## Number theory in science and communication : with applications in cryptography, physics, digital information, computing, and self-similarity 2nd enl. ed

000 | 00971camuuu200277 a 4500 | |

001 | 000000110700 | |

005 | 19980706103212.0 | |

008 | 850718s1986 gw a b 001 0 eng | |

010 | ▼a 85017260 //r90 | |

020 | ▼a 0387158006 (U.S. : pbk.) | |

040 | ▼a DLC ▼c DLC ▼d FTV | |

049 | 1 | ▼l 421041362 ▼f 과학 |

050 | 0 0 | ▼a QA241 ▼b .S318 1986 |

082 | 0 0 | ▼a 512/.7 ▼2 19 |

090 | ▼a 512.7 ▼b S381n | |

100 | 1 | ▼a Schroeder, M. R. ▼q (Manfred Robert), ▼d 1926- |

245 | 1 0 | ▼a Number theory in science and communication : ▼b with applications in cryptography, physics, digital information, computing, and self-similarity / ▼c M.R. Schroeder. |

250 | ▼a 2nd enl. ed. | |

260 | ▼a Berlin ; ▼a New York : ▼b Springer-Verlag, ▼c c1986. | |

300 | ▼a xix, 374 p. : ▼b ill. ; ▼c 24 cm. | |

440 | 0 | ▼a Springer series in information sciences ; ▼v v. 7. |

500 | ▼a Includes indexes. | |

504 | ▼a Bibliography: p. [353]-362. | |

650 | 0 | ▼a Number theory. |

650 | 7 | ▼a Nombres, theorie des. ▼2 ram. |

### Holdings Information

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No. 1 | Location Science & Engineering Library/Sci-Info(Stacks2)/ | Call Number 512.7 S381n | Accession No. 421041362 | Availability Available | Due Date | Make a Reservation | Service |

