A Journey Through The Realm of Numbers: From Quadratic Equations to Quadratic Reciprocity PDF
Preview A Journey Through The Realm of Numbers: From Quadratic Equations to Quadratic Reciprocity
SUMS Readings Menny Aka Manfred Einsiedler Thomas Ward A Journey Through The Realm of Numbers From Quadratic Equations to Quadratic Reciprocity Springer Undergraduate Mathematics Series SUMS Readings Advisory Editors Mark A. J. Chaplain, St. Andrews, UK Angus Macintyre, Edinburgh, UK Simon Scott, London, UK Nicole Snashall, Leicester, UK Endre Süli, Oxford, UK Michael R. Tehranchi, Cambridge, UK John F. Toland, Bath, UK SUMS Readings is a collection of books that provides students with opportunities to deepen understanding and broaden horizons. Aimed mainly at undergraduates, the series is intended for books that do not fit the classical textbook format, from leisurely-yet-rigorous introductions to topics of wide interest, to presentations of specialised topics that are not commonly taught. Its books may be read in parallel with undergraduate studies, as supplementary reading for specific courses, background reading for undergraduate projects, or out of sheer intellectual curiosity. The emphasis of the series is on novelty, accessibility and clarity of exposition, as well as self- study with easy-to-follow examples and solved exercises. More information about this series at http://www.springer.com/series/16607 Menny Aka • Manfred Einsiedler • Thomas Ward A Journey Through The Realm of Numbers From Quadratic Equations to Quadratic Reciprocity Menny Aka Manfred Einsiedler Department of Mathematics Department of Mathematics ETH ETH Zürich, Switzerland Zürich, Switzerland Thomas Ward School of Mathematics University of Leeds Leeds, UK ISSN 1615-2085 ISSN 2197-4144 (electronic ) Springer Undergraduate Mathematics Series ISSN 2730-5813 ISSN 2730-5821 (electronic) SUMS Readings ISBN 978-3-030-55232-9 ISBN 978-3-030-55233-6 (eB ook) https://doi.org/10.1007/978-3-030-55233-6 Mathematics Subject Classification (2020): 11A15, 11A51, 11D09, 11D25, 11E25, 11R04, 97E60 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland To our teachers S.K., E.T., and A.W. Preface “DieKunst istlang, dasLebenkurz,das Urteilschwierig, dieGelegenheit flu¨chtig. Handeln ist leicht, Denken schwer; nach dem Gedanken handeln unbequem. Aller Anfangistheiter,dieSchwelleistderPlatzderErwartung.”(JohannWolfgangvon Goethe, in‘WilhelmMeistersWanderjahre’)1 These notes do not require any prerequisites beyond school-level mathe- matics,andareintended forseveraldifferentaudiences.The bookoriginated in courses aimed at enthusiastic and talented high school students who want togetasenseofnumbertheoryinaction,andtogainasenseofwhatfurther study in mathematics might entail. The book could be used as a basis for a seminarformathematicsundergraduatesonaclassicalpartofnumbertheory usingChapters3,4,5,and7foranintermediatecourse,andincludingChap- ters 6 and 8 for a more advanced course. Moreover, undergraduate students may use the book as a self-study reading course. Inordertoconnectwith someofthe targetaudiences,Chapter1develops somefamiliarideasinformally.Ouremphasishereisonconvincingarguments thathelpdevelopafeelformathematics.Chapter2isconceptuallyimportant, asitshowshowsimplemethodsofproofcanproducepowerfulandsurprising consequences. It is, apart from some basic definitions concerning sets and functions,notdirectlyusedlater.FromChapter3onwardsthemaindirection is pursued, and the arguments are developed with increasing rigour. There are several motivations for this book. One is to introduce some beautiful mathematical results for their own interest. Another is to try and give some insight into what more advanced mathematics is about, and how mathematicsisdone.Athirdistousetheseresultstoillustratethatthinking carefully and precisely about quite simple questions generates interesting ideas and problems. We do this by tracing one of many possible journeys through a simple, classical, idea: Various number systems can be used to 1 Thisistranslatedinmultipleways,including“Artislong,lifeshort,judgmentdifficult, opportunity transient. Toact iseasy, to think ishard; to act according toour thought is inconvenient. Everybeginningischeerful;thethresholdistheplaceofexpectation.” vii viii Preface solve equations and problems concerning patterns in the natural numbers, mostlyinvolvingsquaresandprimes.Alongthewaywetrytointroducesome of the language, notation, and culture of