Ebook: Number Theory in Science and Communication: With Applications in Cryptography, Physics, Digital Information, Computing, and Self-Similarity

Number Theory in Science and Communication is an introduction for non-mathematicians. The book stresses intuitive understanding rather than abstract theory and highlights important concepts such as continued fractions, the golden ratio, quadratic residues and Chinese remainders, trapdoor functions, pseudoprimes and primituve elements. Their applications to problems in the real world is one of the main themes of the book. This third edition is augmented by recent advances in primes in progressions, twin primes, prime triplets, prime quadruplets and quintruplets, factoring with elliptic curves, quantum factoring, Golomb rulers and "baroque" integers.

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Number Theory in Science and Communication is an introduction for non-mathematicians. The book stresses intuitive understanding rather than abstract theory and highlights important concepts such as continued fractions, the golden ratio, quadratic residues and Chinese remainders, trapdoor functions, pseudoprimes and primituve elements. Their applications to problems in the real world is one of the main themes of the book. This third edition is augmented by recent advances in primes in progressions, twin primes, prime triplets, prime quadruplets and quintruplets, factoring with elliptic curves, quantum factoring, Golomb rulers and "baroque" integers.
Content:
Front Matter....Pages I-XXII
Introduction....Pages 1-15
The Natural Numbers....Pages 16-24
Primes....Pages 25-37
The Prime Distribution....Pages 38-61
Fractions: Continued, Egyptian and Farey....Pages 62-94
Linear Congruences....Pages 95-101
Diophantine Equations....Pages 102-117
The Theorems of Fermat, Wilson and Euler....Pages 118-124
Euler Trap Doors and Public-Key Encryption....Pages 125-134
The Divisor Functions....Pages 135-141
The Prime Divisor Functions....Pages 142-153
Certified Signatures....Pages 154-155
Primitive Roots....Pages 156-172
Knapsack Encryption....Pages 173-176
Quadratic Residues....Pages 177-189
The Chinese Remainder Theorem and Simultaneous Congruences....Pages 190-198
Fast Transformation and Kronecker Products....Pages 199-202
Quadratic Congruences....Pages 203-204
Pseudoprimes, Poker and Remote Coin Tossing....Pages 205-215
The M?bius Function and the M?bius Transform....Pages 216-223
Generating Functions and Partitions....Pages 224-231
Cyclotomic Polynomials....Pages 232-246
Linear Systems and Polynomials....Pages 247-249
Polynomial Theory....Pages 250-255
Galois Fields....Pages 256-268
Spectral Properties of Galois Sequences....Pages 269-282
Random Number Generators....Pages 283-288
Waveforms and Radiation Patterns....Pages 289-300
Number Theory, Randomness and “Art”....Pages 301-310
Self-Similarity, Fractals, Deterministic Chaos and a New State of Matter....Pages 311-335
Back Matter....Pages 336-364

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Number Theory in Science and Communication is an introduction for non-mathematicians. The book stresses intuitive understanding rather than abstract theory and highlights important concepts such as continued fractions, the golden ratio, quadratic residues and Chinese remainders, trapdoor functions, pseudoprimes and primituve elements. Their applications to problems in the real world is one of the main themes of the book. This third edition is augmented by recent advances in primes in progressions, twin primes, prime triplets, prime quadruplets and quintruplets, factoring with elliptic curves, quantum factoring, Golomb rulers and "baroque" integers.
Content:
Front Matter....Pages I-XXII
Introduction....Pages 1-15
The Natural Numbers....Pages 16-24
Primes....Pages 25-37
The Prime Distribution....Pages 38-61
Fractions: Continued, Egyptian and Farey....Pages 62-94
Linear Congruences....Pages 95-101
Diophantine Equations....Pages 102-117
The Theorems of Fermat, Wilson and Euler....Pages 118-124
Euler Trap Doors and Public-Key Encryption....Pages 125-134
The Divisor Functions....Pages 135-141
The Prime Divisor Functions....Pages 142-153
Certified Signatures....Pages 154-155
Primitive Roots....Pages 156-172
Knapsack Encryption....Pages 173-176
Quadratic Residues....Pages 177-189
The Chinese Remainder Theorem and Simultaneous Congruences....Pages 190-198
Fast Transformation and Kronecker Products....Pages 199-202
Quadratic Congruences....Pages 203-204
Pseudoprimes, Poker and Remote Coin Tossing....Pages 205-215
The M?bius Function and the M?bius Transform....Pages 216-223
Generating Functions and Partitions....Pages 224-231
Cyclotomic Polynomials....Pages 232-246
Linear Systems and Polynomials....Pages 247-249
Polynomial Theory....Pages 250-255
Galois Fields....Pages 256-268
Spectral Properties of Galois Sequences....Pages 269-282
Random Number Generators....Pages 283-288
Waveforms and Radiation Patterns....Pages 289-300
Number Theory, Randomness and “Art”....Pages 301-310
Self-Similarity, Fractals, Deterministic Chaos and a New State of Matter....Pages 311-335
Back Matter....Pages 336-364
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