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Jones J.A., Jones J.M. — Elementary Number Theory
Jones J.A., Jones J.M. — Elementary Number Theory

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Название: Elementary Number Theory

Авторы: Jones J.A., Jones J.M.

Аннотация:

This book gives an elementary undergraduate-level introduction to Number Theory, with the emphasis on carefully explained proofs and worked examples; exercises (with solutions) are integrated into the text as part of the learning process. The first few chapters, covering divisibility, prime numbers and modular arithmetic, assume only basic school algebra, and are therefore suitable for first or second year students as an introduction to the methods of pure mathematics. Elementary ideas about groups and rings (summarised in an appendix) are then used to study groups of units, quadratic residues and arithmetic functions with applications to enumeration and cryptography. The final part, suitable for third-year students, uses ideas from algebra, analysis, calculus and geometry to study more advanced topics such as Dirichlet series and sums of squares; in particular, the last chapter gives a concise account of Fermat's Last Theorem, from its origin in the ancient Babylonian and Greek study of Pythagorean triples to its recent proof by Andrew Wiles.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1998

Количество страниц: 301

Добавлена в каталог: 11.04.2008

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
Abelian group      97 243
Abscissa of absolute convergence      186
Abscissa of convergence      186
Absolute residue, least      42
Adleman, L.M.      35
Alford, W.R.      77
Algorithm, division      2
Algorithm, Euclid’s      5
Algorithm, least remainders      17
Analytic function      187
Apery, R.      176
Apostol, T.M.      174
Arithmetic function      143
Arithmetic progression, primes in      29
Arithmetic, fundamental theorem of      22
Arithmetic, modular      37
Arithmetica of Diophantos      217 223 226
Armengaud, J.      31
Associates      198
Babylonian mathematics      220 223
Bachet, C.G.      217 226
Basis of a lattice      207
Bernoulli numbers      177
Bernoulli, J.      177
Bezout, E.      7
Bezout’s identity      7
Brahmagupta      13
Carmichael number      76 117
Carmichael, R.D.      76 77
Cayley, A.      206
Centrally symmetric      210
Chen, Jing-run      29
Chinese remainder theorem      53 59
Clarkson, R.      31
Class number      234
Class, congruence      40
Code      92
Common divisor      5
Common divisor, greatest      5 23
Common factor      5
Common factor, highest      5
Common multiple      12
Common multiple, least      12 23
Comparison Test      247
Complete set of residues      42
Completely multiplicative      182
Complex variable      185
Composite      20 32
Congruence class      40
Congruence, linear      46
Congruence, quadratic      119
Congruences, simultaneous linear      52
Congruences, simultaneous non-linear      57
Congruent      38
Conjecture, Goldbach’s      29
Conjecture, Taniyama — Shimura      236
Conjugate of a quaternion      206
Convergence, abscissa of      186
Convergence, abscissa of absolute      186
Convergence, absolute      247
Convergence, conditional      247
Convergence, half-plane of absolute      186
Convergence, uniform      248
Convex      210
Convolution      157
Coprime      10
Coprime, mutually      10
Criterion, Eisenstein’s      21
Criterion, Euler’s      125
Critical strip      188
Cryptography      92
Cubic lattice, simple      207
Curve, elliptic      236
Curve, Prey      236
Cyclic group      244
Cyclotomic      233
Cyclotomic integer      233
Cyclotomic polynomial      21
de l’Hopital, G.F.A.      28
Dedekind, R.      222
Descent, Fermat’s method of      222
Diophantine equations      13 226
Diophantine equations, linear      13
Diophantos      13 217 223 226
Diophantos, Arithmetica of      217 223 226
Direct product      244
Dirichlet inverse      159
Dirichlet product      157
Dirichlet series      179
Dirichlet, P.G.L.      29 157 159 179
Dirichlet’s Theorem      29
Discrete logarithm problem      94
divides      3
Division algorithm      2
Divisor function      144
Divisor, common      5
Divisor, geatest common      5 23
Domain, Euclidean      196
Domain, integral      196
Eisenstein, G.      21 133
Eisenstein’s Criterion      21
Elements, Euclid’s      25 147 223
Elliptic curve      236
Equation, Diophantine      13 226
Equation, linear diophantine      13
Equivalence relation      39
Equivalent, modulo a lattice      208
Eratosthenes      34
Eratosthenes, sieve of      34
Euclid      5 19 25 147 223
Euclidean Domain      196
Euclid’s algorithm      5
Euclid’s Elements      25 147 223
Euclid’s theorem      25
Euler product      182
Euler, L.      30 67 83 86 125 130 147 164 167 182 193 232
Euler’s criterion      125
Euler’s function      85 92
Euler’s theorem      86 92 95 104 105 107 110 116
EXPONENT      116
