Burton D.M. — Elementary Number Theory :: Электронная библиотека попечительского совета мехмата МГУ

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Burton D.M. — Elementary Number Theory

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Название: Elementary Number Theory

Автор: Burton D.M.

Аннотация:

This text provides a simple account of classical number theory, as well as some of the historical background in which the subject evolved. It is intended for use in a one-semester, undergraduate number theory course taken primarily by mathematics majors and students preparing to be secondary school teachers. Although the text was written with this audience in mind, very few formal prerequisites are required. Much of the text can be read by students with a sound background in high school mathematics.

Язык:

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Издание: sixth edition

Год издания: 2005

Количество страниц: 434

Добавлена в каталог: 28.02.2007

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID

Предметный указатель

Exponent, universal      162
Factor bases      360
Factorials, inductively defined      5
Factorization into primes, canonical form      42
Factorization into primes, computers and      353
Factorization into primes, continued fraction factoring algorithm      357—360
Factorization into primes, divisors from      104—105
Factorization into primes, Euler’s phi-function and      132 134
Factorization into primes, Fermat method      97—101
Factorization into primes, Fermat numbers      240—242
Factorization into primes, Fibonacci numbers      287 298—299
Factorization into primes, Fundamental Theorem of Arithmetic      40 41—42
Factorization into primes, Kraitchik method      100—101 360
Factorization into primes, Mersenne numbers      226
Factorization into primes, Pollard’s methods      354—357
Factorization into primes, quadratic sieve algorithm      360—362
Factorization into primes, remote coin flipping      367—370
Factorization into primes, RSA cryptosystem and      203 204 205—206
Factors (divisors)      20
Faltings, Gerd      254
Fermat numbers, defined      236—237
Fermat numbers, factorization into primes      240—242 353 356 357 360
Fermat numbers, primality tests      238—240
Fermat numbers, regular polygons and      237—238
Fermat numbers, table of      241
Fermat — Kraitchik factorization method      97—101 360
Fermat — Kraitchik factorization method, defined      236
Fermat — Kraitchik factorization method, regular polygons and      237—238
Fermat, Pierre de (1601—1665), amicable pair discovery      234
Fermat, Pierre de (1601—1665), biographical information      85—87
Fermat, Pierre de (1601—1665), citations      97 98 129 228 235
Fermat, Pierre de (1601—1665), Fermat numbers work      236—237
Fermat, Pierre de (1601—1665), marginal notes      245—246
Fermat, Pierre de (1601—1665), Mersenne correspondence      218
Fermat, Pierre de (1601—1665), on Pythagorean triangles      250 257
Fermat, Pierre de (1601—1665), on sum of three squares      273
Fermat, Pierre de (1601—1665), Pell’s equation and      334—336
Fermat, Pierre de (1601—1665), photo of      86
Fermat, Pierre de (1601—1665), primes as sums of two squares      265—267
Fermat’s Last Theorem, case      252—253
Fermat’s Last Theorem, case      253—254
Fermat’s Last Theorem, case      256—258
Fermat’s Last Theorem, defined      245—246
Fermat’s Last Theorem, history of proof of      254—255
Fermat’s Little Theorem, citations      171 172
Fermat’s Little Theorem, defined      87—89
Fermat’s Little Theorem, Euler’s generalization of      137—138 139—140
Fermat’s Little Theorem, Euler’s proof of      87 136
Fermat’s Little Theorem, falseness of converse of      89—90
Fermat’s Little Theorem, Lucas’s converse of      363—364
Fermat’s method of infinite descent      252 254 258 273 335
Fermat’s test for nonprimality      362—363
Fibonacci (c.1170—after 1240), biographical information      283—285
Fibonacci (c.1170—after 1240), citations      286
Fibonacci (c.1170—after 1240), continued fractions      306
Fibonacci (c.1170—after 1240), image of      284
Fibonacci numbers, as relatively prime numbers      286—288
Fibonacci numbers, basic identities      292—293
Fibonacci numbers, Binet formula      296—298
Fibonacci numbers, continued fractions representation      310—311 322
