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Burton D.M. Elementary Number Theory
Burton D.M.  Elementary Number Theory









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: Elementary Number Theory

: Burton D.M.

:

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.


: en

: /

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ed2k: ed2k stats

: sixth edition

: 2005

: 434

: 28.02.2007

: | | ID
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$\mu$-function (Mobius)      112113 407408
$\phi(n)$      see Eulers phi-function
$\pi$ as irrational number      329
$\pi$, continued fractions representation      327328 331332
$\pi$, 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      373376
$\pi(x)$ (prime counting function), defined      371
$\pi(x)$ (prime counting function), Prime Number Theorem proof and      375378
$\pi(x)$ (prime counting function), properties of      371373
$\prod$ notation      106107
$\rho$ factorization method      354356
$\sigma(n)$      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)      350351
Absolute pseudoprime numbers      91 363
Abundant numbers      235
Adleman, Leonard      203
Agrawal, Manindra (1966)      354
Alcuin (c.732804)      38 219
Alembert, Jean Le Rond d (17171783)      63 262
Alexandrian Museum      1415
Algebraic numbers      254
Amicable numbers      233
Amicable pairs      233235
Amicable triples      236
An Elementary Proof of the Prime Number Theorem (Selberg)      378
Anthoniszoon, Adriaen (15271617)      332
Apery, Roger (19161994)      373
Arabic numerals      284
Archimedean property      2
Archimedean value of $\pi$      331
Archimedes (c.287212 B.C.)      331 346
Area of Pythagorean triangles      250 257
Arithmetic functions      see Number-theoretic functions
Arithmetic progressions of numbers, primes      5455 375
Arithmetic progressions of numbers, pseudoprimes      90
Arithmetica (Diophantus), Bombelli and      307
Arithmetica (Diophantus), Fermat and      245246 257 346
Arithmetica (Diophantus), history of      32
Arithmetica (Diophantus), recovery of script      8586
Artin, Emil (18981962)      157
Artins conjecture      157
Aryabhata I (476C.550 a.d.)      15
Augustine, Saint (354430)      219
Authentication of messages      215216
Autokey cryptosystems      200201
Bachet, Claude (15811638)      86 273
Barlow, Peter (17761862)      228229
Bases for number systems      70
Basis for induction      4
Baudot code      202
Baudot, Jean-Maurice-Emile (18451903)      202
Bennett, G.      237
Bernoulli inequality      7
Bernoulli, Daniel (17001782)      129 130
Bernoulli, Johann (16671748)      129
Bernoulli, Nicolaus (16951726)      129 130
Bertrand, Joseph (18221900)      48
Bertrands conjecture      48 352 371
Bhaskara II(1114c.1185)      83
Binary exponential algorithm      7071
Binary number representation      7071
Binet formula      296298
Binet, Jacques-Philippe-Marie (17861856)      296
Binomial coefficients as integers      119
Binomial coefficients, defined      8
Binomial coefficients, Fibonacci numbers formula      302
Binomial coefficients, identities      810
Binomial congruences      164
Binomial theorem      810
Blum integers      369
Blum, Manuel (1938)      367
Blums coin flipping game      367370
Bombelli, Rafael (15261572)      307
Bonses inequality      47
Bracket function      117 119121
Brahmagupta (598C.665)      83 346
Brent, Richard      240 241 356
Brillhart, John (1930)      240 357 360
Brouncker, William (16201684)      332 335336
Brun, Viggo (18821978)      375
Bruns constant      375
Caesar, Julius (10044 B.C.), cipher system      197198
Caesar, Julius (10044 B.C.), Julian calendar      122 123
Calendars      122124
Canonical form      42
Carlyle, Thomas (17951881)      175
Carmichael numbers      91 363
Carmichael, Robert D. (18791967)      91
Catalan equation      257258
Catalan numbers      12
Catalan, Eugene (18141894)      12 258
Catalans conjecture      258
Cataldi, Pietro (15481626)      222
Cattle problem      346
Chain of inequalities (continued fractions)      317318
Chang Chiu-chien (6th century a.d.)      36 37
