Sequential Experiments with Primes

277 Pages·2017·15.053 MB·english

by  Mihai Caragiu

Mihai Caragiu
Sequential Experiments
with Primes
123
Mihai Caragiu
Department ofMathematics andStatistics
OhioNorthern University
Ada,OH
USA
ISBN978-3-319-56761-7 ISBN978-3-319-56762-4 (eBook)
DOI 10.1007/978-3-319-56762-4
LibraryofCongressControlNumber:2017937523
©SpringerInternationalPublishingAG2017
ThisSpringerimprintispublishedbySpringerNature
TheregisteredcompanyisSpringerInternationalPublishingAG
Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland
Preface
Why This Book?
Thisbookisactuallyaboutthemathematicallifeanddestinyofmathematicsfaculty
andtheirtalentedstudentsinsmallundergraduatecolleges(notnecessarilytheelite
ones, which are different) who wish to obtain a glimpse into the ethereal world of
“higher mathematics.” How can this be done, in spite of daily pressures such as
high teaching and service loads for faculty and the heterogeneous and
career-oriented curricular schedules for students? How can these students learn to
see the value of higher mathematics? These things need to be figured out for an
education that will prepare students for lifetime of learning.
MyexperienceasamathematicsteacheratOhioNorthernUniversityhasledme
to an answer that I would like to share by means of this book with other faculty
members and talented students at small undergraduate colleges: it is about exper-
imentingwithelementarynumbertheory(primenumbersandrelatedfunctions)and
witnessingtheamazingbehaviorofspecialintegersequences.Throughelementary
means we managed in six years to get from scribbling a recurrence formula on a
piece of paper to being spoken of at a 2012 international conference on Fibonacci
numbers. For a small undergraduate college, that was a big deal.
Ada, OH, USA Mihai Caragiu
January 2017
Contents
1 Introduction.... .... .... ..... .... .... .... .... .... ..... .... 1
1.1 Low-Budget Space Travel . .... .... .... .... .... ..... .... 1
1.2 A Topical Overview. ..... .... .... .... .... .... ..... .... 6
2 Warming Up: Integers, Sequences, and Experimental
Mathematics.... .... .... ..... .... .... .... .... .... ..... .... 13
2.1 From the Lebombo Bone to OEIS ... .... .... .... ..... .... 13
2.2 Experimental Mathematics . .... .... .... .... .... ..... .... 14
2.3 Primes ... .... .... ..... .... .... .... .... .... ..... .... 15
2.3.1 Harmonic Numbers Revisited .... .... .... ..... .... 17
2.4 Periodic Sequences: Visualization, Periods, Preperiods,
Floyd’s Cycle-Finding Algorithm.... .... .... .... ..... .... 21
2.5 Mathematical Beauty at the Addition/Multiplication Interface ... 26
2.6 Some Classical Recurrent Sequences. Ducci Games.. ..... .... 28
2.7 Deeper into the Randomness ... .... .... .... .... ..... .... 35
2.8 The Greatest Prime Factor Function.. .... .... .... ..... .... 39
2.9 Overview of Some Other Number-Theoretic Functions
and Sequences . .... ..... .... .... .... .... .... ..... .... 40
2.10 MATLAB Too! .... ..... .... .... .... .... .... ..... .... 45
2.11 An Experiment with Pairs of Primitive Roots Modulo Primes ... 51
2.12 Traffic Flow and Quadratic Residues . .... .... .... ..... .... 55
3 Greatest Prime Factor Sequences.... .... .... .... .... ..... .... 67
3.1 The Prehistory: GPF Sequences, First Contact.. .... ..... .... 67
3.2 GPF-Fibonacci: Toward a Generalized GPF Conjecture.... .... 74
3.3 Vector-Valued MGPF Sequences .... .... .... .... ..... .... 84
3.4 The Ubiquitous 2,3, 5, 7 andan Interesting Magma Structure ... 92
3.5 Solvability: A Surprising Property of a Class of Infinite-Order
GPF Recurrences ... ..... .... .... .... .... .... ..... .... 103
3.5.1 The Case of an Arbitrary Seed ... .... .... ..... .... 109
3.6 GPF Ducci Games: A Combinatorial Unleashing
of 2, 3, 5, 7 ... .... ..... .... .... .... .... .... ..... .... 114
3.6.1 GPF Ducci Period Fishing with a Monte Carlo Rod.... 126
3.6.2 An Infinite-Dimensional Analogue .... .... ..... .... 126
3.7 All Primes in Terms of a Single Prime and Related Puzzles .... 130
3.7.1 Prologue: An Exercise with Commuting Pairs..... .... 130
3.7.2 A Cyclicity Conjecture . .... .... .... .... ..... .... 131