mathematics, as well as the results themselves. To help with this initially, a short table of notation sufficient to getthereaderstartedisincludedonpagexix.Moreover,throughoutthetext important terminology is indicated with bold text when it is introduced. Part of this story is concerned with proofs and methods of proof. Two of these methods are so important that they deserve a special mention. The mathematical principle of induction is formally discussed in Section 3.3. The 37 places where it is used—sometimes without this being explicitly stated—are marked with vertical dots in the margin. Once the method has been formally introduced,the readershould makesure they understandwhy the argumentsindicateddoindeeduse aninductionargument.Similarly,the method introduced by Eudoxus known as ‘reductio ad absurdum’, or proof by contradiction, is used at least 21 times. More substantial contradiction arguments are labelled with a cloud in the margin to mark the indirect assumption made, and a lightning strike in the margin where the contra- diction that is a consequence of the erroneous assumption is reached.In fact mathematicians often use contradiction arguments without formally saying so,orinanabbreviatedway.The readershouldensurethey understandwhy these do indeed contain proofs by contradiction. At certain points we do something tricky or come close to difficult prob- lems. These are markedwith a ‘dangerousbend’ symbol in the margin,in atraditionstartedbyNicolasBourbaki.Thereadermayenjoylookingupthe extraordinary story of the mathematical author Bourbaki on the internet. Someofthe resultswepresenthavemoreelementaryproofsthanthe ones we have chosen to include. One reason for this choice is that ‘elementary’ often entails more calculations, with a less clear picture of why something is true. Our preference is to aim for the clear picture rather than calculations without the big picture in mind. The second reason is that we would like to use the problems considered as a way to introduce mathematics and its structures. Avoiding naturally emerging mathematical structures would run counter to that objective. There are 310 exercises in the text, 181 of them having hints in an ap- pendix. These exercises are part of the text, and the reader should attempt many of them. The later parts of the book are more advanced, and some of the exercisesflagged as ‘Challenges’could be used as the basis of projects in number theory. Computers—and, specifically, the widespread availability of sophisticatedsoftwarepackagesformathematics—havebecomeanimportant part of how mathematics is taught. The ability to rapidly experiment and test or even formulate hypotheses, together with powerful methods to illus- trate mathematical phenomena, is so useful that we have included a small amountofprogramming.Becauseofits mixofhighportabilityanduniversal availability, the open source package SageMath is used, and SageMath com- mands are indicated with use of the typewriter font. Unfortunately, it is Preface ix clearthat the precisesyntax for SageMaththat we use may become obsolete before long, but we hope that the reader will be able to find help on the internet when this happens. One of many attractive features of mathematics is that it is naturally expressed in multiple ways. One of these is algebraic and symbolic, another is geometric and visual. The extent to which both aspects play a role may be seen in the 88 figures used to illustrate the mostly algebraic and number- theoretic arguments we present. More advanced material related to the topics considered here are men- tioned as ‘Outlooks’ in the text. The 25 outlooks vary in the level of math- ematical sophistication involved, which is indicated as follows. Those simply labelled as ‘Outlook’ with an indication of titles of related courses might be accessible for an interested reader in the first year of university; those flagged ‘(through binoculars)’ use some more advanced university-level ma- terial;finally,thoseflaggedwith‘(throughatelescope)’ involvesophisticated post-graduate methods, and even unsolved problems. All outlooks can be skipped, as they will not be essential for our later discussions. Really un- derstanding the mathematics behind all of them is a near impossible task, certainly beyond the ability of the authors. Outlooks involving substantial mathematics have references for further reading, either in the body of the text or within the title, but these carry the health warning that they