### Contents information

#### Table of Contents

권호명 : 1 CONTENTS Part Ⅰ A Few Fundamentals 1. Introduction = 1 1.1 Fibonacci, Continued Fractions and the Golden Ratio = 4 1.2 Fermat, Primes and Cyclotomy = 7 1.3 Euler, Totients and Cryptography = 9 1.4 Gauss, Congruences and Diffraction = 10 1.5 Galois, Fields and Codes = 12 2. The Natural Numbers = 17 2.1 The Fundamental Theorem = 17 2.2 The Least Common Multiple = 18 2.3 Planetary "Gears" = 19 2.4 The Greatest Common Divisor = 20 2.5 Human Pitch Perception = 22 2.6 Octaves, Temperament, Kilos and Decibels = 22 2.7 Coprimes = 25 2.8 Euclid's Algorithm = 25 3. Primes = 26 3.1 How Many Primes are There? = 26 3.2 The Sieve of Eratosthenes = 27 3.3 A Chinese Theorem in Error = 29 3.4 A Formula for Primes = 29 3.5 Mersenne Primes = 30 3.6 Repunits = 34 3.7 Perfect Numbers = 35 3.8 Fermat Primes = 37 3.9 Gauss and the Impossible Heptagon = 38 4. The Prime Distribution = 40 4.1 A Probabilistic Argument = 40 4.2 The Prime-Counting Function π(x) = 43 4.3 David Hilbert and Large Nuclei = 47 4.4 Coprime Probabilities = 48 4.5 Twin Primes = 51 4.6 Primeless Expanses = 53 4.7 Square-Free and Coprime Integers = 54 Part Ⅱ Some Simple Applications 5. Fractions : Continued, Egyptian and Farey = 55 5.1 A Neglected Subject = 55 5.2 Relations with Measure Theory = 60 5.3 Periodic Continued Fractions = 60 5.4 Electrical Networks and Squared Squares = 64 5.5 Fibonacci Numbers and the Golden Ratio = 65 5.6 Fibonacci, Rabbits and Computers = 70 5.7 Fibonacci and Divisibility = 72 5.8 Generalized Fibonacci and Lucas Numbers = 73 5.9 Egyptian Fractions, Inheritance and Some Unsolved Problems = 76 5.10 Farey Fractions = 77 5.11 Fibonacci and the Problem of Bank Deposits = 80 5.12 Error-Free Computing = 81 Part Ⅲ Congruences and the Like 6. Linear Congruences = 87 6.1 Residues = 87 6.2 Some Simple Fields = 90 6.3 Powers and Congruences = 92 7. Diophantine Equations = 95 7.1 Relation with Congruences = 95 7.2 A Gaussian Trick = 96 7.3 Nonlinear Diophantine Equations = 98 7.4 Triangular Numbers = 100 7.5 Pythagorean Numbers = 102 7.6 Exponential Diophantine Equations = 103 7.7 Format's Last "Theorem" = 104 7.8 The Demise of a Conjecture by Euler = 105 7.9 A Nonlinear Diophantine Equation in Physics and the Geometry of Numbers = 106 7.10 Normal-Mode Degeneracy in Room Acoustics (A Number-Theoretic Application) = 108 7.11 Waring's Problem = 109 8. The Theorems of Format, Wilson and Euler = 111 8.1 Format's Theorem = 111 8.2 Wilson's Theorem = 112 8.3 Euler's Theorem = 113 8.4 The Impossible Star of David = 115 8.5 Dirichlet and Linear Progression = 116 Part Ⅳ Cryptography and Divisors 9. Euler Trap Doors and Public-Key Encryption = 118 9.1 A Numerical Trap Door = 118 9.2 Digital Encryption = 119 9.3 Public-Key Encryption = 121 9.4 A Simple Example = 123 9.5 Repeated Encryption = 123 9.6 Summary and Encryption Requirements = 125 10. The Divisor Functions = 127 10.1 The Number of Divisors = 127 10.2 The Average of the Divisor Function = 130 10.3 The Geometric Mean of the Divisors = 131 10.4 The Summatory Function of the Divisor Function = 131 10.5 The Generalized Divisor Functions = 132 10.6 The Average Value of Euler's Function = 133 11. The Prime Divisor Functions = 135 11.1 The Number of Different Prime Divisors = 135 11.2 The Distribution of w(n) = 138 11.3 The Number of Prime Divisors = 141 11.4 The Harmonic Mean of Ω(n) = 144 11.5 Medians and Percentiles of Ω(n) = 146 11.6 Implications for Public-Key Encryption = 147 12. Certified Signatures = 149 12.1 A Story of Creative Financing = 149 12.2 Certified Signature for Public-Key Encryption = 149 13. Primitive Roots = 151 13.1 Orders = 151 13.2 Periods of Decimal and Binary Fractions = 154 13.3 A Primitive Proof of Wilson's Theorem = 157 13.4 The Index - A Number-Theoretic Logarithm = 158 13.5 Solution of Exponential Congruences = 159 13.6 What is the Order Tm of an Integer m Modulo a Prime p? = 161 13.7 Index "Encryption" = 162 13.8 A Fourier Property of Primitive Roots and Concert Hall Acoustics = 163 13.9 More Spacious-Sounding Sound = 164 13.10 A Negative Property of the Fermat Primes = 167 14. Knapsack Encryption = 168 14.1 An Easy Knapsack = 168 14.2 A Hard Knapsack = 169 Part Ⅴ Residues and Diffraction 15. Quadratic Residues = 172 15.1 Quadratic Congruences = 172 15.2 Euler's Criterion = 173 15.3 The Legendre Symbol = 175 15.4 A Fourier Property of Legendre Sequences = 176 15.5 Gauss Sums = 177 15.6 Pretty Diffraction = 179 15.7 Quadratic Reciprocity = 179 15.8 A Fourier Property of Quadratic-Residue Sequences = 180 15.9 Spread Spectrum Communication = 183 15.10 Generalized Legendre Sequences Obtained Through Complexification of the Euler Criterion = 183 Part Ⅵ Chinese and Other Fast Algorithms 16. The Chinese Remainder Theorem and Simultaneous Congruences = 186 16.1 