3.7.3 (cid:1)Explori(cid:3)ng the Possible Cyclicity of a General Magma
P;fa;b : Necessary Conditions and Computational
Evidence ... ..... .... .... .... .... .... ..... .... 135
4 Conway’s Subprime Function and Related Structures
with a Touch of Fibonacci Flavor ... .... .... .... .... ..... .... 151
4.1 An Euler–Fibonacci Sequence .. .... .... .... .... ..... .... 151
4.1.1 A Kepler Moment? .... .... .... .... .... ..... .... 155
4.1.2 The Euler–Fibonacci Sequence Modulo 4... ..... .... 157
4.2 Conway’s Subprime Fibonacci Sequences . .... .... ..... .... 158
4.2.1 A Monte Carlo Approach to Subprime Fib Period
Search. .... ..... .... .... .... .... .... ..... .... 160
4.2.2 General Second-Order Subprime Sequences . ..... .... 161
4.3 Subprime Tribonacci Sequences and Beyond... .... ..... .... 162
4.3.1 What Lies Beyond?.... .... .... .... .... ..... .... 164
4.3.2 Are All of Them Ultimately Periodic?.. .... ..... .... 164
4.4 Conway Subprime Ducci Games .... .... .... .... ..... .... 165
4.5 Conway Subprime Magmas, a Remarkable Cyclicity Result,
and an Unexpected Sighting of the Golden Ratio.... ..... .... 170
4.5.1 On a Class of Nontrivial Finite Submagmas
of ðN;(cid:2)Þ... ..... .... .... .... .... .... ..... .... 177
4.5.2 Substructures with Two Elements. .... .... ..... .... 178
4.5.3 Substructures with Three Elements .... .... ..... .... 179
4.5.4 Concluding the Proof of Theorem 4.5.. .... ..... .... 180
4.5.5 On a Class of Nontrivial Infinite Submagmas
of ðN;(cid:2)Þ... ..... .... .... .... .... .... ..... .... 181
5 Going All Experimental: More Games and Applications. ..... .... 185
5.1 The Greatest Prime Factor and “Nonassociative” Quaternary
Cellular Automata... ..... .... .... .... .... .... ..... .... 186
5.1.1 Two-Dimensional Nonassociative Quaternary Cellular
Automata... ..... .... .... .... .... .... ..... .... 190
5.2 Complex Evolution for a Class of Integer-Valued
Nonassociative Automata .. .... .... .... .... .... ..... .... 191
5.2.1 Taking the Boundary into Account.... .... ..... .... 194
5.2.2 MATLAB for 2D GPF Automata . .... .... ..... .... 196
5.2.3 2D Conway Subprime Automata.. .... .... ..... .... 199
5.3 Walks fromGreatestPrimeFactorSequencesand aMysterious
Chebyshev-Like Bias ..... .... .... .... .... .... ..... .... 203
5.3.1 Self-correlations... .... .... .... .... .... ..... .... 214
5.3.2 A Quasirandomness Test for the Limit Cycles
of GPF Sequences. .... .... .... .... .... ..... .... 217
5.3.3 The Curious Negative-Leaning Trend .. .... ..... .... 219
5.3.4 Limitations and Opportunities.... .... .... ..... .... 225
5.3.5 2D and 3D Walks. .... .... .... .... .... ..... .... 225
5.3.6 WYSIWYG (Well, Almost…).... .... .... ..... .... 232
5.4 Linear Complexity of Bitstreams Derived
from GPF Sequences ..... .... .... .... .... .... ..... .... 234
Appendix A: Review of Frequently Used Functions,
Hands-On Visualization... .... .... .... .... ..... .... 249
Appendix B: Review of Floyd’s Algorithm and Floyd–Monte
Carlo Data Acquiring for Periods... .... .... ..... .... 255
Appendix C: Julia Programs Used in Exploring GPF
and Conway Magmas .... .... .... .... .... ..... .... 259
Appendix D: What’s Next? Epilogue and Some Reflections . ..... .... 269
References.... .... .... .... ..... .... .... .... .... .... ..... .... 273
Chapter 1
Introduction
1.1 Low-Budget Space Travel
A frequent key phrase often heard in undergraduate colleges is “high-impact
learning.” Indeed, the Association of American Colleges and Universities
(AAC&U) issued alistof ten “high impact educational practices”(Kuh 2008) that
wouldarguablybooststudentsuccess,outofwhichtwoareofparticularinterestto
this work: undergraduate research, and capstone courses and projects.
The present book will consider ways of boosting success in mathematics for
studentsatsmallundergraduatecolleges.Itwillconsistofagrabbagofnew(tothe
bestknowledgeoftheauthor)mathematicalideasandproblems(mostofthemopen
problems, with some of them proved) involving prime numbers and related
sequences that the author hopes will boost the enthusiasm for exploration in pure
mathematics of both students and faculty at small undergraduate colleges.