Exponent, universal      116
Factor      3
Factor, common      5
Factor, highest common      5
Factorisation, prime-power      22 34
Faltings, G.      235
Fermat number      30 132
Fermat prime      30 132
Fermat, P.      30 67 193 217 218 222 226 227
Fermat, S.      226
Fermat’s Last Theorem      217 226
Fermat’s Little Theorem      67 86
Fermat’s method of descent      222
fibonacci      16
Fibonacci number      16
Field      245
Four squares theorem      203
Four-group, Klein      99
Fourier series      175
Fourier, J.B.J.      175
Frey curve      236
Frey, G.      236
Frobenius, G.      206
Function, analytic      187
Function, arithmetic      143
Function, completely multiplicative      182
Function, divisor      144
Function, Euler’s      85 92
Function, identity      149
Function, Liouville’s      182
Function, Mangoldt      162
Function, Mobius      149 154
Function, multiplicative      144
Function, polynomial      82
Function, Riemann zeta      164 185
Function, unit      145
Fundamental region      208
Fundamental Theorem of Arithmetic      22
Gauss, C.F., vi      27 30 127 130 232
Gaussian integers      196
Gauss’s Lemma      127
Generator      244
Genus      235
Geometric series      247
Germain, S.      231
Goldbach, C      29
Goldbach’s conjecture      29
Granville, A.      77
Graves, J.T.      206
Greatest common divisor      5 23
Group      243
Group homomorphism      244
Group isomorphism      244
Group of units      85 98
Group, abelian      243
Group, cyclic      244
Hadamard, J.      27
Half-plane of absolute convergence      186
Hamilton, W.R.      205 206
Hardy, G.H.      vi 188
Harmonic series      163
Height      16
Hensel, K.      81
Hensel’s Lemma      81
Highest common factor      5
Hippasus of Metapontum      222
Homomorphism, group      244
Homomorphism, ring      114 245
Hurwitz, A.      206
Hypothesis, Riemann      164 188
Ideal      234
Identity function      149
Identity, Bezout’s      7
Inclusion-Exclusion Principle      90 153
Independent, statistically      171
Induction      239
Induction, strong      240
Infinite product      247
Integer lattice      207
Integer lattice point      15
Integer point      170
Integer, cyclotomic      233
Integer, Gaussian      196
Integer, random      170
Integral domain      196
Integral test      247
Inverse, Dirichlet      159
Inverse, multiplicative      84
Inversion formula, Mobius      150 152
Irrational number      24 222
Irreducible element      198
Irreducible polynomial      21
Isomorphism, group      244
Isomorphism, ring      245
Julius Caesar      93
Kernel      122
Key      93 94
Key, public      95
Klein four-group      99
Klein, F.      99
Kummer, E.E.      234
L-series      179
Lagrange, J.L.      65 70 86 203 244
Lagrange’s theorem (for groups)      244
Lagrange’s theorem (for polynomials)      65
Lame, G.      16 232 233
Lame’s theorem      16
Last theorem, Fermat’s      217 226
Lattice      207
Lattice point, integer      15
Lattice, basis of      207
Lattice, equivalent modulo      208
Lattice, integer      207
Lattice, simple cubic      207
Lattice, square      207
Laurent series      248
Laurent, P.-A.      176
Law of quadratic reciprocity      130
Least absolute residue      42
Least common multiple      12 23
Least non-negative residue      42
Least remainders algorithm      17
Legendre symbol      123
Legendre, A-M.      123 130 232
Leibniz, G.W.      67
Lemma, Gauss’s      127
Lemma, Hensel’s      81
Length of a quaternion      206
Lindemann, F.      179
Linear congruence      46
Linear congruences, simultaneous      52
Linear Diophantine equation      13
Liouville, J.      182
Liouville’s function      182
Little theorem, Fermat’s      67 86
Logarithm, discrete problem      94
L’Hopital’s Rule      28
Mangoldt function      162
Mersenne number      31 98
Mersenne prime      31 147
Mersenne, M.      31 147
Method of descent, Fermat’s      222
Method, Newton’s      81
Minkowski, H.      206 211 212 213 214
Minkowski’s theorem      211
Mirimanoff, D.      235
Mobius function      149 154
Mobius inversion formula      150 152
Mobius, A.F.      143 148 149
Modular arithmetic      37
Modular curve      236
Modulus      37
Mordell, L.J.      235
Multiple      3
Multiple, common      12
Multiple, least common      12 23
Multiplicative function      144
Multiplicative inverse      84
Multiplicative, completely      182
Mutually coprime      10
Newton, I.      81 236
Newton’s method      81
Non-linear congruences, simultaneous      57
Non-negative residue, least      42
Non-trivial zero      188
Norm      197
Number, bernoulli      177
Number, Carmichael      76 117
Number, class      234
Number, Fermat      30 132
Number, Fibonacci      16
Number, irrational      24 222
Number, Mersenne      31 98
Number, p-adic      81
Number, perfect      146
Number, rational      24
Octonion      206
Order      98 244
p-adic number      81
Pepin, T.      132
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