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

Главная Ex Libris Книги Журналы Статьи Серии Каталог Wanted Загрузка ХудЛит Справка Поиск по индексам Поиск Форум

Авторизация

Поиск по указателям

Burton D.M. — Elementary Number Theory

Обсудите книгу на научном форуме

Нашли опечатку?
Выделите ее мышкой и нажмите Ctrl+Enter

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

$\mu$ -function (Mobius)      112—113 407—408
$\phi(n)$       see “Euler’s phi-function”
$\pi$ as irrational number      329
$\pi$ , continued fractions representation      327—328 331—332
$\pi$ , decimal expansion of      353
$\pi$ , infinite series representation      306
$\pi(x)$ (prime counting function) for p = an + b      53
$\pi(x)$ (prime counting function), approximations of      373—376
$\pi(x)$ (prime counting function), defined      371
$\pi(x)$ (prime counting function), Prime Number Theorem proof and      375—378
$\pi(x)$ (prime counting function), properties of      371—373
$\prod$ notation      106—107
$\rho$ factorization method      354—356
$\sigma(n)$       see “Sum of divisors”
$\sum$ notation, defined      104
$\sum$ notation, multiplicative property and      109 115
$\tau(n)$       see “Number of divisors”
A Course in Pure Mathematics (Hardy)      350—351
Absolute pseudoprime numbers      91 363
Abundant numbers      235
Adleman, Leonard      203
Agrawal, Manindra (1966—)      354
Alcuin (c.732—804)      38 219
Alembert, Jean Le Rond d’ (1717—1783)      63 262
Alexandrian Museum      14—15
Algebraic numbers      254
Amicable numbers      233
Amicable pairs      233—235
Amicable triples      236
An Elementary Proof of the Prime Number Theorem (Selberg)      378
Anthoniszoon, Adriaen (1527—1617)      332
Apery, Roger (1916—1994)      373
Arabic numerals      284
Archimedean property      2
Archimedean value of $\pi$       331
Archimedes (c.287—212 B.C.)      331 346
Area of Pythagorean triangles      250 257
Arithmetic functions      see “Number-theoretic functions”
Arithmetic progressions of numbers, primes      54—55 375
Arithmetic progressions of numbers, pseudoprimes      90
Arithmetica (Diophantus), Bombelli and      307
Arithmetica (Diophantus), Fermat and      245—246 257 346
Arithmetica (Diophantus), history of      32
Arithmetica (Diophantus), recovery of script      85—86
Artin, Emil (1898—1962)      157
Artin’s conjecture      157
Aryabhata I (476—C.550 a.d.)      15
Augustine, Saint (354—430)      219
Authentication of messages      215—216
Autokey cryptosystems      200—201
Bachet, Claude (1581—1638)      86 273
Barlow, Peter (1776—1862)      228—229
Bases for number systems      70
Basis for induction      4
Baudot code      202
Baudot, Jean-Maurice-Emile (1845—1903)      202
Bennett, G.      237
Bernoulli inequality      7
Bernoulli, Daniel (1700—1782)      129 130
Bernoulli, Johann (1667—1748)      129
Bernoulli, Nicolaus (1695—1726)      129 130
Bertrand, Joseph (1822—1900)      48
Bertrand’s conjecture      48 352 371
Bhaskara II(1114—c.1185)      83
Binary exponential algorithm      70—71
Binary number representation      70—71
Binet formula      296—298
Binet, Jacques-Philippe-Marie (1786—1856)      296
Binomial coefficients as integers      119
Binomial coefficients, defined      8
Binomial coefficients, Fibonacci numbers formula      302
Binomial coefficients, identities      8—10
Binomial congruences      164
Binomial theorem      8—10
Blum integers      369
Blum, Manuel (1938—)      367
Blum’s coin flipping game      367—370
Bombelli, Rafael (1526—1572)      307
Bonse’s inequality      47
Bracket function      117 119—121
Brahmagupta (598—C.665)      83 346
Brent, Richard      240 241 356
Brillhart, John (1930—)      240 357 360
Brouncker, William (1620—1684)      332 335—336
Brun, Viggo (1882—1978)      375
Brun’s constant      375
Caesar, Julius (100—44 B.C.), cipher system      197—198
Caesar, Julius (100—44 B.C.), Julian calendar      122 123
Calendars      122—124
Canonical form      42
Carlyle, Thomas (1795—1881)      175
Carmichael numbers      91 363
Carmichael, Robert D. (1879—1967)      91
Catalan equation      257—258
Catalan numbers      12
Catalan, Eugene (1814—1894)      12 258
Catalan’s conjecture      258
Cataldi, Pietro (1548—1626)      222
Cattle problem      346
Chain of inequalities (continued fractions)      317—318
Chang Ch’iu-chien (6th century a.d.)      36 37
Check digits      72—73
Chinese Remainder Theorem in Blum’s coin flipping game      368 370
Chinese Remainder Theorem, defined      79—81
Chinese Remainder Theorem, Euler’s generalization of Fermat’s theorem and      139—140
Cicero, Marcus Tullius (106—43 B.C.)      197 198
Ciphers, autokey systems      200—201
Ciphers, Caesar      198
Ciphers, defined      197
Ciphers, Elgamal      213—216
Ciphers, Hill      201
Ciphers, Merkle — Hellman      209—212
Ciphers, Vigenere      199—200
Clavius, Christopher (1537—1612)      123
Cogitata Physica-Mathematica (Mersenne)      225
Coin flipping, remote      367—370
Cole Prize      351
Cole, Frank Nelson (1861—1926)      226
Common divisors      20—21
Common multiples      29
Complete set of residues modulo n      64
Composite numbers      39 305 “Primality
Computational number theory      353
Computers in number theory, cryptography and      197 205—206
Computers in number theory, Mersenne primes and      229—230
Computers in number theory, prime number factorization and      353
Congruences      63—82 (see also “Quadratic congruences”)
Congruences in Caesar cipher      198
Congruences, basic properties      63—67
Congruences, binomial      164
Congruences, check digits      72—73
Congruences, Chinese Remainder Theorem      79—81
Congruences, days of the week and      123
Congruences, defined      63
Congruences, indices for solving      164—167
Congruences, linear in two variables      81—82
Congruences, partition function and      305
Congruences, place-value notation systems      69—71
Congruences, polynomial functions      71—72
Congruences, simultaneous linear      78—79
Congruences, single linear      76—78
Congruences, to perfect squares      100
Congruent modulo n, defined      63
Constant function f(n) = n      107 110
Continued fraction algorithm (irrational numbers)      326—328
Continued fraction factoring algorithm      357—360
Continued fractions, defined      306 (see also “Finite continued fractions; infinite continued fractions” “Pell’s
Convergents of continued fractions, $\pi$       327—328
Convergents of continued fractions, finite continued fractions      311—315 317—318
Convergents of continued fractions, infinite continued fractions      321—322 325—326

Convergents of continued fractions, Pell’s equation and      336 337—341
Critical line of the zeta function      376
Cryptography      197—216
Cryptography, defined      197
Cryptography, Elgamal system      213—216
Cryptography, knapsack problems      208—209
Cryptography, Merkle — Hellman knapsack system      209—212
Cryptography, monoalphabetic systems      197—198
Cryptography, poly alphabetic systems      198—201
Cryptography, RSA system      203—206
Cryptography, Verman one-time pad system      202—203
Cunningham, Allen Joseph (1848—1928)      229
Day of the week, determining      123—126
de Polignac, Alphonse (1817—1890)      58
Decimal number representation      71
Deciphering/decrypting, defined      197 (see also “Cryptography”)
Decomposition into primes      see “Factorization into primes”
Deficient numbers      235
Denominator of Legendre symbol      175
Descartes, Rene (1596—1650), amicable pair discovery      234
Descartes, Rene (1596—1650), citations      235 273
Descartes, Rene (1596—1650), Mersenne and      217 218
Dickson, Leonard Eugene (1874—1954)      278 351
Difference of two squares      269—270
Digital alphabet for RSA cryptosystem      203—204
Digital signatures      215—216
Digits of a number, defined      71
Diophantine equations,       245—250
Diophantine equations,       252—253
Diophantine equations,       253—254
Diophantine equations,       256—258
Diophantine equations,       245—246 254—255
Diophantine equations, ax + by + cz = d      36—37
Diophantine equations, ax + by = c      32—35
Diophantine equations, defined      32
Diophantine equations, Fibonacci work on      283
Diophantine equations, linear congruences and      76 78
Diophantine equations, linear in two unknowns      32—35
Diophantine equations, word problems      35—37
Diophantus of Alexandria (3rd century a.d.) on sum of three squares      273
Diophantus of Alexandria (3rd century a.d.), biographical information      32
Dirichlet, Peter Gustav Lejeune (1805—1859), citations      54 172 186 375
Dirichlet, Peter Gustav Lejeune (1805—1859), Fermat’s Last Theorem work      254
Dirichlet, Peter Gustav Lejeune (1805—1859), pigeonhole principle      264
Dirichlet’s Theorem      54
Discrete logarithm problems, cryptography and      213
Disquisitiones Arithmeticae (Gauss), citations      157 163 175 354
Disquisitiones Arithmeticae (Gauss), history of      61 63
Disquisitiones Arithmeticae (Gauss), Quadratic Reciprocity Law      186
Disquisitiones Arithmeticae (Gauss), regular polygons      237
Divergent series      374—375
Divisibility theory      13—38
Divisibility theory by 9 or 11      71—72
Divisibility theory, Diophantine equations      32—37
Divisibility theory, Division Algorithm      17—19
Divisibility theory, early number theory      13—16
Divisibility theory, Euclidean algorithm      26—31
Divisibility theory, greatest common divisor      19—24
Divisibility theory, Mersenne and      218
Divisibility theory, symbols for      20
Division algorithm      17—19
divisors      see also “Number of divisors” “Sum
Divisors from prime factorizations      104—105
Divisors, common      20—21
Divisors, defined      20
Divisors, greatest common      21 24
Divisors, Mersenne numbers      228—229
Double Wieferich primes      258
e (continued fractions representation)      328—329
e-prime numbers      42
Early number theory      13—16
Eisenstein, Ferdinand Gottfried Max (1823—1852)      186
El Madschriti of Madrid (11th century)      234
Elements (Euclid), Diophantine equations and      32
Elements (Euclid), Euclidean algorithm      26
Elements (Euclid), Euclid’s theorem      45
Elements (Euclid), Fundamental Theorem of Arithmetic      39—40
Elements (Euclid), history of      15
Elements (Euclid), Legendre revision of      175
Elements (Euclid), perfect numbers work      220
Elements (Euclid), translations of      85
Elements de Geometrie (Legendre)      175
Elgamal cryptosystem      213—216
Elgamal, Taher (1955—)      213
Elkies, Noam      279
Elliptic curves      255
Enciphering exponent      203 204
Enciphering modulus      203
Enciphering/encrypting, defined      197 (see also “Cryptography”)
Encke, Johann Franz (1791—1865)      374
Equality, congruence and      65
Eratosthenes of Cyrene (c.276—c.194 B.C.)      45 346
Erdos, Paul (1913—1996)      351—353
Essai sur la Theorie des Nombres (Legendre)      175 186 373
Essaipour les Coniques (Pascal)      217
Euclid (c.300 B.C.), citations      246
Euclid (c.300 B.C.), early number theory      15
Euclid (c.300 B.C.), perfect numbers work      220
Euclidean algorithm, defined      26—28
Euclidean algorithm, least common multiple and      29—30
Euclidean algorithm, more than two integers      30—31
Euclidean algorithm, number of steps      28—29
Euclidean numbers      46
Euclid’s lemma      24
Euclid’s theorem, defined      45—48
Euclid’s theorem, Euler’s phi-function and      134—135
Euler polynomial      55—56
Euler, Leonhard (1707—1783), $\pi$ , symbol      327
Euler, Leonhard (1707—1783), amicable pair work      234 235
Euler, Leonhard (1707—1783), biographical information      129—131 262
Euler, Leonhard (1707—1783), citations      55 57 63 87 185 221 265 279
Euler, Leonhard (1707—1783), e continued fractions representation      328
Euler, Leonhard (1707—1783), Fermat numbers work      237 240
Euler, Leonhard (1707—1783), Fermat’s Last Theorem work      254
Euler, Leonhard (1707—1783), Goldbach conjecture and      51
Euler, Leonhard (1707—1783), Mersenne numbers work      225—226
Euler, Leonhard (1707—1783), on Catalan equation      258
Euler, Leonhard (1707—1783), on odd perfect numbers      231 232
Euler, Leonhard (1707—1783), on triangular numbers      15
Euler, Leonhard (1707—1783), Pell’s equation and      336
Euler, Leonhard (1707—1783), photo of      130
Euler, Leonhard (1707—1783), primitive roots for primes      162
Euler, Leonhard (1707—1783), proof of Fermat’s theorem      87 136
Euler, Leonhard (1707—1783), sum of four squares      273
Euler, Leonhard (1707—1783), Waring’s problem      351
Euler, Leonhard (1707—1783), word problems      38
Euler, Leonhard (1707—1783), zeta function formula      373
Euler’s criterion, citations      180
Euler’s criterion, defined      171—172
Euler’s criterion, Dirichlet’s proof of      172—173
Euler’s generalization of Fermat’s theorem, applications of      139—140
Euler’s generalization of Fermat’s theorem, defined      137—138
Euler’s generalization of Fermat’s theorem, Fermat’s Little Theorem as proof of      139
Euler’s identity      273 277
Euler’s phi-function $\phi(n)$ as even integer      134
Euler’s phi-function $\phi(n)$ as multiplicative function      132 133 142
Euler’s phi-function $\phi(n)$ , defined      131—132
Euler’s phi-function $\phi(n)$ , Euclid’s theorem and      134—135
Euler’s phi-function $\phi(n)$ , Gauss’ theorem and      141—143
Euler’s phi-function $\phi(n)$ , Mobius inversion formula and      144—145
Euler’s phi-function $\phi(n)$ , sum of integers identity      143
Euler’s phi-function $\phi(n)$ , table of      407—408
Even-numbered convergents      317—318 321
Even-numbered convergents in Pythagorean triples      247
Even-numbered convergents, defined      18
Even-numbered convergents, Euler’s phi-function as      134
Exponent of a prime in n! factorization      117—118
Exponent to which a belongs modulo n      147—150
Exponent, enciphering      203 204
Exponent, recovery      204

1 2 3 4

О проекте