Intheopinionoftheauthor,disseminatingandadvertisingthecoremathematical
ideas(thoseroutinelygroupedunderthe“puremathematics”label)faceavarietyof
objective challenges in today’s context that need to be acknowledged. The rapid
growth of business, investment, consulting, and insurance companies (needless to
say, well funded and offering attractive salaries) creates a high demand for grad-
uates “good with numbers,” implicitly generating changes in the expectations of
mathematics(andstatistics)majors(forexample,therapidgrowthoftheinsurance
industry tilts the demand scale towards bachelor’s degrees in mathematics or
statisticswithanactuarialscienceconcentration).Relativelyfewstudentstodayare
eager to engage in “hopelessly pure” areas such as number theory, geometry,
combinatorics, or analysis. For incoming freshmen, “being good with numbers” in
thecontextofapossiblecareeraftergraduationis,mostofthetime,associatedwith
careersinaccounting,business,andmanagement:beinggoodwithnumbersisonly
marginally associated with, say, prime numbers.
2 1 Introduction
That is why mathematics faculty engaged in undergraduate education need to
find ways for a “new beginning” that can jump-start the enthusiasm for that
amazing body of knowledge commonly known as “pure mathematics.”
For applied-minded students who are pursuing a degree in computer engineer-
ing,onecanemphasize,forexample,thetremendoussuccessofnumbertheoryand
related cryptosystems in the “hot areas” (by the standards of today’s society) of
informationsecurity—anactivitythatmaybeconstruedasasignificant“outreach”
onbehalfofthe“queenofmathematics”(numbertheory,asseenbyCarlFriedrich
Gauss).
For those students with interests in the fundamental physical sciences, we can
emphasize the important role of geometry, topology, and algebra in the areas of
physics in which the “big questions” reside (quantum theory and
gravitation/cosmology);forexample,allthree2016Nobellaureatesinphysicsused
topological ideas to guide their explorations involving new phases of matter and
topological phase transitions (Nobelprize.org 2016).
For students interested in applications of statistics, the mathematics teacher can
suggestthattheymaywishtofocustheirtalentonfoundationalissuesofprobability
theory and random processes (after all, modern probability theory, a pure mathe-
matical theory, was founded by Kolmogorov in the 1930s), and possibly—to offer
justtwoexamples—tofollowupbyapplyingthesefundamentalresultstothestudy
oftheimmutablerealityofprimenumbers[e.g.,thespecialPoissoniancharacterof
the distribution of primes (Gallagher 1976)] orto mathematical economics (if they
have seen the movie “A Beautiful Mind,” that would definitely be a plus).
In any case, this is not about trying to get students to “switch” their major or
concentration of study to pure mathematics. There are students with genuine
interest in business, analytics, applied statistics, actuarial science, physics, engi-
neering, environmental science, chemistry, etc., which is great. At the same time,
however, a passionate mathematics teacher should always try to provide students
withopportunitiestowitnessfirsthandsomeaspectsofthebeautyandthedepthof
mathematics.Gettingthemtoexperiencepuremathematicswouldbeenriching,and
forthosestudentshaving,infact,a“pure”mathematicssensibility,thatwouldbea
genuine moment of self-discovery.
Everything sounds nice… if it weren’t so difficult. The typical small under-
graduate college is not in the “top 20” elite, does not have a huge financial
endowment, and is faced with many challenges, especially when it comes to
admissions.Gettinganystudentsatalltomajorinmathematicsisattimesdifficult.
Manyincomingstudentsareconfused,unsureaboutthepaththeyshouldtake,and
the subtleties of upper-level mathematics courses make them uncomfortable.
The dissemination of the beauty of the pure mathematics among students is
generally harmed by a variety offactors. In the opinion of the author, the leading
such factor, especially when it comes to mathematics or statistics majors, is
downplaying—or simply not even emphasizing enough—the importance of
extracurricular activities such as group projects involving solving problems pro-
posed in various mathematics journals or discussing various important mathemat-
ical ideas, participating in summer Research Experiences for Undergraduates,
1.1 Low-BudgetSpaceTravel 3
presenting at mathematical conferences, or simply “writing” mathematics.
Neglecting such extracurricular activities arguably limits a student’s creativity and
the desire to pursue graduate work in mathematics.
Also,therearethesuggestionsandpromisesofgenerallywell-paiddeskjobsfor
new graduates (in business, banking, management, insurance, etc.) after an
accordinglynarrowcourseofstudythatavoidsthedeeperendofthemathematical
content. This makes courses such as abstract algebra, real analysis, geometry,
topology, combinatorics, advanced random processes, and number theory, appear
unnecessary,burdensome,uninteresting,irrelevant,andunnecessarilycumbersome.
