Undergraduate Texts in Mathematics Cam McLeman Erin McNicholas Colin Starr Explorations in Number Theory Commuting through the Numberverse Undergraduate Texts in Mathematics Undergraduate Texts in Mathematics Series Editors Pamela Gorkin Mathematics Department, Bucknell University, Lewisburg, PA, USA Jessica Sidman Mathematics and Statistics, Mount Holyoke College, South Hadley, MA, USA Advisory Board Colin Adams, Williams College, Williamstown, MA, USA Jayadev S. Athreya, University of Washington, Seattle, WA, USA Nathan Kaplan, University of California, Irvine, CA, USA Lisette G. de Pillis, Harvey Mudd College, Claremont, CA, USA Jill Pipher, Brown University, Providence, RI, USA Jeremy Tyson, University of Illinois at Urbana-Champaign, Urbana, IL, USA Undergraduate Texts in Mathematics are generally aimed at third- and fourth-year undergraduate mathematics students at North American universities. Thesetextsstrivetoprovidestudentsandteacherswithnewperspectivesandnovel approaches.Thebooksincludemotivationthatguidesthereadertoanappreciation ofinterrelations among different aspectsofthesubject.Theyfeature examples that illustrate key concepts as well as exercises that strengthen understanding. Cam McLeman (cid:129) Erin McNicholas (cid:129) Colin Starr Explorations in Number Theory Commuting through the Numberverse 123 CamMcLeman ErinMcNicholas Department ofMathematics Department ofMathematics University of Michigan–Flint Willamette University Flint, MI,USA Salem, OR,USA Colin Starr Department ofMathematics Willamette University Salem, OR,USA ISSN 0172-6056 ISSN 2197-5604 (electronic) Undergraduate Texts inMathematics ISBN978-3-030-98930-9 ISBN978-3-030-98931-6 (eBook) https://doi.org/10.1007/978-3-030-98931-6 MathematicsSubjectClassification: 11-01 ©SpringerNatureSwitzerlandAG2022 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart 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 orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfrom therelevantprotectivelawsandregulationsandthereforefreeforgeneraluse. The publisher, the authors, and the editorsare safeto assume that the adviceand informationin 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 hereinorforanyerrorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregard tojurisdictionalclaimsinpublishedmapsandinstitutionalaffiliations. ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Preface ...wherein we welcome readers of all sorts to our journey. Who is This Text’s Audience? We have written this book for you, our inquisitive, delightful reader! Your dis- cerningeyeformathematics,interestinnumbers,willingnesstoexplore,anddesire to understand underlying structures deeply are commendable. While this text examinesnumbersystemsfromanalgebraiclens,nopreviousexposuretoabstract algebra is assumed. We do, however, assume some familiarity with vector spaces andbasicprooftechniques.Calculus:MultivariablebyMcCallumetal.,andProofs and Fundamentals: A First Course in Abstract Mathematics by Ethan Bloch are excellent references for the assumed background material. To the Student: Dear Student, Welcome to the start of your journey! Consider this book as your official companionguide—itisinsomewaysatraditionalmathematicstextbook(ithas,for example, lots of math in it), but deviates from standard practice in several notable ways as well. First, it has multiple authors coming from a variety of backgrounds andexperienceswithinthegeneralrealmofalgebraandnumbertheory.Youmight see sparks of more excited writing as one of us encountered a favorite piece of numbertheory,forexample,thoughwegenerallyattemptedtoconformourwriting to a single voice1. Second, and more importantly, we call particular attention to the existence of “Explorations” sprinkled through the chapters. These worksheets are the manifes- tations of the authors’ zeal for inquiry-based learning—the process offiguring out mathematics for yourself like our mathematical forebears out in the unexplored 1Well,someofusdid. v vi Preface wilderness. Thereareno solutionsprovided totheworksheets, astheirpoint isnot typicallyto“gettheanswer” butrather tobring up questions that we’ll spend time discussinginthefollowingsection.Sodon’tskipthem!Infact,youmightwantto photocopy those pages for repeated use. Or better yet, buy several copies of the book, just to be safe. To the Instructor: Dear Instructor, Phew! Now that the students have stopped reading, we can be frank. Two adjectives that stand out as highly desirable when choosing a textbook are com- prehensiveandself-contained.Thisbookdoesnotattempttoachieveeitherofthose ambitions. In exchange, we stake a claim to the partially opposite adjectives of being modern and flexible. As to comprehensiveness, the over-arching goal of the texthasbeentohookthereader’sinterestsinthemysteryofnumbers,anditisthe opinion of the authors that no book this size can do justice to the vast wealth of classical number-theoretic pearls while simultaneously working toward contem- porarynumbertheory.Consequently,our“toincludeornottoinclude”philosophy hasbeentofocusondevelopingafewcorethemesthatpersistthroughthestudyof number theory at all levels: (cid:129) The study of Diophantine equations, partly for their own sake, but more typi- cally as catalysts for introducing and developing bigger structural ideas. (cid:129) The notion of what a “number” is, and the premise that it takes familiarity in quite a large variety of number systems to fully explore number theory. (cid:129) The use of abstract algebra in number theory, and in particular the extent to which it provides the “Fundamental Theorem of Arithmetic” for various new number systems. Inadditiontothecorethemes,otheraspectsofmodernnumbertheoryarepresentin smaller but persistent threads woven through chapters and exercise sets, e.g., the studyofellipticcurves,theanalogiesbetweenintegerandpolynomialarithmetic,p- adic valuations,relationshipsbetweenthespectrum ofprimes invariousrings,etc. Thepropersequencingofnumbertheoryandabstractalgebraisaconundrumwe have chosen to embrace. While an introduction to number theory provides a good preparationforthestudyofabstractalgebra,studentswhohavehadabstractalgebra arebetterabletofullygrasptheconceptsinanintroductorynumbertheorycourse. Our solution is to develop and define algebraic concepts as needed to summarize and formalize the patterns observed in various number systems. In this text, stu- dents encounter groups, rings, fields, ideals, and more. While we have taken uncharacteristic care to make the coverage of these concepts self-contained, it is certainly not comprehensive. In our experience, the result is a text that simulta- neously serves both students with and without any previous exposure to abstract algebra. Students with little to no abstract algebra experience are able to grasp the Preface vii salient details by applying the intuition and experience gained from the adjacent number theory. These students gain a solid foundation on which to build in a subsequentabstractalgebracourse.Forstudentswhohavehadacourseinabstract algebra, the pithy and targeted coverage of algebraic concepts is a good refresher that helps them apply their algebraic knowledge to gain a deeper understanding of the number theory content. For these students, the text provides not only an introductiontonumbertheorybutalsoastrengthenedappreciationforthepowerof an algebraic perspective. As to self-containedness more broadly, we have been content to assume some reasonable prerequisites and avoid compulsively writing appendices to provide basic familiarity with complex numbers, matrix algebra, and vector spaces. In additiontodirectingstudentstogoodtextsourcesforthisbackgroundmaterial,we take advantage of a thoroughly modern approach, namely, the ability of readers to go online and look things up. The skill of finding and processing mathematics onlineisanincreasinglycrucialoneandshouldbefosteredratherthandiscouraged. This philosophy makes itself explicit in the form of a section of each chapter’s homeworkproblems(“GeneralNumberTheoryAwareness”problems)whosegoal is to get students to conduct research online on biographical information about mathematicians and mathematical content that space prevented us from including, or pieces of mathematical culture that don’t fit strictly in the traditional academic viewofwhatamathematicstextbookshouldcover.Exercisesinthesesectionsalso leadstudentstopracticeoneofthemostfundamentalmathematicalresearchskills: seeing,uponmakingamathematicaldiscovery,ifsomeoneelsehasalreadydoneit. The broader claim to modernity is an over-arching belief in inquiry-based learning. In addition to these research-based homework questions, a second sub- sectionofexerciseshasstudentsnumericallyinvestigateconjecturesandmaketheir own. Sometimes, the tools to prove these conjectures will be within a student’s grasp, and sometimes they will be cutting-edge research questions to which humanity does not know the answer. While we do not prescribe a