Simultaneous Congruences = 186 16.2 The Sino-Representation : A Chinese Number System = 187 16.3 Applications of the Sino-Representation = 189 16.4 Discrete Fourier Transformation in Sino = 190 16.5 A Sino-Optical Fourier Transformer = 191 16.6 Generalized Sino-Representation = 192 16.7 Fast Prime-Length Fourier Transform = 194 17. Fast Transformations and Kronecker Products = 196 17.1 A Fast Hadamard Transform = 196 17.2 The Basic Principle of the Fast Fourier Transforms = 199 18. Quadratic Congruences = 201 18.1 Application of the Chinese Remainder Theorem(CRT) = 201 Part Ⅶ Pseudoprimes, M o ·· bius Transform, and Partitions 19. Pseudoprimes, Poker and Remote Coin Tossing = 203 19.1 Pulling Roots to Ferret Out Composites = 203 19.2 Factors from a Square Root = 205 19.3 Coin Tossing by Telephone = 206 19.4 Absolute and Strong Pseudoprimes = 209 19.5 Fermat and Strong Pseudoprimes = 211 19.6 Deterministic Primality Testing = 212 19.7 A Very Simple Factoring Algorithm = 213 20. The M o ·· bius Function and the M o ·· bius Transform = 215 20.1 The M o ·· bius Transform and Its Inverse = 215 20.2 Proof of the Inversion Formula = 217 20.3 Second Inversion Formula = 218 20.4 Third Inversion Formula = 219 20.5 Fourth Inversion Formula = 219 20.6 Riemann's Hypothesis and the Disproof of the Mertens Conjecture = 219 20.7 Dirichlet Series and the M o ·· bius Function = 220 21. Generating Functions and Partitions = 223 21.1 Generating Functions = 223 21.2 Partitions of Integers = 225 21.3 Generating Functions of Partitions = 226 21.4 Restricted Partitions = 227 Part Ⅷ Cyclotomy and Polynomials 22. Cyclotomic Polynomials = 232 22.1 How to Divide a Circle into Equal Parts = 232 22.2 Gauss's Great Insight = 235 22.3 Factoring in Different Fields = 240 22.4 Cyclotomy in the Complex Plane = 240 22.5 How to Divide a Circle with Compass and Straightedge = 242 22.5.1 Rational Factors of zN - 1 = 243 22.6 An Alternative Rational Factorization = 244 22.7 Relation Between Rational Factors and Complex Roots = 245 22.8 How to Calculate with Cyclotomic Polynomials = 247 23. Linear Systems and Polynomials = 249 23.1 Impulse Responses = 249 23.2 Time-Discrete Systems and the z Transform = 250 23.3 Discrete Convolution = 251 23.4 Cyclotomic Polynomials and z Transform = 251 24. Polynomial Theory = 253 24.1 Some Basic Facts of Polynomial Life = 253 24.2 Polynomial Residues = 254 24.3 Chinese Remainders for Polynomials = 256 24.4 Euclid's Algorithm for Polynomials = 257 Part Ⅸ Galois Fields and More Applications 25. Galois Fields = 259 25.1 Prime Order = 259 25.2 Prime Power Order = 259 25.3 Generation of GF( 24 ) = 262 25.4 How Many Primitive Elements? = 263 25.5 Recursive Relations = 264 25.6 How to Calculate in GF( pm ) = 266 25.7 Zech Logarithm, Doppler Radar and Optimum Ambiguity Functions = 267 25.8 A Unique Phase-Array Based on the Zech Logarithm = 271 25.9 Spread-Spectrum Communication and Zech Logarithms = 272 26. Spectral Properties of Galois Sequences = 274 26.1 Circular Correlation = 274 26.2 Application to Error-Correcting Codes and Speech Recognition = 277 26.3 Application to Precision Measurements = 278 26.4 Concert Hall Measurements = 279 26.5 The Fourth Effect of General Relativity = 280 26.6 Toward Better Concert Hall Acoustics = 281 26.7 Higher-Dimensional Diffusors = 287 26.8 Active Array Applications = 287 27. Random Number Generators = 289 27.1 Pseudorandom Galois Sequences = 290 27.2 Randomness from Congruences = 291 27.3 "Continuous" Distributions = 292 27.4 Four Ways to Generate a Gaussian Variable = 293 27.5 Pseudorandom Sequences in Cryptography = 295 28. Waveforms and Radiation Patterns = 296 28.1 Special Phases = 297 28.2 The Rudin-Shapiro Polynomials = 299 28.3 Gauss Sums and Peak Factors = 300 28.4 Galois Sequences and the Smallest Peak Factors = 302 28.5 Minimum Redundancy Antennas = 305 29. Number Theory, Randomness and "Art" = 307 29.1 Number Theory and Graphic Design = 307 29.2 The Primes of Gauss and Eisenstein = 309 29.3 Galois Fields and Impossible Necklaces = 310 Part Ⅹ Self-Similarity, Fractals and Art 30. Self-Similarity, Fractals, Deterministic Chaos and a New State of Matter = 315 30.1 Fibonacci, Noble Numbers and a New State of Matter = 319 30.2 Cantor Sets, Fractals and a Musical Paradox = 324 30.3 The Twin Dragon : a Fractal from a Complex Number System = 330 30.4 Statistical Fractals = 331 30.5 Some Crazy Mappings = 333 30.6 The Logistic Parabola and Strange Attractors = 337 30.7 Conclusion = 340 Appendix = 341 A. A Calculator Program for Exponentiation and Residue Reduction = 341 B. A Calculator Program for Calculating Fibonacci and Lucas Numbers = 345 C. A Calculator Program for Decomposing an Integer According to the Fibonacci Number System = 346 Glossary of Symbols = 349 References = 353 Name Index = 363 Subject Index = 367