Solving Diophantine Equations

2014·3.2 MB·

by  Octavian Cira

Solving Diophantine Equations
OctavianCiraandFlorentinSmarandache
2014
2
Preface
InrecenttimesthewewitnessedanexplosionofNumberTheoryprob-
lems that are solved using mathematical software and powerful comput-
ers. The observation that the number of transistors packed on integrated
circuitsdoubleseverytwoyearsmadebyGordonE.Moorein1965isstill
accurate to this day. With ever increasing computing power more and
moremathematicalproblemscanbetackedusingbruteforce. Atthesame
timetheadvancesinmathematicalsoftwaremadetoolslikeMaple,Math-
ematica, Matlab or Mathcad widely available and easy to use for the vast
majority of the mathematical research community. This tools don’t only
perform complex computations at incredible speeds but also serve as a
great tools for symbolic computation, as proving tools or algorithm de-
sign.
Theonlinemeetingofthetwoauthorsleadtolivelyexchangeofideas,
solutionsandobservationonvariousNumberTheoryproblems. Theever
increasing number of results, solving techniques, approaches, and algo-
rithms led to the the idea presenting the most important of them in in
this volume. The book offers solutions to a multitude of η–Diophantine
equationproposedbyFlorentinSmarandacheinpreviousworks[Smaran-
dache,1993,1999b,2006]overthepasttwodecades. Theexpertiseintack-
lingNumberTheoryproblemswiththeaidofmathematicalsoftwaresuch
as [Cira and Cira, 2010], [Cira, 2013, 2014a, Cira and Smarandache, 2014,
Cira, 2014b,c,d,e] played an important role in producing the algorithms
and programs used to solve over 62 η–Diophantine equation. There are
numerousotherimportantpublicationsrelatedtoDiophantineEquations
i
ii
thatoffervariousapproachesandsolutions. However,thisbookisdiffer-
entfromotherbooksofnumbertheorysinceitdedicatesmostofitsspace
tosolvingDiophantineEquationsinvolvingtheSmarandachefunction. A
search for similar results in online resources like The On-Line Encyclopedia
ofIntegerSequencesrevealsthelackofaconcentratedeffortinthisdirection.
Thebruteforceapproachforsolvingη–Diophantineequationisawell
knowntechniquethatchecksallthepossiblesolutionsagainsttheproblem
constrains to select the correct results. Historically, the proof of concept
was done by Appel and Haken [1977] when they published the proof for
the four color map theorem. This is considered to be the the first theorem
thatwasprovenusingacomputer. Theapproachusedboththecomputing
powerofmachinesaswellastheoreticalresultsthatnarroweddowninfi-
nitesearchspaceto1936mapconfigurationsthathadtobecheck. Despite
some controversy in the ’80 when a masters student discovered a series
of errors in the discharging procedure, the initial results was correct. Ap-
pel and Haken went on to publish a book [Appel and Haken, 1989] that
containedtheentireandcorrectprofthateveryplanarmapisfour-colorable.
Recently,in2014anempiricalresultsofGoldbachconjecturewaspub-
lishedinMathematicsofComputationwhereOliveiraeSilvaetal.[2013],
[Oliveira e Silva, 2014], confirm the theorem to be true for all even num-
bersnotlargerthan4×1018.
The use of Smarandache function η that involves the set of all prime
numbers constitutes one of the main reasons why, most of the problems
proposed in this book do not have a finite number of cases. It could be
possible that the unsolved problems from this book could be classified in
classesofunsolvedproblems,andthussolvingasingleproblemwillhelp
insolvingalltheunsolvedproblemsinitsclass. Buttheauthorscouldnot
classify them in such classes. The interested readers might be able to do
that. Inthegivencircumstancestheauthorsfocusedonprovidingthemost
comprehensivepartialsolutionpossible,similartoothersuchsolutionsin
theliteraturelike:
• Goldbach’s conjecture. In 2003 Oliveira e Silva announced that all
even numbers ≤ 2×1016 can be expressed as a sum of two primes.
iii
In 2014 the partial result was extended to all even numbers smaller
then4×1018,[OliveiraeSilva,2014].
• For any positive integer n, let f(n) denote the number of solutions
totheDiophantineequation4/n = 1/x+1/y+1/z withx,y,z posi-
tiveintegers. TheErdo˝s-Strausconjecture,[Obla´th,1950,Rosati,1954,
Bernstein, 1962, Tao, 2011], asserts that f(n) ≥ 1 for every n ≥ 2.
Swett[2006]establishedthattheconjectureistrueforallintegersfor
any n ≤ 1014. Elsholtz and Tao [2012] established some related re-
sultsonf andrelatedquantities,forinstanceestablishedthebound
f(p) (cid:28) p3/5+O(cid:0)1/log(log(p))(cid:1)forallprimesp.
