Table Of ContentList of Symbols
Symbol Meaning Page Symbol Meaning Page
Z setofintegers (3) a|b aisafactorofb (74)
x∈S xbelongstosetS (3) a(cid:2)b aisnotafactorofb (74)
x∈/S xdoesnotbelongtosetS (3) |A| thenumberofelementsinsetA (76)
Z+ setofpositiveintegers (3) A∪B theunionofsetsAandB (76)
N setofpositiveintegers (3) A∩B theintersectionofsetsAandB (76)
(cid:11)
W setofwholenumbers (4) A thecomplementofsetA (76)
a<b aislessthanb (4) N=(akak−1...a1a0)b base-brepresentationofN (83)
a>b aisgreaterthanb (4) Rn repunitwithnones (96)
a≤b a<bora=b (5) π(x) thenumberofprimes≤x (110)
a≥b a>bora=b (5) Fn thenthFibonaccinumber (129)
min{x,y} theminimumofxandy (5) Ln thenthLucasnumber (136)
max{x,y} themaximumofxandy (5) |A| thedeterminantofmatrixA (138)
|x| theabsolutevalueofx (5) fn thenthFermatnumber (139)
(cid:5)x(cid:6) theflooroftherealnumberx (6) (a,b) thegreatestcommonfactorofaandb (155)
(cid:7)x(cid:8) theceilingoftherealnumberx (6) (a1,a2,...,an) thegreatestcommonfactorofa1,a2,...,andan (162)
i(cid:2)=m (cid:2)m (cid:2)m pa(cid:12)n paexactlydividesn (183)
ai= ai= ai ak+ak+1+···+am (9) [a,b] theleastcommonmultipleofaandb (184)
(cid:2)i=k i=k k [a1,a2,...,an] theleastcommonmultipleofa1,a2,...,andan (187)
ai thesumofthevaluesofaiasirunsoverthevariousvaluesinI (11) a≡b(modm) aiscongruenttobmodulom (212)
(cid:2)i∈I a(cid:14)≡b(modm) aisnotcongruenttobmodulom (212)
aij thesumofthevaluesofaij,whereiandjsatisfypropertiesP (11) [r] thecongruenceclassrepresentedbyr (216)
P a−1 aninverseofamodulom (234)
i(cid:3)=m (cid:3)m (cid:3)m
ai= ai= ai akak+1···am (13) ρ(n) thedigitalrootofn (291)
i=k i=k k In theidentitymatrixofordern (316)
n(cid:4)! (cid:5) nfactorial (13) n# theproductofprimes≤n (325)
n
binomialcoefficient (33) ϕ(n) Euler’sphifunction (342)
r τ(n) thenumberofpositivefactorsofn (365)
tn triangularnumber (40) σ(n) thesumofthepositivefactorsofn (366)
sn squarenumber (44) Mp Mersennenumber2p−1 (381)
pn pentagonalnumber (46) μ(n) Möbiusfunction (398)
hn hexagonalnumber (48) λ(n) Liouvillefunction (405)
Tn tetrahedralnumber (49) ordma theorderofamodulom (456)
Sn squarepyramidalnumber (50) ψ(d) thenumberofincongruentresiduesoforderdmodulop (470)
Pn pentagonalpyramidalnumber (51) indαa theindexofatothebaseα (483)
Hn hexagonalpyramidalnumber (51) (a/p) Legendresymbol (501)
adivb thequotientwhenaisdividedbyb (71) (a/m) Jacobisymbol (527)
amodb theremainderwhenaisdividedbyb (71) (a/n) Kroneckersymbol (549)
Elementary Number Theory with
Applications
Second Edition
Elementary Number Theory with
Applications
Second Edition
Thomas Koshy
AMSTERDAM•BOSTON•HEIDELBERG•LONDON
NEWYORK•OXFORD•PARIS•SANDIEGO
SANFRANCISCO•SINGAPORE•SYDNEY•TOKYO
AcademicPressisanimprintofElsevier
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LibraryofCongressCataloging-in-PublicationData
Koshy,Thomas.
Elementarynumbertheorywithapplications/ThomasKoshy.–2nded.
p.cm.
Includesbibliographicalreferencesandindex.
ISBN978-0-12-372487-8(alk.paper)
1.Numbertheory.I.Title.
QA241.K672007
512.7–dc22
2007010165
BritishLibraryCataloguing-in-PublicationData
AcataloguerecordforthisbookisavailablefromtheBritishLibrary.
ISBN:978-0-12-372487-8
ForinformationonallAcademicPresspublications
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PrintedintheUnitedStatesofAmerica
07 08 09 10 9 8 7 6 5 4 3 2 1
Dedicatedto
mysister,AleyammaZachariah,andmybrother,
M.K.Tharian;andtothememoryof
ProfessorEdwinWeiss,ProfessorDonaldW.Blackett,
andViceChancellorA.V.Varughese
Contents
Preface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii
AWordtotheStudent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxi
1 Fundamentals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 FundamentalProperties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 TheSummationandProductNotations . . . . . . . . . . . . . . . . . . . . . 9
1.3 MathematicalInduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.4 Recursion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.5 TheBinomialTheorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
1.6 PolygonalNumbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
1.7 PyramidalNumbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
1.8 CatalanNumbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
ChapterSummary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
ReviewExercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
SupplementaryExercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
ComputerExercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
EnrichmentReadings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
2 Divisibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
2.1 TheDivisionAlgorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
(cid:2)2.2 Base-bRepresentations(optional). . . . . . . . . . . . . . . . . . . . . . . . . 80
(cid:2)2.3 OperationsinNondecimalBases(optional). . . . . . . . . . . . . . . . . . . 89
2.4 NumberPatterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
2.5 PrimeandCompositeNumbers. . . . . . . . . . . . . . . . . . . . . . . . . . . 103
2.6 FibonacciandLucasNumbers . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
2.7 FermatNumbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
ChapterSummary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
ReviewExercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
SupplementaryExercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
ComputerExercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
EnrichmentReadings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
vii