Check digits      7273
Chinese Remainder Theorem in Blums coin flipping game      368 370
Chinese Remainder Theorem, defined      7981
Chinese Remainder Theorem, Eulers generalization of Fermats theorem and      139140
Cicero, Marcus Tullius (10643 B.C.)      197 198
Ciphers, autokey systems      200201
Ciphers, Caesar      198
Ciphers, defined      197
Ciphers, Elgamal      213216
Ciphers, Hill      201
Ciphers, Merkle Hellman      209212
Ciphers, Vigenere      199200
Clavius, Christopher (15371612)      123
Cogitata Physica-Mathematica (Mersenne)      225
Coin flipping, remote      367370
Cole Prize      351
Cole, Frank Nelson (18611926)      226
Common divisors      2021
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 205206
Computers in number theory, Mersenne primes and      229230
Computers in number theory, prime number factorization and      353
Congruences      6382 (see also Quadratic congruences)
Congruences in Caesar cipher      198
Congruences, basic properties      6367
Congruences, binomial      164
Congruences, check digits      7273
Congruences, Chinese Remainder Theorem      7981
Congruences, days of the week and      123
Congruences, defined      63
Congruences, indices for solving      164167
Congruences, linear in two variables      8182
Congruences, partition function and      305
Congruences, place-value notation systems      6971
Congruences, polynomial functions      7172
Congruences, simultaneous linear      7879
Congruences, single linear      7678
Congruences, to perfect squares      100
Congruent modulo n, defined      63
Constant function f(n) = n      107 110
Continued fraction algorithm (irrational numbers)      326328
Continued fraction factoring algorithm      357360
Continued fractions, defined      306 (see also Finite continued fractions; infinite continued fractions Pells
Convergents of continued fractions, $\pi$      327328
Convergents of continued fractions, finite continued fractions      311315 317318
Convergents of continued fractions, infinite continued fractions      321322 325326
Convergents of continued fractions, Pells equation and      336 337341
Critical line of the zeta function      376
Cryptography      197216
Cryptography, defined      197
Cryptography, Elgamal system      213216
Cryptography, knapsack problems      208209
Cryptography, Merkle Hellman knapsack system      209212
Cryptography, monoalphabetic systems      197198
Cryptography, poly alphabetic systems      198201
Cryptography, RSA system      203206
Cryptography, Verman one-time pad system      202203
Cunningham, Allen Joseph (18481928)      229
Day of the week, determining      123126
de Polignac, Alphonse (18171890)      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 (15961650), amicable pair discovery      234
Descartes, Rene (15961650), citations      235 273
Descartes, Rene (15961650), Mersenne and      217 218
Dickson, Leonard Eugene (18741954)      278 351
Difference of two squares      269270
Digital alphabet for RSA cryptosystem      203204
Digital signatures      215216
Digits of a number, defined      71
Diophantine equations, $x^2 + y^2 = z^2$      245250
Diophantine equations, $x^4 + v^4 = z^2$      252253
Diophantine equations, $x^4 + v^4 = z^4$      253254
Diophantine equations, $x^4 - v^4 = z^2$      256258
Diophantine equations, $x^n + v^n = z^n$      245246 254255
Diophantine equations, ax + by + cz = d      3637
Diophantine equations, ax + by = c      3235
Diophantine equations, defined      32
Diophantine equations, Fibonacci work on      283
Diophantine equations, linear congruences and      76 78
Diophantine equations, linear in two unknowns      3235
Diophantine equations, word problems      3537
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 (18051859), citations      54 172 186 375
Dirichlet, Peter Gustav Lejeune (18051859), Fermats Last Theorem work      254
Dirichlet, Peter Gustav Lejeune (18051859), pigeonhole principle      264
Dirichlets 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      374375
Divisibility theory      1338
Divisibility theory by 9 or 11      7172
Divisibility theory, Diophantine equations      3237
