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 . .