The Little Book of Bigger Primes

1. Front Matter

   Pages i-xvii

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2. Introduction

   • Paulo Ribenboim

   Pages 1-2

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3. How Many Prime Numbers Are There?

   • Paulo Ribenboim

   Pages 3-9

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4. How to Recognize Whether a Natural Number is a Prime

   • Paulo Ribenboim

   Pages 11-104

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5. Are There Functions Defining Prime Numbers?

   • Paulo Ribenboim

   Pages 105-118

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6. How Are the Prime Numbers Distributed?

   • Paulo Ribenboim

   Pages 119-160

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7. Which Special Kinds of Primes Have Been Considered?

   • Paulo Ribenboim

   Pages 161-178

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8. Heuristic and Probabilistic Results about Prime Numbers

   • Paulo Ribenboim

   Pages 179-191

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9. Conclusion

   • Paulo Ribenboim

   Pages 193-194

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10. Back Matter

    Pages 195-237

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This book could have been called "Selections from the Book of Prime Number Records." However, I prefered the title which propelled you on the first place to open it, and perhaps (so I hope) to buy it! Richard K. Guy, with his winning ways, suggested the title to me, and I am grateful. But the book isn't very different from its parent. Like a bonsai, which has all the main characteristics of the full-sized tree, this little paperback should exert the same fatal attraction. I wish it to be as dangerous as the other one, in the saying of John Brillhart. I wish that you, young student, teacher or retired mathematician, engineer, computer buff, all of you who are friends of numbers, to be driven into thinking about the beautiful theory of prime numbers, with its inherent mystery. I wish you to exercise your brain and fingers-not vice-versa. But I do not wish you, specialist in number theory to look at this little book-most likely you have been eliminated from this shorter version-what a terrible feeling. But do not cry, you had your book already. This one is for those who will be taking over and should put their steps forward, mostly little, occasionally giant, to develop the science of numbers. Paulo Ribenboim Contents Preface vii Guiding the Reader xii Index of Notations xiii Introduction 1 1 How Many Prime Numbers Are There? 3 I. Euclid's Proof .. 3 11. Kummer's Proof 4 II. P6lya's Proof . .

• DEX
• Prime
• Propel
• character
• computer
• number theory
• prime number
• proof
• selection

From the reviews of the second edition:

P. Ribenboim

The Little Book of Big Primes

"Everyone has at one time or another taken an interest, however fleeting, in prime numbers . . . Ribenboim, however, is clearly a ‘prime nut,’ and this excellent, good-humored book is written for other (actually or potentially) incurable aficionados . . . it is a masterly presentation . . . This genially reader-friendly tour de force, by a scientist whit an encyclopedic and up-to-the-minute knowledge of the subject, is a wholly admirable addition to anyone’s bookshelf."—AMERICAN SCIENTIST

"The book presents a wealth of material and references on fundamental theorems, challenging open problems, and the most recent computational records in a language without secrets." (Zentralblatt für Didaktik und Mathematik, November, 2004)

"This book is the substantially expanded and updated second edition of the 1991 ‘Little book of big primes’. … The author’s thorough knowledge of and passion for primes comes through clearly. Most of the book is accessible even to interested amateurs or undergraduate students … . At the same time, a specialist can also find it useful to have so many results, open problems and references collected together." (Gábor Megyesi, Acta Scientiarum Mathematicarum, Vol. 72, 2006)

"One could say that Paul Ribenboim created a new genre of mathematical writing with the publication of ‘The Book of Prime Number Records.’. … This together with the author’s charming style, made it a success. The book under review is the abridged version of its successor ‘The New Book of Prime Number Records.’ … The reviewer believes that this book really has the ‘fatal attraction’ its author wishes it to exert." (Ch. Baxa, Monatshefte für Mathematik, Vol. 148 (1), 2006)

"This is a book of well documented records. … But this is a book of much more than records, providing as good an introduction as any to the theory of primes … . The first edition of the book appeared in 1991 … . It is here expanded and updated … . Throughout the book history, theory, specific cases and other results are woven into a compact but comprehensive presentation. A most useful resource for both teaching and research into the prime numbers." (Alan Sutcliffe, The Mathematical Gazette, Vol. 89 (516), 2005)

• Department of Mathematics and Statistics, Queen’s University, Kingston, Canada

  Paulo Ribenboim

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