Divisibility theory, Division Algorithm      1719
Divisibility theory, early number theory      1316
Divisibility theory, Euclidean algorithm      2631
Divisibility theory, greatest common divisor      1924
Divisibility theory, Mersenne and      218
Divisibility theory, symbols for      20
Division algorithm      1719
divisors      see also Number of divisors Sum
Divisors from prime factorizations      104105
Divisors, common      2021
Divisors, defined      20
Divisors, greatest common      21 24
Divisors, Mersenne numbers      228229
Double Wieferich primes      258
e (continued fractions representation)      328329
e-prime numbers      42
Early number theory      1316
Eisenstein, Ferdinand Gottfried Max (18231852)      186
El Madschriti of Madrid (11th century)      234
Elements (Euclid), Diophantine equations and      32
Elements (Euclid), Euclidean algorithm      26
Elements (Euclid), Euclids theorem      45
Elements (Euclid), Fundamental Theorem of Arithmetic      3940
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      213216
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 (17911865)      374
Equality, congruence and      65
Eratosthenes of Cyrene (c.276c.194 B.C.)      45 346
Erdos, Paul (19131996)      351353
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      2628
Euclidean algorithm, least common multiple and      2930
Euclidean algorithm, more than two integers      3031
Euclidean algorithm, number of steps      2829
Euclidean numbers      46
Euclids lemma      24
Euclids theorem, defined      4548
Euclids theorem, Eulers phi-function and      134135
Euler polynomial      5556
Euler, Leonhard (17071783), $\pi$, symbol      327
Euler, Leonhard (17071783), amicable pair work      234 235
Euler, Leonhard (17071783), biographical information      129131 262
Euler, Leonhard (17071783), citations      55 57 63 87 185 221 265 279
Euler, Leonhard (17071783), e continued fractions representation      328
Euler, Leonhard (17071783), Fermat numbers work      237 240
Euler, Leonhard (17071783), Fermats Last Theorem work      254
Euler, Leonhard (17071783), Goldbach conjecture and      51
Euler, Leonhard (17071783), Mersenne numbers work      225226
Euler, Leonhard (17071783), on Catalan equation      258
Euler, Leonhard (17071783), on odd perfect numbers      231 232
Euler, Leonhard (17071783), on triangular numbers      15
Euler, Leonhard (17071783), Pells equation and      336
Euler, Leonhard (17071783), photo of      130
Euler, Leonhard (17071783), primitive roots for primes      162
Euler, Leonhard (17071783), proof of Fermats theorem      87 136
Euler, Leonhard (17071783), sum of four squares      273
Euler, Leonhard (17071783), Warings problem      351
Euler, Leonhard (17071783), word problems      38
Euler, Leonhard (17071783), zeta function formula      373
Eulers criterion, citations      180
Eulers criterion, defined      171172
Eulers criterion, Dirichlets proof of      172173
Eulers generalization of Fermats theorem, applications of      139140
Eulers generalization of Fermats theorem, defined      137138
Eulers generalization of Fermats theorem, Fermats Little Theorem as proof of      139
Eulers identity      273 277
Eulers phi-function $\phi(n)$ as even integer      134
Eulers phi-function $\phi(n)$ as multiplicative function      132 133 142
Eulers phi-function $\phi(n)$, defined      131132
Eulers phi-function $\phi(n)$, Euclids theorem and      134135
Eulers phi-function $\phi(n)$, Gauss theorem and      141143
Eulers phi-function $\phi(n)$, Mobius inversion formula and      144145
Eulers phi-function $\phi(n)$, sum of integers identity      143
Eulers phi-function $\phi(n)$, table of      407408
Even-numbered convergents      317318 321
Even-numbered convergents in Pythagorean triples      247
Even-numbered convergents, defined      18
Even-numbered convergents, Eulers phi-function as      134
Exponent of a prime in n! factorization      117118
Exponent to which a belongs modulo n      147150
Exponent, enciphering      203 204
Exponent, recovery      204
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