Fibonacci numbers, defined      284—286
Fibonacci numbers, greatest common divisors of      288—291
Fibonacci numbers, prime factors of      287 298—299
Fibonacci numbers, square/rectangle geometric deception      293—294
Fibonacci numbers, table of      294
Fibonacci numbers, Zeckendorf representation      295—296
Fibonacci sequence, defined      284 285
Fields medal      352 378
Finite continued fractions, convergents of      311—315 317—318
Finite continued fractions, defined      306—307
Finite continued fractions, linear Diophantine equation solutions      315—317
Finite continued fractions, rational numbers as      307—310
First day of the month, determining      125
First Principle of Finite Induction      2
Frederick the Great (1712—1786)      130 262
Frenicle de Bessy, Bernhard (1605—1675)      87 334—336
Friday the Thirteenth      126
Functions      see also “Euler’s phi-function” “Multiplicative “Number-theoretic
Functions, $\pi(x)$ (prime counting function)      53 371—378
Functions, greatest integer      117 119—121
Functions, Liouville $\lambda$ -function      116
Functions, Mangoldt $\Lambda$ -function      116
Functions, Mobius $\mu$ -function      112—113 407—408
Functions, polynomial      55—57 71—72
Functions, prime-producing      55—57
Functions, zeta      373 376
Fundamental solution (Pell’s equation)      343—344
Fundamental theorem of algebra      63
Fundamental Theorem of Arithmetic      39—42
Galileo, Galilei (1564—1642)      218
Gauss, Carl Friedrich (1777—1855), $\pi(x)$ approximation      374
Gauss, Carl Friedrich (1777—1855), 17-sided polygon discovery      62 238
Gauss, Carl Friedrich (1777—1855), biographical information      61—63
Gauss, Carl Friedrich (1777—1855), citations      163 169 242 273 354
Gauss, Carl Friedrich (1777—1855), congruence concept      63
Gauss, Carl Friedrich (1777—1855), motto      352
Gauss, Carl Friedrich (1777—1855), notation methods      94 132
Gauss, Carl Friedrich (1777—1855), on primitive roots      157 162
Gauss, Carl Friedrich (1777—1855), photo of      62
Gauss, Carl Friedrich (1777—1855), Quadratic Reciprocity Law work      186
Gauss’ lemma, citations      183
Gauss’ lemma, defined      179—181
Gauss’ lemma, Quadratic Reciprocity Law and      187
Gauss’ Theorem      141—143
Generalized Quadratic Reciprocity Law      192
Germain primes      182
Germain, Sophie (1776—1831)      182
Gershom, Levi ben (1288—1344)      258
Girard, Albert (1595—1632)      63 265 286
Goldbach conjecture      51—52
Goldbach, Christian (1690—1764), Euler and      130
Goldbach, Christian (1690—1764), Euler correspondence      51
Goldbach, Christian (1690—1764), on odd integers      57
Greatest common divisor      see also “Euclidean algorithm”
Greatest common divisor, defined      21 24
Greatest common divisor, divisibility relations      19—21
Greatest common divisor, Fibonacci numbers      288—291
Greatest common divisor, least common multiple and      30
Greatest common divisor, linear combination representation      21—22
Greatest common divisor, more than two integers      30—31
Greatest common divisor, relatively prime numbers      22—23
Greatest integer function      117 119—121
Gregorian calendar      122—123
Gregory XIII, Pope (1572—1585)      122
Hadamard, Jacques-Salomon (1865—1963)      376
Hagis, Peter      231
Halley, Edmund (1656—1742)      92 261
Halley’s comet      92
Hardy — Littlewood conjecture      372—373 375
Hardy, Godfrey Harold (1877—1947), biographical information      349—351
Hardy, Godfrey Harold (1877—1947), on Goldbach conjecture      52
Hardy, Godfrey Harold (1877—1947), on Littlewood’s $\pi$ (x) approximation      377
Hardy, Godfrey Harold (1877—1947), photo of      350
Hardy, Godfrey Harold (1877—1947), Ramanujan collaboration      272 303—305 320
Hardy, Godfrey Harold (1877—1947), Riemann hypothesis and      376
Harmonic mean H(n)      225
Haselgrove, C. B.      353
Hellman, Martin      209
Highly composite numbers      305
Hilbert, David (1862—1943)      278 350
Hill cipher      201
Hill, Lester (1890—1961)      201
History of the Theory of Numbers (Dickson)      351
Holzmann, Wilhelm      see “Xylander”
Hundred fowls problem      36—37
Hurwitz, Alexander      240 333
Hypothesis, induction      4
Iamblichus of Chalcis (c.250—c.330 a.d.)      234
Ideal numbers      254
Identification numbers, check digits for      72—73

