*(English)*Zbl 1084.11002

This is the fourth edition of the classical title *Number Theory in Science and Communication* (for a review of the previous editions see Zbl 0997.11501, Zbl 0613.10001 and Zbl 0542.10001) written by a physicist “inclined toward mathematics”.

Obviously his idea of Number Theory is opposed to the famous of G. H. Hardy (“...there is one science [number theory] whose very remoteness from ordinary human activities should keep it gentle and clean”, A Mathematician’s Apology). As the title makes clear the author considers Number Theory related to “ordinary human activities” and the aim of the book is to show some of such interrelations in domains like Information Transmission, Concert Hall Acoustics and many others.

The book has 30 chapters divided into ten parts whose titles allow an image of the contents of the book: Part I. A Few Fundamentals (Natural Numbers, Primes); Part II. Some Simple Applications (Fractions: Continued, Egyptians and Farey); Part III. Congruences and the Like; Part IV. Cryptography and Divisors (Public-Key Cryptography, Primitive Roots); Part V. Residues and Diffraction; Part VI. Chinese and other Fast Algorithms; Part VII. Pseudoprimes, Möbius Transform and Partitions); Part VIII. Cyclotomy and Polynomials; Part IX. Galois Fields and More Applications (Spectral Properties of Galois Sequences, Random Number Generators, Waveforms and Radiation Patterns, Number Theory, Randomness and “Art”); Part X. Self-Similarity, Fractals and Art.

In this fourth edition new materials have been added on Fibonacci Numbers, Divisibility Tests for 7, 13, 17 and 19, Two Square Theorem, new Factoring Methods (the Lenstra’s Factoring with Elliptic Curves) and Primality Tests.

The book intends to be self-contained and the definitions and materials of Number Theory are introduced and studied when they are needed, always with a view to showing their relationship and applications in the “real world”. References to the contents of each chapter are provided at the end of the book.

##### MSC:

11-01 | Textbooks (number theory) |

68-01 | Textbooks (computer science) |

94-01 | Textbooks (information and communication) |

11Axx | Elementary number theory |

11A41 | Elementary prime number theory |

11T22 | Cyclotomy |

11T71 | Algebraic coding theory; cryptography |

11Y05 | Factorization |