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Andrews G.E. — Number Theory

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Название: Number Theory

Автор: Andrews G.E.

Аннотация:

Most mathematics majors first encounter number theory in courses on abstract algebra, for which number theory provides numerous examples of algebraic systems, such as finite groups, rings, and fields. The instructor of undergraduate number theory thus faces a predicament. He must interest advanced mathematics students, who have previously studied congruences and the fundamental theorem of arithmetic, as well as other students, mostly from education and liberal arts, who usually need a careful exposition of these basic topics.

Язык:

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1971

Количество страниц: 259

Добавлена в каталог: 02.02.2010

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Предметный указатель

"Even prime"      28
Abel's lemma      190—191 195
Absolute convergence, for infinite products      217
Additivity      139
Alder, H.L.      229—231
Algebra, abstract      v
Algorithm, Euclidean      16
Andrews, G.E.      227 229—231
Arithmetic functions      75—92
Arithmetic functions, multiplicative      85—86 88
Arithmetic mean      92
Ayoub, R.      230—231
Base      8 10
Basis representation      3—11
Basis representation theorem      8—11 26 227
Belongs to the exponent h      93
Bertrand's postulate      111
Binary notation      8
Binomial coefficient $\begin{pmatrix}n\\r\\\end{pmatrix}$       32
Binomial series      42
Binomial theorem      35
Birkhoff, G.      55
Bromwich, T.J.I'A.      229 231
Cancellation law      51
Card shuffling      56—57
Card shuffling, historical note      227
Carmichael's conjecture      82
Carmichael, R.D.      82 227 231
Casting out nines      64
Catalan number (Theorem 3—6)      41
Cauchy sequence      218
Chebyshev, P.      see "Tchebychev P."
Chinese remainder theorem      66—71 228
Circle problem      see "Gauss's Circle Problem"
Combination      32
Combinatorial analysis      see "Combinatorics"
Combinatorics      v 227
Common denominator      22
Complementary divisors      243
Complete residue system      52—53
Complex conjugate      143
Complex number      143
Complex number, absolute value of      143
Computers and binary numbers      10
Computers and numerical examples      v 44—48
Computers in number theory      30 227
Congruences      v 1
Congruences, definition of      49
Congruences, fundamentals of      49—57
Congruences, linear      58—61
Congruences, mutually incongruent solutions of      59—60
Congruences, polynomial      71—74
Congruences, quadratic      113
Congruences, solving      58—74
Conjugate partitions      151
Convergence absolute      223
Convergence absolute of double series      223
Convergence absolute of products      217
Convergence by columns      221
Convergence by rows      221
coordinates      23
de La Vallee Poussin, C.      100
Decimal system      8
Derivative, $k^{th}$       73
Dickson, L.E.      227—229 231
Diophantine equations      146—147
Diophantine equations, linear      23—26 58—59
Diophantus of Alexandria      141
Dirichlet's divisor problem      207—210 229
divides      15
Divisibility      1 15—23
Division lemma      see "Euclid's division lemma"
Divisor      15
Divisor function and Dirichlet's Divisor Problem      207—210 229
Divisor function and geometric number theory      199
Divisor function and Moebius pairs      89—90
Divisor function and numerical table      47
Divisor function and sum of divisors      92
Divisor function as arithmetic function      1 75
Divisor function as multiplicative function      85—86
Divisor function, average value of      201
Divisor function, formulae for      82—85
Divisor function, historical note      228
Divisor function, table of values      77
Divisor problem      see "Dirichlet's Divisor Problem"
Divisor, common      15
Divisor, greatest common      15
Double series      221—225
Double series, absolute convergence of      223
Double series, convergence of      223
Empty partition      149 181 192
Empty product      184
Empty set      33
Equivalence relation      50
Erdoes, P.      81 100
Euclid      100
Euclid's division lemma      12—15
Euclidean algorithm      16
Euler's $\phi$ -function and Moebius pairs      89—91
Euler's $\phi$ -function and numerical table      48
Euler's $\phi$ -function as multiplicative function      85—86
Euler's $\phi$ -function, combinatorial study of      75—82
Euler's $\phi$ -function, definition of      54
Euler's $\phi$ -function, historical note      228
Euler's $\phi$ -function, table of values      77
Euler's criterion      115—117
Euler's partition theorem      150 154—156 164—165
Euler's partition theorem, searching for      155—156
Euler's pentagonal number theorem      see "Pentagonal Number Theorem"
Euler's theorem      62
Euler, L.      45 111 118 149—150 175
Exponent, to which a belongs      93
Factorial      32
Factorizations into primes, table of      26
Faro shuffle, modified perfect      56—57
Fermat numbers      65—66 71 104
Fermat's conjecture      147 229
Fermat's last theorem      147
Fermat's little theorem as a congruence      49
Fermat's little theorem, combinatorial proof      36—38
Fermat's little theorem, historical notes      227 228
Fermat's little theorem, proof by congruences      61—66
Fermat, P.      36 147
Fibonacci numbers      6—7 35 44
Fibonacci numbers, generalized      23
Fields, finite      v
Flow chart      45—46
Flow diagram      45—46
Four-square problem      see "Sums of four squares"
Fractional part of x      207
Fundamental Theorem of Arithmetic      v 12—29
Gauss's Circle Problem      201—207 229
Gauss's lemma      119
Gauss, C.F.      49 100 113 118—120
General combinatorial principle      31
Generating functions      30 40—44
Generating functions for $D_{1}(n)$       163—164
Generating functions for $d_{m}(n)$       163
Generating functions for $p(S_{d}, n)$       162—163
Generating functions for $Q_{3}(n)$       165
Generating functions for $\pi_{m}(n)$       160—161
Generating functions for p(n)      162
Generating functions for partitions      vi 160—174
Generating functions for the sequence A      40
Generating functions, applications of      164—167
Generating functions, infinite products as      160—167
Geometric mean      92
Geometric series and generating functions      41
Geometric series and partitions      161
Geometric series and primes      102