Identity function (f(n) = 1)      107 110
Incongruence modulo n      64 76
Indeterminate problems      see “Puzzle problems”
Indicator      see “Euler’s phi-function”
Indices (index of a relative to r) for solving congruences      164—166
Indices (index of a relative to r), defined      163—164
Indices (index of a relative to r), solvability criterion      166—167
Induction      see “Mathematical induction”
Induction hypothesis      4
Induction step      4
Infinite continued fractions, $\pi$ representation      327—328 329 331—332
Infinite continued fractions, continued fraction algorithm      326—328
Infinite continued fractions, defined      319—322
Infinite continued fractions, e representation      328—329
Infinite continued fractions, irrational numbers as      323—326
Infinite continued fractions, irrational numbers representation      329—331 332
Infinite continued fractions, periodic      322
Infinite descent, Fermat’s method of      252 254 258 273 335
Infinite series, $\pi$ representation      306
Infinite series, e representation      328
Infinite series, partition function      305
Infinitude of primes of the form 8k - 1      182
Infinitude of primes, Dirichlet’s theorem and      54
Infinitude of primes, Euclid’s theorem and      45—48 53
Infinitude of primes, Euler’s phi-function and      134—135
Infinitude of primes, Euler’s zeta function and      373
Infinitude of primes, Fermat numbers and      238
Infinitude of primes, Fibonacci numbers and      291
Infinitude of primes, Legendre symbol and      177—178
Infinitude of primes, pseudoprimes      92
Integer factorization      see “Factorization into primes”
Integers      see “Numbers”
Integral solutions (Pell’s equation)      345—346
International Standard Book Numbers (ISBNs)      75
Introductio Arithmeticae (Nichomachus)      79 219
Inverse of a modulo n      11
Irrational numbers, $\pi$ as      329
Irrational numbers, $\sqrt{2}$       42
Irrational numbers, e as      328—329
Irrational numbers, infinite continued fractions as      323—326
Irrational numbers, zeta function values      373
ISBNs (International Standard Book Numbers)      75
Jacobi, Carl Gustav Jacob (1804—1851)      278
Jensen, K. L.      254
Julian calendar      122 123
k-perfect numbers      224
k-perfect numbers, cryptosystems using      209—212
k-perfect numbers, defined      208—209
Kanold, Hans-Joachim      231
Kayal, Neeraj      354
Keys for cryptosystems, automatic      200
Keys for cryptosystems, Elgamal system      213—214
Keys for cryptosystems, Merkle — Hellman knapsack system      210—211 212
Keys for cryptosystems, public-key systems      203—204 210—211 212 213—214
Keys for cryptosystems, RSA cryptosystem      203—204
Keys for cryptosystems, running      200
Keys for cryptosystems, Verman system      202
Keys for cryptosystems, Vigenere system      199 200
Kraitchik factorization method      100—101 360
Kraitchik, Maurice (1882—1957)      100
Kronecker, Leopold (1823—1891)      1 61
Kulp, G. W.      198—199
Kummer, Ernst Eduard (1810—1893)      254
Lagrange, Joseph-Louis (1736—1813), biographical information      261—263
Lagrange, Joseph-Louis (1736—1813), on odd integers      57
Lagrange, Joseph-Louis (1736—1813), Pell’s equation and      336
Lagrange, Joseph-Louis (1736—1813), photo of      262
Lagrange, Joseph-Louis (1736—1813), Wilson’s theorem and      94
Lagrange’s four-square theorem      263 273 277
Lagrange’s polynomial congruence theorem      152—154 162 170
Lambert, J. H. (1728—1777)      329
Lame, Gabriel (1795—1870), Euclidean algorithm work      28
Lame, Gabriel (1795—1870), Fermat’s Last Theorem work      254
Lame, Gabriel (1795—1870), Fibonacci numbers work      288
Landau, Edmund (1877—1938)      52