• Tutescu[1996]statedthatη(n) (cid:54)= η(n+1)foranyn ∈ N∗. OnMarch
3rd, 2003 Weisstein published a paper stating that all the relation is
validforallnumbersupto109,[SondowandWeisstein,2014].
• Anumbernisk–hyperperfectforsomeintegersk ifn = 1+k·s(n),
wheres(n)isthesumoftheproperdivisorsofn. Allk–hyperperfect
numbers less than 1011 have been computed. It seems that the con-
jecture ”all k–hyperperfect numbers for odd k > 1 are of the form p2 ·q,
with p = (3k + 4)/4 prime and q = 3k + 4 = 2p + 3 prime” is false
[McCranie,2000].
Thisresultsdonotofferthesolutionstotheproblemsbuttheyareimpor-
tantcontributionsworthmentioning.
The emergence of mathematical software generated a new wave of
mathematicalresearchaidedbycomputers. Nowadaysitisalmostimpos-
sibletoconductresearchinmathematicswithoutusingsoftwaresolutions
such as Maple, Mathematica, Matlab or Mathcad, etc. The authors used
extensively Mathcad to explore and solve various Diophantine equations
becauseoftheveryfriendlynatureoftheinterfaceandthepowerfulpro-
gramming tools that this software provides. All the programs presented
inthefollowingchaptersareintheircompletesyntaxasusedinMathcad.
Thecompactnatureofthecodeandeaseofinterpretationmadethechoice
ofthisparticularsoftwareevenmoreappropriateforuseinawrittenpre-
sentationofsolvingtechniques.
iv
Theempiricalsearchprogramsinthisbookwheredevelopedandexe-
cutedinMathcad. Thesourcecodeofthisalgorithmscanbeinterpretedas
pseudo code (the Mathcad syntax allows users to write code that is very
easytoread)andthustranslatedtootherprogramminglanguages.
Although the intention of the authors was to provide the reader with
acomprehensivebooksomeofthenotionsarepresentedoutoforder. For
example the book the primality test that used Smarandache’s function is
extensivelyused. Thefirstoccurrencesofthistestprecededthedefinition
the actual functions and its properties. However, overall, the text covers
all definition and proves for each mathematical construct used. At the
sametimethereferencespointtothemostrecentpublicationsinliterature,
while results are presented in full only when the number of solutions is
reasonable. For all other problems, that generate in excess of 100 double,
tripleorquadruplepairs,onlypartialresultsarecontainedinthesections
of this book. Nevertheless, anyone interested in the complete list should
contacttheauthorstoobtainaelectroniccopyofit. Runningtheprograms
inthisbookwillalsogeneratethesamecompletelistofpossiblesolutions
foranyoddtheproblemsinthisbook.
Authors
Acknowledgments
We would like to thank all the collaborators that helped putting to-
getherthisbook,especiallytoCodru¸taStoicaandCristianMihaiCira,for
theimportantcommentsandobservations.
Contents
Preface v
Contents ix
Listoffigure x
Listoftable xi
Introduction xii
1 Primenumbers 1
1.1 Generatingprimenumbers . . . . . . . . . . . . . . . . . . . 2
1.2 Primalitytests . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.2.1 Thetestofprimalityη . . . . . . . . . . . . . . . . . . 14
1.2.2 Deterministictests . . . . . . . . . . . . . . . . . . . . 15
1.2.3 Smarandache’scriteriaofprimality . . . . . . . . . . 24
1.3 Decompositionproductofprimefactors . . . . . . . . . . . . 32
1.3.1 Directfactorization . . . . . . . . . . . . . . . . . . . . 35
1.3.2 Othermethodsoffactorization . . . . . . . . . . . . . 37
1.4 Countingoftheprimenumbers . . . . . . . . . . . . . . . . . 39
1.4.1 Programofcountingoftheprimenumbers . . . . . . 39
1.4.2 Formulaofcountingoftheprimenumbers . . . . . . 40
v
vi CONTENTS
2 Smarandache’sfunctionη 42
2.1 Thepropertiesoffunctionη . . . . . . . . . . . . . . . . . . . 45
2.2 ProgramsforKempner’salgorithm . . . . . . . . . . . . . . . 50
