Number Theory in Science and Communication

1. Front Matter

   Pages I-XIX

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2. A Few Fundamentals

   1. Introduction

      • Manfred R. Schroeder

      Pages 1-16

   2. The Natural Numbers

      • Manfred R. Schroeder

      Pages 17-25

   3. Primes

      • Manfred R. Schroeder

      Pages 26-39

   4. The Prime Distribution

      • Manfred R. Schroeder

      Pages 40-54

3. Some Simple Applications

   1. Fractions: Continued, Egyptian and Farey

      • Manfred R. Schroeder

      Pages 55-86

4. Congruences and the Like

   1. Linear Congruences

      • Manfred R. Schroeder

      Pages 87-94

   2. Diophantine Equations

      • Manfred R. Schroeder

      Pages 95-110

   3. The Theorems of Fermat, Wilson and Euler

      • Manfred R. Schroeder

      Pages 111-117

5. Cryptography and Divisors

   1. Euler Trap Doors and Public-Key Encryption

      • Manfred R. Schroeder

      Pages 118-126

   2. The Divisor Functions

      • Manfred R. Schroeder

      Pages 127-134

   3. The Prime Divisor Functions

      • Manfred R. Schroeder

      Pages 135-148

   4. Certified Signatures

      • Manfred R. Schroeder

      Pages 149-150

   5. Primitive Roots

      • Manfred R. Schroeder

      Pages 151-167

   6. Knapsack Encryption

      • Manfred R. Schroeder

      Pages 168-171

6. Residues and Diffraction

   1. Quadratic Residues

      • Manfred R. Schroeder

      Pages 172-185

7. Chinese and Other Fast Algorithms

   1. The Chinese Remainder Theorem and Simultaneous Congruences

      • Manfred R. Schroeder

      Pages 186-195

   2. Fast Transformations and Kronecker Products

      • Manfred R. Schroeder

      Pages 196-200

   3. Quadratic Congruences

      • Manfred R. Schroeder

      Pages 201-202

8. Pseudoprimes, Möbius Transform, and Partitions

   1. Pseudoprimes, Poker and Remote Coin Tossing

      • Manfred R. Schroeder

      Pages 203-214

"Beauty is the first test: there is no permanent place in the world for ugly mathematics. " - G. H. Hardy Number theory has been considered since time immemorial to be the very paradigm of pure (some would say useless) mathematics. In fact, the Chinese characters for mathematics are Number Science. "Mathematics is the queen of sciences - and number theory is the queen of mathematics," according to Carl Friedrich Gauss, the lifelong Wunderkind, who himself enjoyed the epithet "Princeps Mathematicorum. " What could be more beautiful than a deep, satisfying relation between whole numbers. (One is almost tempted to call them wholesome numbers') In fact, it is hard to come up with a more appropriate designation than their learned name: the integers - meaning the "untouched ones". How high they rank, in the realms of pure thought and aesthetics, above their lesser brethren: the real and complex number- whose first names virtually exude unsavory involvement with the complex realities of everyday life! Yet, as we shall see in this book, the theory of integers can provide totally unexpected answers to real-world problems. In fact, discrete mathematics is taking on an ever more important role. If nothing else, the advent of the digital computer and digital communication has seen to that. But even earlier, in physics the emergence of quantum mechanics and discrete elementary particles put a premium on the methods and, indeed, the spirit of discrete mathematics.

• Astrophysics
• Biology
• Digital Signal Processing
• Electrical Engineering
• Geophysics
• Medicine
• Physics and Mathematics
• communication system
• cryptography
• number theory
• sets

..."A lighthearted and readable volume with a wide range of applications to which the author has been a productive contributor - useful mathematics given outside the formalities of theorem and proof"... #Scientific American#1

• Drittes Physikalisches Institut, Universität Göttingen, Göttingen, Fed. Rep. of Germany

  Manfred R. Schroeder

• Acoustics Speech and Mechanics Research, Bell Laboratories, Murray Hill, USA

  Manfred R. Schroeder

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