Lander, L. J.      279
Landry, Fortune      239 242
Laplace, Pierre-Simon de (1749—1827)      63
Lattice points, in Quadratic Reciprocity Law proof      186 187
leap years      123 124
Least absolute remainder      28
Least common multiple      29—30
Least nonnegative residues modulo n      64
Least positive primitive root      156 393
Lebesgue, V. A.      258
Legendre formula      118
Legendre symbol (alp), defined      175—176
Legendre symbol (alp), for odd integer a      183—184
Legendre symbol (alp), infinitude of primes and      177—178
Legendre symbol (alp), primitive roots and      181—182
Legendre symbol (alp), properties of      176—179
Legendre symbol (alp), quadratic congruences with composite moduli      189—190 192—195
Legendre, Adrien-Marie (1752—1833), $\pi(x)$ , approximation      373—374
Legendre, Adrien-Marie (1752—1833), amicable pair discovery      234
Legendre, Adrien-Marie (1752—1833), biographical information      175
Legendre, Adrien-Marie (1752—1833), citations      273 376
Legendre, Adrien-Marie (1752—1833), continued fraction factoring algorithm      357
Legendre, Adrien-Marie (1752—1833), Fermat’s Last Theorem work      254
Legendre, Adrien-Marie (1752—1833), primitive roots for primes      162
Legendre, Adrien-Marie (1752—1833), Quadratic Reciprocity Law work      185—186
Lehman, R. S.      353
Lehmer, Derrick (1905—1991)      231
Leibniz, Gottfried Wilhelm (1646—1716)      87 94
Length of period (continued fraction expansions)      322 339
Lenstra, W. Hendrik, Jr.      241
Leonardo of Pisa      see “Fibonacci”
Les Mecaniques de Galilee (Mersenne)      218
Levi ben Gershom (1288—1344)      258
Li (x) (logarithmic integral function)      374 376
Liber Abaci (Fibonacci)      283 284 285 306
Liber Quadratorum (Fibonacci)      283
Linear combination, defined      21
Linear congruences in two variables      81—82
Linear congruences, defined      76
Linear congruences, simultaneous      78—79
Linear congruences, single      76—78
Linear Diophantine equations, defined      32—33
Linear Diophantine equations, finite continued fractions for solving      315—318
Linear Diophantine equations, solvability criteria      33—35
Linear Diophantine equations, traditional word problems      35—37
Linnik, Y. V. (1915—1972)      278
Liouville $\lambda$ -function      116
Liouville, Joseph (1809—1882)      278
Littlewood, John E. (1885—1977), $\pi(x)$ approximation      377
Littlewood, John E. (1885—1977), Goldbach conjecture and      52 53
Littlewood, John E. (1885—1977), Hardy collaboration      350—351 372 375
Logarithmic integral function li (x)      374 376
Lucas numbers      301
Lucas sequence      6
Lucas — Lehmer test      230—231
Lucas, Edouard (1842—1891), Fermat numbers work      240
Lucas, Edouard (1842—1891), Fibonacci numbers and      284 288 302
Lucas, Edouard (1842—1891), primality tests      226 363
Lucas, Edouard (1842—1891), search for larger Mersenne primes      229
Lucas’s Converse of Fermat’s Theorem      363—364
L’Algebra Opera (Bombelli)      307
Mahaviracarya (9th century a.d.)      38
Manasse, M. S.      241
Mangoldt $\Lambda$ -function      116
Mathematical Classic (Chang)      36
Mathematical induction      1—6
Mathematical induction, binomial theorem proof      9—10
Mathematical induction, Fermat’s method of infinite descent      252 254 258 273 335
Mathematical induction, First Principle of Finite Induction      2
Mathematical induction, induction hypotheses      4
Mathematical induction, induction step      4
Mathematical induction, Second Principle of Finite Induction      5—6
McDaniel, Wayne      231
Measurement of a Circle (Archimedes)      331
Mecanique Analytique (Lagrange)      262

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