes. Sincep appearsinthe expression,the lemmaisproved. Incidentally we have provedhere that the expressionform1 is also unique. Proofofthefundamentaltheorem byH. Davenport. Ifpossible, assume thatn is theleast numberwhichhas at leasttwodistinctrepresentations as aproduct of primes,say n=p1p2p3-..pk, k>2; (1) and n =pip'zp’a -- 41}. 1'? 2. (2) Evidentlythennop canbe ap', forifp1 =p'l thenn/pl, whichiscertainlylessthan n,willhavetwo distinctexpressions as aproduct ofprimes.Withoutloss ofgeneral- ity,we can assume that p1<l72 <Pa g ' ' ' <Pk; and p3 <11; <p'3 < ‘ ' <1);- Hence,we musthave n>pi, n>p32. (3) Sincepl 95p'l,the sign ofequality cannot holdinbothplacesin (3). Therefore n>p1p'1. (4) Letn —plp'l =N. Sincep1l n andalsop'1ln,bothp, andp'1 divideN. Since N<n,the theoremholds forNandthe lemma shows that we canwrite N=p1plm. (5) Thisgives n=p1pi(m+1)=p1p'1qlqzmqh, (6) say, where theq’s are primes,not necessarily distinct fromothersonthe right of (6). From(1) and (6),we nowhave Pi4142---qh=n/P1=P2P3---Pk- (7)