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Author Ribenboim, Paulo. author The Book of Prime Number Records [electronic resource] / by Paulo Ribenboim New York, NY : Springer New York : Imprint: Springer, 1988 http://dx.doi.org/10.1007/978-1-4684-9938-4 XXIII, 476 p. online resource

SUMMARY

This text originated as a lecture delivered November 20, 1984, at Queen's University, in the undergraduate colloquium series established to honour Professors A. J. Coleman and H. W. Ellis and to acknowledge their long-lasting interest in the quality of teaching undergraduate students. In another colloquium lecture, my colleague Morris Orzech, who had consulted the latest edition of the Guinness Book oj Records, reminded 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 also include 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. Paulo Ribenboim CONTENTS Preface vii Guiding the Reader xiii Index of Notations xv Introduction Chapter 1. How Many Prime Numbers Are There? 3 I. Euclid's Proof 3 II

CONTENT

1. How Many Prime Numbers Are There? -- I. Euclidโ{128}{153}s Proof -- II. Kummerโ{128}{153}s Proof -- III. Pรณlyaโ{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. Fรผrstenbergโ{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 -- A. Fermatโ{128}{153}s Little Theorem and Primitive Roots Modulo a Prime -- B. The Theorem of Wilson -- C. The Properties of Giuga, Wolstenholme and Mann & Shanks -- D. The Power of a Prime Dividing a Factorial -- E. The Chinese Remainder Theorem -- F. Eulerโ{128}{153}s Function -- G. Sequences of Binomials 31 -- H. Quadratic Residues -- 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 -- 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} 2โ{128}{153} 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 -- Gallimawfries

Mathematics Number theory Mathematics Number Theory

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