are in some cases at a much more advanced level than this text. Withfewexceptions,thehistoryofsomeonewithadeepinterestinmathe- matics featuresanexceptionalmathematical influence, oftena teacher,play- ing a roleatsomeearlypoint. The authorsarenotexceptions inthis regard. Menny: My greatest influence was actually a fellow student at my high • school,S.K.WecamefromsimilarbackgroundsbutwhileI,asmyteach- ers put it, had great potential but a lack of motivation, he had both the potential and enough motivation for ten people, and in particular for me. He made me sit down and study for important exams, was more excitedthanmewhenIsolvedadifficultproblemandactuallymadesure I finished high school (following the strict orders of our beloved physics teacher). Since he had started his university studies long before I did, when my time came to go to university, he was the one enrolling me to study physicsandinsistingthatItakethemathcoursesfromthe mathe- maticsbachelor’sprogram.Fromthispointonhisworkwasdone:Ifellin lovewiththemathematicsIwasexposedtoinuniversityalmostimmedi- ately,beingexposedtothetruenatureofthesubjectviathecoursesthat S.K. chose for me. His advice directed me onward into graduate school. I think he would have been happy to have read this book when we were together in high school. Manfred: My mathematics teacher E.T. in high school did two very im- • portantthingsforme.Firstly,heapologizedfornotbeingabletoexplain calculus fully in classbecauseoftime constraints—buthe offeredto lend ananalysisbooktointerestedstudents.Idonotthinkhewasaskedoften, x Preface and I do not think that he got asked more than once for the second or thethirdvolume.2 Thisofferchangedmylife.Secondly,hehookedmeup with other students that needed help with mathematics. Certainly one can earn some money this way, but the confidence and fluency one ob- tains in dealing with mathematics after having explained it many times is probably impossible to come by in any other way, and was by far the bigger gain for me. From these two simple things I found what I wanted todoinmylifeandteachingmademesucceedatit.Insteadofbecoming a gardener in a small Austrian town, I became a mathematics professor in Zu¨rich. Thomas: Three early influences made me a mathematician. My mother • introduced me to many unconventionalbooks, including one by Thomp- son [83]. She helped me navigate this dense, opinionated, entertaining, and non-rigoroustreatment of ‘calculus’, and the power of the ideas was immediately intriguing to me. An older sister was studying some mathe- matics at university level and helped place this in a wider context. The third influence was a high school mathematics teacher, A.W. At some point the syllabus was full of ‘mechanics’ problems that were really ar- tificially designed to be solvable in elementary terms using the tools of calculus,andhewentthroughasimpleexamplefromhisownbackground in engineering of a single-hinged flap opening under an aeroplane. With the assumption of no friction and no air, this was readily solvable using our methods. Then he showed us something closer to reality, a double- hinged flap with one end sliding horizontally. We could write down the relevantequations but couldnot solvethem in elementary terms,and he explained that in fact simple problems of this sort really would not fall to our methods. This was my firstglimpse into what mathematics might really be about—interesting and powerfultheories,but all aroundus de- ceptively simple problems that were more difficult than they appeared. WearegratefultoMeikeAkveld,EmilioCorso,RuthHolland,KevinHous- ton, Tomoko Kitagawa, Matthias Koeppe, S´ebastien Labb´e, Roland Pro- haska, Lorena Schwerzmann, C¸a˘gri Sert, Shaun Stevens, Andreas Wieser, andDonZagierforcomments andsuggestionsondrafts ofthe text.We took the treatment in Exercises 4.75 and 4.76 from lecture notes of Peter Steven- hagen [76], and thank him for allowing us to include them here. The third named author learned the ‘loaf of bread’ image on page 264 in lectures by StewartStonehewer.WealsothankthestudentswhoparticipatedintheETH ‘Studienwochen fu¨r Gymnasiastinnen und Gymnasiasten’ in 2018 and 2019 fortheirinterest,whichmotivatedustostart—andtocomplete—thisproject. Menny Aka, Zu¨rich Manfred Einsiedler, Zu¨rich Thomas Ward, Leeds 2Truthbetold,IknowhewasaskedtwiceforthesecondandthirdvolumebecauseIwent throughthecompleteseriestwicewhileastudentathisschool.
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