A Journey Through The Realm of Numbers: From Quadratic Equations to Quadratic Reciprocity

356 Pages·2020·4.6 MB·English

by  Menny Aka, Manfred Einsiedler, Thomas Ward

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

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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.