Of course, the fact that in the “age of media streams” the average human attention
span dropped from 12 s (in 2000) to 8 s (in 2015) (McSpadden 2015) does not
reallyhelp(ofcourse,theauthordoesnotmakeanyclaimofnotbeingpartofthis
trend).
So, the real question is, what can we do (as college teachers), in this particular
context, to increase students’ exposure to mathematical ideas that might awaken
their “researcher within” and subsequently send them on a path of discovery in
mathematics?
I don’t think that asking students to read renowned classic mathematical
monographs (or voluminous collections of classical articles) is a feasible solution,
especially today, and especially at the “typical” small undergraduate college.
Instead, we should exploit one particularity of the current age that has already
proved time and again to be beneficial to the research mathematician: the use of
computers.
Theapproachtakeninthisworkisthatifthemathematicsprofessormanagesto
communicate the spirit of “experimental mathematics” to an interested student,
indicatingaproblemthatiselementaryandeasytoformulate(butgenerallyhardto
solve), then the computing environment becomes an extremely beneficial “instru-
ment of dialogue” between the professorand the student.Mathematical reality can
beinvestigatedmuchasphysicalrealityisinvestigatedinparticlephysics:athigher
andhigherenergies,manyinterestingandunexpectedphenomenaandparticlesare
generated. In the same way, using (generally simple) programming and computer
algebra systems by “mathematical experimentalists” (student and teacher) can
generate interesting new mathematical ideas and conjectures.
In addition, when some of these conjectures (suggested as a result of compu-
tational analysis) can be proved through an approach that is reasonable for an
undergraduate mathematics major (generally elementary, albeit intricate at places),
the satisfaction is so much the greater. In the author’s educational experience,
students who participate in the process of discovering new and interesting mathe-
matical knowledge (about prime numbers and related sequences in the case of the
presentwork) through computation, tosay nothingaboutsome ofit being actually
proved,willhaveafirst-hand,significant,high-impactlearningexperiencethatwill
change their views on pure mathematics for the better.
At the same time, the instructor can say, figuratively speaking, that a “low-cost
space travel” experience has been made available to the student. “Low-cost”
becauseitinvolveselementaryideasthataresimpletoformulateandthentheuseof
4 1 Introduction
a computing environment requiring generally simple programing and a convenient
user interface. “Space travel” because the conclusions of the experiment are
beautiful, interesting, and represent new mathematical truth, with some parts of it
proved, while most other parts await further exploration.
Thisbookismeanttoinspirethepure-mathematics-mindededucatorinatypical
small undergraduate college by offering a variety of elementary number theory
insights involving sequences essentially built from prime numbers and associated
number-theoretic functions, together with related conjectures and proofs. It is not
meant to be “complete,” nor can it ever be so, since one can imagine new
number-theoretic functions and use appropriate programming to investigate other
topics,incollaborationwithstudentslookingforasignificanthigh-impactlearning
experience in mathematics.
Ourlifeascollegemathematicsteachersseekingsignificantresearchexperiences
to offer our students is not exactly easy. Indeed, it is not easy to be in a
“research-intensive” mode when the regular curriculum teaching load is 12 credit
hours per semester. Yet I hope that this computational/experimental approach will
help to make it easier.
The basic requirements for a typical participating student are an interest in
mathematical discovery, an eagerness to face new ideas, an ability to read and do
somebasicproofsinelementarynumbertheory,andawillingnesstousecomputers
to test the hypotheses that appear along the way.
Theeagernesstofacenewideasshouldbereflectedinapassionatedesiretotest
new results and discoveries using computers. Just to give an example, while
Michelle Haver, one of our undergraduate students and a “pure” mathematics
major, was considering a topic of presentation to the Ohio MAA “Centennial
Meeting” that took place on April 8–9, 2016, at Ohio Northern University, we
discovered the celebrated “prime conspiracy” result of Kannan Soundararajan and
Robert Lemke Oliver (Klarreich 2016). Since we liked the topic and didn’t have
much time left before the upcoming MAA conference, we decided to try a sort of
“Monte Carlo” simulation to study the interesting “self-avoidance” phenomenon
between the congruence classes of pairs of consecutive primes discussed in the
amazing paper (Lemke Oliver and Soundararajan 2016), to the effect that, for
example,ifaprimeendsin1,thefollowingprimeislesslikelytoendin1.Thuswe
decided to verify the “self-avoidance” conjecture using MAPLE by randomly
selecting pairs of consecutive large primes up to 12 digits and analyzing the set of
corresponding last-digit pairs. We wanted the output to be in the form of a his-
togram, because visualizing is believing. The four groups of four bins reflect the
distributionoflast-digitpairsaftertenthousandrandomtrials.Inthefirstgroup,we
seethecountsfor11,13,17,and19,inthesecondgroupthecountsfor31,33,37,
and 39, etc. The simple instruction line in MAPLE was as follows: