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

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Предметный указатель

Meditationes Algebraicae (Waring)      277
Merkle — Hellman knapsack cryptosystem      209—212
Merkle, Ralph      209
Mersenne numbers, defined      225
Mersenne numbers, divisor properties      228—229
Mersenne numbers, primality tests      226 227—228 230—231
Mersenne numbers, search for larger numbers      226 229—230 231
Mersenne primes, defined      225
Mersenne primes, table of      230
Mersenne, Marin (1588—1648), biographical information      217—218
Mersenne, Marin (1588—1648), citations      97 102 237
Mersenne, Marin (1588—1648), correspondence on amicable pairs      234
Mersenne, Marin (1588—1648), Fermat correspondence      265
Mersenne, Marin (1588—1648), Mersenne numbers work      225 226
Mersenne, Marin (1588—1648), photo of      219
Mertens conjecture      115—116
Mertens, Franz (1840—1927)      115
Mihailescu, Preda      258
Miller — Rabin primality test      365—366
Mills, WH. (1921—)      57
Mobius $\mu$ -function      112—113 407—408
Mobius inversion formula, defined      113—115
Mobius inversion formula, Euler’s phi-function and      144—145 155
Modulo n (congruence), check digits and      73
Modulo n (congruence), defined      63—64
Monoalphabetic cryptosystems      197—198
Monte Carlo factorization method      354—356
Morain, Francois      241
Morehead, J. C      239 240
Morrison, Michael A.      240 357 360
Muller, Johannes      see “Regiomontanus (1436—1476)”
Multiples, defined      20
Multiplicative functions, $\tau$ and $\sigma$ as      108 109—110
Multiplicative functions, defined      107—108
Multiplicative functions, Euler’s phi-function as      132 133 142
Multiplicative functions, Mobius $\mu$ -function as      112 144 145
Multiplicative inverse of a modulo n      11
Multiplicatively perfect numbers      224
Multiply perfect numbers      224
Natural numbers, defined      1
Newton’s identity      10
Nickel, Laura      229
Nicomachus of Gerasa (c.100 a.d.)      15 79 219
Noll, Curt      229
Nonalphabetic cryptosystems      202—203
Nonnegative residues modulo n      64
Nonresidues, quadratic      171 172—173 178—179
Notation systems, $\prod$ notation      106—107
Notation systems, $\sum$ notation      104 109 115
Notation systems, binary numbers      69—71
Notation systems, decimal numbers      71
Notation systems, finite continued fractions      310
Notation systems, infinite continued fractions      321
Number of divisors $\tau(n)$ as multiplicative function      108 109—110
Number of divisors $\tau(n)$ , basic properties      103—107
Number of divisors $\tau(n)$ , greatest integer function and      120
Number of divisors $\tau(n)$ , table of      407—408
Number-theoretic functions      103—126 (see also “Euler’s phi-function”)
Number-theoretic functions, calendar applications      122—126
Number-theoretic functions, defined      103
Number-theoretic functions, greatest integer function and      119—121
Number-theoretic functions, Mobius inversion formula      112—116
Number-theoretic functions, multiplicative functions      107—110
Number-theoretic functions, number of divisors      103—107
Number-theoretic functions, sum of divisors      103—107
Numbers      see also “Composite numbers” “Fermat “Fibonacci “Mersenne “Perfect “Prime “Relatively
Numbers, absolute pseudoprime      91 363
Numbers, abundant      235
Numbers, algebraic      254
Numbers, amicable      233
Numbers, Catalan      12
Numbers, deficient      235
Numbers, e-prime      42
Numbers, Euclidean      46
Numbers, even      18 134 247
Numbers, Germain      182
Numbers, highly composite      305
Numbers, ideal      254
Numbers, k-perfect      224
Numbers, Lucas      301
Numbers, multiplicatively perfect      224
Numbers, multiply perfect      224
Numbers, natural, defined      1
Numbers, odd      18 160—162 231—233 247 394—403
Numbers, palindromes      75
Numbers, Pell      348
Numbers, pentagonal      16
Numbers, pseudoprime      90—92 242
Numbers, regular prime      254
Numbers, repunit      48—49
Numbers, Skewes      377
Numbers, square-free      43 91
Numbers, square-full      43
Numbers, strong pseudoprime      367
Numbers, superperfect      225
Numbers, triangular      15—16 257 295
Numerator of Legendre symbol      175
Odd-numbered convergents      317—318 321
Odd-numbered convergents in Pythagorean triples      247
Odd-numbered convergents, defined      18
Odd-numbered convergents, perfect      231—233
Odd-numbered convergents, prime factors, table of      394—403
Odd-numbered convergents, primitive roots for      160—162
Odlyzko, Andrew M.      116
One-time pad cryptosystems, Verman      202—203
One-time pad cryptosystems, Vigenere      200
Opera Mathematica (Wallis)      336
Order of a modulo n      147—150
Pairs of quadratic residues      171
Palindromes      75
Parkin, Thomas      279
Partial denominators      307 310
Partial quotients      310
Partition theory      304—305
Pascal, Blaise (1623—1662), mathematical induction work      10
Pascal, Blaise (1623—1662), scholarly gatherings      217 218
Pascal’s rule      8
Pascal’s triangle      9
Pell numbers      348
Pell, John (1611—1685)      336
Pell’s equation, continued fraction expansions      337—341
Pell’s equation, fundamental solution      343—344
Pell’s equation, history of      334—336 346
Pell’s equation, integral solutions      345—346
Pell’s equation, positive solutions      337 342—343 344—345
Pentagonal numbers      16
Pepin, Theophile (1826—1904)      238
Pepin’s test      238—240
Perfect numbers, defined      219—220
Perfect numbers, discovery of larger numbers      226 229—230 231
Perfect numbers, final digits of      223
Perfect numbers, general form      220—222
Perfect numbers, odd      231—233
Period (continued fraction expansions)      322 338 339
Periodic continued fractions      322 338—340
Personal identification numbers, check digits for      72—73
Peter the Great (1672—1725)      130
Phi-function $\phi(n)$       see “Euler’s phi-function”
Piazzi, Giuseppi (1746—1826)      63
Pigeonhole Principle      264
Place-value notation systems      69—71
Plaintext      197
Plutarch (c.46—after 119 a.d.)      15
Pocklington, Henry (1870—1952)      364
Pocklington’s theorem      364—365
Poe, Edgar Allan (1809—1849)      198—199
Polignac, Alphonse de (1817—1890)      58
Pollard, John M., citations      240 241
Pollard, John M., p - 1 factorization method      356—357
Pollard, John M., rho factorization method      354—356

Polya conjecture      353
Polya, George (1888—1985)      353
Polyalphabetic cryptosystems      198—201
POLYGONS      62 237—238
Polynomial congruences, divisibility tests      71—72
Polynomial congruences, Lagrange’s theorem and      152—154
Positive solutions (Pell’s equation)      337 342—343 344—345
Powerful numbers      43
Powers, R. E.      229
Primality tests, computers and      229—230
Primality tests, efficient algorithms for      354
Primality tests, Fermat’s Little Theorem methods      89 362—365
Primality tests, Mersenne numbers      226 227—228 230—231
Primality tests, Miller — Rabin test      365—366
Primality tests, Pepin’s test      238—240
Primality tests, Wilson’s theorem      95
prime factors      see “Factorization into primes”
Prime Number Theorem, arithmetic proofs of      352 378
Prime Number Theorem, complex proofs of      375—377
Prime Number Theorem, defined      371
Prime numbers      39—57 (see also “Infinitude of primes” “Pseudoprime
Prime numbers as sum of four squares      275—277
Prime numbers of the form       240 242
Prime numbers of the form       46
Prime numbers of the form 4n + 1      53 54 177—178 188 265—267
Prime numbers of the form 4n + 3      53—54 188 264 267—268
Prime numbers of the form 8k + 1/3/5/7      181
Prime numbers of the form 8k - 1      181
Prime numbers of the form an + b      53
Prime numbers of the form n! + 1      96
Prime numbers, arithmetic progressions of      54—55 90 375
Prime numbers, defined      39
Prime numbers, double Wieferich      258
Prime numbers, Euclid’s theorem      45—48
Prime numbers, Fundamental Theorem of Arithmetic      39—42
Prime numbers, gaps between      50—51
Prime numbers, Germain      182
Prime numbers, Goldbach conjecture      51—52
Prime numbers, length of intervals      377—378
Prime numbers, of the form $3n \pm 1$       53
Prime numbers, prime-producing functions      55—57
Prime numbers, primitive roots      154—157
Prime numbers, repunit      48—49
Prime numbers, sieve of Eratosthenes      44—45
Prime numbers, tables of      404—406
Prime numbers, twin      50 375 406
Prime-producing functions      55—57
Prime-triplets      58
Primers (autokey cryptosystems)      200
Primitive Pythagorean triples defined      246
Primitive Pythagorean triples defined, properties of      247—249
Primitive Pythagorean triples defined, table of      249
Primitive roots, composite numbers      158—162
Primitive roots, cryptography application      213
Primitive roots, defined      150—151
Primitive roots, Legendre symbol and      181—182