2.2.1 Applications . . . . . . . . . . . . . . . . . . . . . . . . 53
2.2.2 Calculationtheofvaluesη function . . . . . . . . . . 54
3 Divisorfunctionsσ 58
3.1 Thedivisorfunctionσ . . . . . . . . . . . . . . . . . . . . . . 58
3.1.1 Computingthevaluesofσ functions . . . . . . . . . 62
k
3.2 k–hyperperfectnumbers . . . . . . . . . . . . . . . . . . . . . 63
4 Euler’stotientfunctionϕ 64
4.1 Thepropertiesoffunctionϕ . . . . . . . . . . . . . . . . . . . 65
4.1.1 Computingthevaluesofϕfunction . . . . . . . . . . 67
4.2 AgeneralizationofEuler’stheorem . . . . . . . . . . . . . . 68
4.2.1 Analgorithmtosolvecongruences . . . . . . . . . . . 72
4.2.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . 73
5 Generalizationofcongruencetheorems 75
5.1 Notionsintroductory . . . . . . . . . . . . . . . . . . . . . . . 75
5.2 TheoremsofcongruenceoftheNumberTheory . . . . . . . 78
5.3 Aunifyingpointofconvergencetheorems. . . . . . . . . . . 81
5.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
6 Analyticalsolving 87
6.1 GeneralDiophantineequations . . . . . . . . . . . . . . . . . 87
6.2 GenerallinearDiophantineequation . . . . . . . . . . . . . . 89
6.2.1 Thenumberofsolutionsofequation . . . . . . . . . . 90
6.2.2 Diophantineequationoffirstorderwithtwounknown 92
6.3 SolvingtheDiophantinelinearsystems . . . . . . . . . . . . 98
6.3.1 Procedureofsolvingwithrow–reducedechelonform 98
6.3.2 SolvingwithSmithnormalform . . . . . . . . . . . . 105
6.4 SolvingtheDiophantineequationofordern . . . . . . . . . 107
6.5 TheDiophantineequationofsecondorder . . . . . . . . . . 113
CONTENTS vii
6.5.1 Existenceandnumberofsolutions . . . . . . . . . . . 113
6.5.2 Methodofsolving . . . . . . . . . . . . . . . . . . . . 115
6.5.3 Procedureforsolving . . . . . . . . . . . . . . . . . . 118
6.5.4 Generalizations . . . . . . . . . . . . . . . . . . . . . . 124
6.6 TheDiophantineequationx2−2y4+1 = 0 . . . . . . . . . . 127
7 Partialempiricalsolving 130
7.1 Empiricaldeterminationofsolutions . . . . . . . . . . . . . . 130
7.1.1 PartialempiricalsolvingofDiophantineequations . 131
7.2 Theη–Diophantineequations . . . . . . . . . . . . . . . . . . 135
7.2.1 Partialempiricalsolvingofη–Diophantineequations 137
7.2.2 Theequation2069 . . . . . . . . . . . . . . . . . . . . 137
7.2.3 Theequation2070 . . . . . . . . . . . . . . . . . . . . 140
7.2.4 Theequation2071 . . . . . . . . . . . . . . . . . . . . 141
7.2.5 Theequation2072 . . . . . . . . . . . . . . . . . . . . 143
7.2.6 Theequation2073 . . . . . . . . . . . . . . . . . . . . 145
7.2.7 Theequation2074 . . . . . . . . . . . . . . . . . . . . 146
7.2.8 Theequation2075 . . . . . . . . . . . . . . . . . . . . 149
7.2.9 Theequation2076 . . . . . . . . . . . . . . . . . . . . 151
7.2.10 Theequation2077 . . . . . . . . . . . . . . . . . . . . 154
7.2.11 Theequation2078 . . . . . . . . . . . . . . . . . . . . 157
7.2.12 Theequation2079 . . . . . . . . . . . . . . . . . . . . 157
7.2.13 Theequation2080 . . . . . . . . . . . . . . . . . . . . 160
7.2.14 Theequation2081 . . . . . . . . . . . . . . . . . . . . 163
7.2.15 Theequation2082 . . . . . . . . . . . . . . . . . . . . 164
7.2.16 Theequation2083 . . . . . . . . . . . . . . . . . . . . 165
7.2.17 Theequation2084 . . . . . . . . . . . . . . . . . . . . 165
7.2.18 Theequation2085 . . . . . . . . . . . . . . . . . . . . 167
7.2.19 Theequation2086 . . . . . . . . . . . . . . . . . . . . 169
7.2.20 Theequation2087 . . . . . . . . . . . . . . . . . . . . 169
7.2.21 Theequation2088 . . . . . . . . . . . . . . . . . . . . 169
7.2.22 Theequation2089 . . . . . . . . . . . . . . . . . . . . 170
7.2.23 Theequation2090 . . . . . . . . . . . . . . . . . . . . 171
7.2.24 Theequation2091 . . . . . . . . . . . . . . . . . . . . 172