programming language for these numerical investigations, the book’s companion website has Python worksheets designed to walk students with little to no programming experience through those exercises. Finally, and perhaps most importantly, inquiry-based learning is built into the text itself: exploration worksheets lovingly placed throughout the chapters let students lead the way by coming up with the pivotal ideas for the upcoming sections while simultaneously improving their flu- encyinpreviouslycoveredtopics.Thefinalchaptercloselymirrorsthatofapurely inquiry-based textbook, allowing students to work individually or in groups— perhaps as a final project—through problem-driven sections covering material begun in the main text (Fermat’s Last Theorem, quaternions, real quadratic units, elliptic curves, cryptography, ideal theory). Exercises designed to pique students’ interest in these topics are interspersed throughout earlier chapters. Thepacingofthecourseisalsoratherflexible.Thedefaultapproach,webelieve, should begin with the standard theoretical development of Z and Z=ðnÞ but using the explicit language of groups and rings, rather than shunting this perspective off asadvancedmaterialneartheendofthecourse.Incorporatingabstractalgebrainto viii Preface thisprocessnecessarilyproceedsmoreslowly,buthasthebenefitofbeingquickly generalizable: once you prove that Z½i(cid:2) has a Euclidean algorithm, for example, proving that it enjoys unique factorization is essentially a matter of copy–pasting theargumentfromZ.GoingquicklythroughChapters5and6leavesenoughtime to do both culminating chapters, on quadratic reciprocity and p-adic numbers, in a singlesemester.Ontheotherhand,aninstructorwhowantedtousethebooktofill twosemesterscoveringbothabstractalgebraandnumbertheorycouldeasilydoso by using all of the in-class explorations, covering the details of abstract Euclidean domains, and providing in-class time for student presentations from Chapter 9. Finally, some closing remarks on the use of the book. As mentioned in the student preface, we highly value the inclusion of the explorations appearing throughout the book. They provide an inquiry-based break from the more tradi- tional class structure during which an instructor can engage in informal formative assessmentofhowtheclassisdoingandallowstudentstoengageinindependentor group discovery. Questions andanswers tothese problemsaretypicallydeveloped more carefully in the sections that follow. Relatedly, the homework addresses a variety of ways in which one becomes a master of number theory. There are computational exercises to promote fluency in arithmetic calculations, proof-based problems to develop theoretical understanding, theaforementioned online research problems, and problems that lend themselves to computer-aided exploration. Suggested Pacing and Content Coverage A note on the use of explorations: we find that different classes take significantly different amounts of time to work through the items on each exploration. Accordingly, we do not necessarily recommend trying to get through a full explorationinclass. Partscould beomittedorassignedasadditional homeworkas needed. For a 14-week semester, 3 hours per week, we suggest the following pacing: (cid:129) Week 1: Sections 1.1–2.3 (leaving most details for students to read), Exploration A. (cid:129) Week 2: Section 3.1, Exploration B, and Section 3.2. (cid:129) Week 3: Exploration C, Sections 3.3 and 3.4, Exploration D, and Section 3.5. (cid:129) Week 4: Sections 4.1–4.3, Exploration E, and Section 4.4. (cid:129) Week 5: Exploration F, Section 4.5, Exploration G, and Section 4.6. (cid:129) Week 6: Exploration H, Sections 4.7 and 4.8. (cid:129) Week 7: Sections 5.1 and 5.2, Exploration I, Sections 5.3 and 5.4, and Exploration J. (cid:129) Week 8: Section 5.5, Exam 1, and Section 5.6. (cid:129) Week 9: Sections 5.6–6.2, Exploration K, and Section 6.3. (cid:129) Week 10: Sections 6.3 and 6.4, Exploration L, and Sections 6.5 and 6.6. Preface ix (cid:129) Week 11: Sections 7.1 and 7.2, Exploration M, Sections 7.3 and 7.5. (cid:129) Week 12: Sections 7.5–7.7. (cid:129) Week 13: Chapter 9 topics. (cid:129) Week 14: Chapter 9 topics. Flint, USA Cam McLeman Salem, USA Erin McNicholas Salem, USA Colin Starr AcknowledgmentsWeappreciatetheeffortsofoureditorstokeepusontrackandthehelpfuland insightfulcommentsofthereviewers.Additionally,eachoftheauthorswouldliketoexpresstheir heartfeltappreciationtotheothertwoauthorsforeachdoingapproximately25%ofthework.