Author | Ribenboim, Paulo. author |
---|---|

Title | The Book of Prime Number Records [electronic resource] / by Paulo Ribenboim |

Imprint | New York, NY : Springer US, 1989 |

Edition | Second Edition |

Connect to | http://dx.doi.org/10.1007/978-1-4684-0507-1 |

Descript | XXIII, 479p. online resource |

SUMMARY

This text originated as a lecture delivered November 20, 1984, at Queen's University, in the undergraduate colloquim series established to honor Professors A. J. Coleman and H. W. Ellis and to acknowยญ ledge their long lasting interest in the quality of teaching underยญ graduate students. In another colloquim lecture, my colleague Morris Orzech, who had consulted the latest edition of the Guilllless Book oj Records, remainded me very gently that the most "innumerate" people of the world are of a certain tribe in Mato Grosso, Brazil. They do not even have a word to express the number "two" or the concept of plurality. "Yes Morris, I'm from Brazil, but my book will contain numbers different from 'one.' " He added that the most boring 800-page book is by two Japanese mathematicians (whom I'll not name), and consists of about 16 million digits of the number 11. "I assure you Morris, that in spite of the beauty of the apparent randomness of the decimal digits of 11, I'll be sure that my text will include also some words." Acknowledgment. The manuscript of this book was prepared on the word processor by Linda Nuttall. I wish to express my appreciation for the great care, speed, and competence of her work

CONTENT

1. How Many Prime Numbers Are There? -- I. Euclidโ{128}{153}s Proof -- II. Kummerโ{128}{153}s Proof -- III. Polyaโ{128}{153}s Proof -- IV. Eulerโ{128}{153}s Proof -- V. Thueโ{128}{153}s Proof -- VI. Two-and-a-Half Forgotten Proofs -- VII. Washingtonโ{128}{153}s Proof -- VIII. Fiirstenbergโ{128}{153}s Proof -- 2. How to Recognize Whether a Natural Number Is a Prime? -- I. The Sieve of Eratosthenes -- II. Some Fundamental Theorems on Congruences -- III. Classical Primality Tests Based on Congruences -- IV. Lucas Sequences -- V. Classical Primality Tests Based on Lucas Sequences -- VI. Fermat Numbers -- VII. Mersenne Numbers -- VIII. Pseudoprimes -- Addendum on the Congruence an?k ? bn?k (mod n) -- IX. Carmichael Numbers -- X. Lucas Pseudoprimes -- XI. Last Section on Primality Testing and Factorization! -- 3. Are There Functions Defining Prime Numbers? -- I. Functions Satisfying Condition (a) -- II. Functions Satisfying Condition (b) -- III. Functions Satisfying Condition (c) -- 4. How Are the Prime Numbers Distributed? -- I. The Growth of ?(x) -- II. The nth Prime and Gaps -- III. Twin Primes -- IV. Primes in Arithmetic Progression -- V. Primes in Special Sequences -- VI. Goldbachโ{128}{153}s Famous Conjecture -- VII. The Waring-Goldbach Problem -- VIII. The Distribution of Pseudoprimes and of Carmichael Numbers -- 5. Which Special Kinds of Primes Have Been Considered? -- I. Regular Primes -- II. Sophie Germain Primes -- III. Wieferich Primes -- IV. Wilson Primes -- V. Repunits and Similar Numbers -- VI. Primes with Given Initial and Final Digits -- VII. Numbers k ร{151} 2n ยฑ 1 -- VIII. Primes and Second-Order Linear Recurrence Sequences -- IX. The NSW-Primes -- 6. Heuristic and Probabilistic Results About Prime Numbers -- I. Prime Values of Linear Polynomials -- II. Prime Values of Polynomials of Arbitrary Degree -- III. Some Probabilistic Estimates -- IV. The Density of the Set of Regular Primes -- Conclusion -- Dear Reader -- Citations for Some Possible Prizes for Work on the Prime Number Theorem -- A. General References -- B. Specific References -- 1 -- 2 -- 3 -- 4 -- 5 -- 6 -- Conclusion -- Primes up to 10,000 -- Index of Names -- Gallimaufries -- Addenda to the Second Edition

Mathematics
Number theory
Mathematics
Number Theory