Primitive roots, number of      151
Primitive roots, prime numbers      154—157
Primitive roots, tables of      156 393
Probabilistic primality tests      366
Progressions of numbers      see “Arithmetic progressions of numbers”
Proth, E.      367
Pseudoprime numbers      90—92 242 363
Public-key crypto systems, defined      203
Public-key crypto systems, Elgamal system      213—216
Public-key crypto systems, Merkle — Hellman knapsack system      209—212
Public-key crypto systems, RSA system      203—206
Puzzle problems, cattle      346
Puzzle problems, congruences      79—80
Puzzle problems, Diophantine equations      35—37
Puzzle problems, hundred fowls      36—37
Puzzle problems, square/rectangle geometric deception      293—294
Pythagoras (c.580—c.500 B.C.) on irrational numbers      42
Pythagoras (c.580—c.500 B.C.) on triangular numbers      15
Pythagoras (c.580—c.500 B.C.), citations      235 246
Pythagoras (c.580—c.500 B.C.), early number theory      13—14
Pythagorean triangles      250 257
Pythagorean triples      246 247—249
Pythagoreans on perfect numbers      219 221
Pythagoreans, amicable pairs and      234
Pythagoreans, history of      14
Pythagoreans, number classification      42
Quadratic congruences in Blum’s coin flipping game      367—369
Quadratic congruences with composite moduli      189—190 192—195
Quadratic congruences, indices for solving      164—165
Quadratic congruences, primitive roots      155—156
Quadratic nonresidues, defined      171
Quadratic nonresidues, Euler’s criterion      172—173
Quadratic nonresidues, Legendre symbol and      178—179
Quadratic Reciprocity Law and      189—190
Quadratic Reciprocity Law and simplification of      169—170
Quadratic Reciprocity Law and solvability criteria      192 194 195
Quadratic Reciprocity Law and Wilson’s theorem and      95—96
Quadratic Reciprocity Law, defined      186
Quadratic Reciprocity Law, generalized      192
Quadratic Reciprocity Law, history of      169 185—186
Quadratic Reciprocity Law, properties of      188—190
Quadratic residues, defined      171
Quadratic residues, Euler’s criterion      171—173
Quadratic residues, Legendre symbol and      178—179
Quadratic residues, sum of four squares problem and      275
Quadratic sieve factoring algorithm      360—362
Quadrivium      13
Quotients      17
Radius of inscribed circle of Pythagorean triangles      250
Ramanujan, Srinivasa Aaiyangar (1887—1920), biographical information      303—306
Ramanujan, Srinivasa Aaiyangar (1887—1920), fraction expansions      320
Ramanujan, Srinivasa Aaiyangar (1887—1920), photo of      304
Ramanujan, Srinivasa Aaiyangar (1887—1920), sum of two cubes      272
Ramanujan’s conjecture      305
Rational numbers as finite continued fractions      307—310
Rational numbers as irrational numbers approximation      329—331
Recovery exponent      204
Rectangle/square Fibonacci number problem      293—294
Recursive sequences      286
Reduced set of residues modulo n      141
Regiomontanus (1436—1476)      83 85 279
Regius, Hudalrichus (fl. 1535)      222
Regular polygons      62 237—238
Regular prime numbers      254 (see also “Prime numbers”)
Relatively prime numbers in Pythagorean triples      247
Relatively prime numbers, convergent numerators and denominators as      314—315
Relatively prime numbers, defined      22—23
Relatively prime numbers, Fermat numbers as      238
Relatively prime numbers, Fibonacci numbers as      286—288
Relatively prime numbers, multiplicative property and      107
Remainder      17 28
Remote coin flipping      367—370
Representation of integers, difference of two squares      269—270
Representation of integers, sum of four squares      263 273—277
Representation of integers, sum of three squares      272—273
Representation of integers, sum of two squares      264—269
Representation of integers, Waring’s problem      277—279 350—351
Representation of integers, Zeckendorf      295—296
Repunit numbers      48—49
Residues, complete set      64
Residues, least nonnegative      64
Residues, quadratic      171—173 178—179 275
Residues, reduced set      141
Rho factorization method      354—356
Riemann hypothesis      376
Riemann, Georg Friedrich Bernhard (1826—1866)      375—376
Rivest, Ronald L.      203
RSA public-key cryptosystem      203—206
RSA-129 cryptosystem      206
RSA-576 cryptosystem      206
Rudolff, Christoff (fl. 1526)      38
Running keys      200
Saxena, Nitin      354
Second Principle of Finite Induction      5—6

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