Geometric series, finite (Theorem 1—2)      5
Geometry, analytic      23 199
Geometry, of numbers      201
Gerstenhaber, M.      118
Gillies, D.      44 112
Goldbach's conjecture      81 85 111
Goldbach, C.      81 111
Golomb, S.W.      227 231
Graphical representation      150—153
Greatest integer function      90—91 101 103—104 207
Grosswald, E.      228 230—231
Group      93
Group, cyclic      93
Group, finite      v
Hadamard, J.      100
Hardy, G.H.      147 149—150 176 229 230—231
Heaslet, M.A.      227 232
Heine's theorem (Exercise 3)      173
Hilbert, D.      147
Ideal      see "Integral ideal"
INDEX      96
Induction hypothesis      5
Inequality of the Geometric and Arithmetic Means      92
Infinite products      160 215—220 229
Infinite products, convergence of      215—219
Infinite products, divergence of      160
Infinite products, identities      167—174
Infinite products, partial products of      160
Infinite series      215—220 229 see "Geometric
Infinite series, convergence of      215
Infinite series, identities      167—174
Infinite series, MacLaurin series      43 219
Infinite series, Taylor series      43 73
Integral ideal      14
Integral part of x      see "Greatest integer function"
Integral test      209 226
Introductio in Analysin Infinitorum      149 175
Inverse modulo c      60
Jacobi symbol      118 124
Jacobi's triple product identity      169—172 176—177 229
Jacobsthal sums, S(m)      135—138 228
Johnson, P.B.      227 231
Knopp, M.I.      230—231
Knuth, D.E.      227 232
l'Hospital's rule      214
Lagniappe      141
Lagrange's theorem      72
Lagrange's theorem and primitive roots      97
Lagrange's theorem, historical note      228
Lagrange, J.      72 144
Landau, E.      230 232
Lattice points      23 121 201—210
Least common multiple      22
Least residue      119
Least-integer principle      14 145
Legendre symbol      117—118
Legendre, A.      118
Lehmer, D.H.      228 232
Leibnitz, G.W.      38
LeVeque, W.J.      230 232
Linear combination, integral      18
Linear Diophantine equation      see "Diophantine equation"
Littlewood, J.E.      147
Lucas number      7
MacLane, S.      55
Maclaurin series      43
MacLaurin series, for infinite products      219—220
MacMahon, P.A.      176
Mamangakis, S.E.      228 232
Mathematical induction      3—8
Mathematical induction, principle of      4 14
Menon, P.K.      229 232
Mersenne primes      see "Primes"
Mersenne, M.      112
Moebius function and Moebius inversion formula      86—91
Moebius function as arithmetic function      75
Moebius function as multiplicative function      85—86
Moebius function, definition of      77
Moebius function, historical note      228
Moebius pair      88
Multiple, least common      22
Multiplicative functions      see "Arithmetic functions"
Multiplicativity      1
Nines, casting out      64
Number theory, additive      vi 139
Number theory, algebraic      230
Number theory, combinatorial      30—44
Number theory, computational      44—48
Number theory, elementary      230
Number theory, geometric      199
Number theory, multiplicative      1
Number-theoretic functions      see "Arithmetic functions"
o-notation      205
p-Gons, regular stellated      39
p-Gons, stellated      38
Partial product      160
Partition function      149
Partition function, generating function for      162
Partition generating functions      160—174 see
Partition identities      150 175—198 229
Partition identities, searching for      155—159 see "Schur's
Partitions      vi 230
Partitions, conjugate      151
Partitions, definition of      149
Partitions, elementary theory of      149—159
Partitions, empty      149 181 192
Partitions, generating functions      160—174
Partitions, self-conjugate      153
Pentagonal number theorem      175—178
Pentagonal numbers      177
Perfect numbers      45 85
Perfect numbers and Mersenne primes      112
Perfect numbers, even      45
Perfect numbers, odd      45
Permutation      32
Phi-function      see "Euler's $\phi$ -function"
Pollard, H.      230 232
POLYGONS      38
Polynomial      72
Polynomial with integral coefficients      72
Polynomial, degree of      72
Polynomial, monic      74
Polynomial, reducible      74
Prime function, $\pi(x)$       100—111 228
Prime number theorem      100
PRIMES      100—112
Primes and computers      44
Primes and fundamental theorem of arithmetic      26—29
Primes and multiplicativity      1
Primes and numerical table      47
Primes, definition of      20
Primes, historical note      228
Primes, Mersenne      44 112
Primitive roots      93—99 228
Primitive roots, definition of      94
Primitive roots, modulo p      97—99
Products      see "Infinite products"
Pseudo-primes (Exercise 19)      65
Quadratic congruences      113
Quadratic nonresidue      128
Quadratic reciprocity law      113 118—127 228
Quadratic Reciprocity Law, applications of      125—127
Quadratic Reciprocity Law, statement of (Theorem 9—4)      120
Quadratic residues      47 115—127 228
Quadratic residues, consecutive pairs      128—133
Quadratic residues, consecutive triples      133—138
Quadratic residues, definition of      115
Quadratic residues, distribution of      128—138
r-Combination      see "Combination"
r-Permutation      see "Permutation"

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