b2530 International Strategic Relations and China’s National Security: World at the Crossroads This page intentionally left blank b2530_FM.indd 6 01-Sep-16 11:03:06 AM Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 SA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 K office: 57 Shelton Street, Covent Garden, London WC2H 9HE Library of Congress Cataloging-in-Publication Data Names: Ribenboim, Paulo. Title: Prime numbers, friends who give problems : a trialogue with Papa Paulo / by Paulo Ribenboim (Queen’s niversity, Canada). Description: New Jersey : World Scientific, 2016. | Includes indexes. Identifiers: LCCN 2016020705| ISBN 9789814725804 (hardcover : alk. paper) | ISBN 9789814725811 (softcover : alk. paper) Subjects: LCSH: Numbers, Prime. Classification: LCC QA246 .R474 2016 | DDC 512.7/23--dc23 LC record available at https://lccn.loc.gov/2016020705 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. Copyright © 2017 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, SA. In this case permission to photocopy is not required from the publisher. Typeset by Stallion Press Email: [email protected] Printed in Singapore YingOi - Prime Numbers, Friends Who Give Problems.indd 1 22-08-16 9:11:29 AM October 4, 2016 8:36 Prime Numbers, Friends Who Give Problems 9in x 6in b2394-fm page v Qu’on ne me dise pas que je n’ai rien dit de nouveau; la disposition des mati`eres est nouvelle. Quand on jouealapaume,c’estunemˆ ` eme balle dont joue l’un et l’autre, mais l’un la place mieux. Blaise Pascal v b2530 International Strategic Relations and China’s National Security: World at the Crossroads This page intentionally left blank b2530_FM.indd 6 01-Sep-16 11:03:06 AM October 4, 2016 8:36 Prime Numbers, Friends Who Give Problems 9in x 6in b2394-fm page vii Preface Mathematicians are sometimes intelligent, but never magicians. Their ways of thinking are natural and intuitive, contorted or opaque as they may seem. Ideas, even when simple in essence, may have to be dressed up, supported by calculations and expressed in a jargon which repulses the layman. The challenge of explaining the theories and discoveries is difficult to surmount. As a result, so many are deprived of the music of mathematics. What you will read revolves around prime numbers. Hardly any kind of numbers is more attractive or mysterious. Not much of what follows will be supported by rigorous proof. But everything will be made credible, even irrefutable. You opened this book. You think that you are smart enough to understand all the discussions reported here. You may be surprised. This book should not be read by specialists, who discover much and explain little. This book is not for anyone lacking in intelligence. You, who are smart enough and even more, curious, must be warned: This is not a novel; what you read here should generate much reflection. vii October 4, 2016 8:36 Prime Numbers, Friends Who Give Problems 9in x 6in b2394-fm page viii viii Prime Numbers, Friends Who Give Problems More than the facts described, the ways mathematicians think should captivate you. To read and to reflect: You must. To consult other books: You will (feel the urge). To reach a supreme intellectual enjoyment: I hope. P.R. October 4, 2016 8:36 Prime Numbers, Friends Who Give Problems 9in x 6in b2394-fm page ix Acknowledgments? Do I have to thank Eric and Paulo? My brain sweats in replying to their incessant and inquisitive questions, prompted by a thirst for knowledge. A thirst for a guaran´a would be easier to satisfy. Do I have to thank Eric and Paulo who convinced me to write this book? Yes, but only if my readers will feel gratified, which remains to be seen. Acknowledgments! Once the crime was perpetrated, it was examined by my virtual friend Reny Montandon. Chosen because he allies the best of Brazil and Switzerland, rigor and friendliness, in whatever order. Thanks! ix b2530 International Strategic Relations and China’s National Security: World at the Crossroads This page intentionally left blank b2530_FM.indd 6 01-Sep-16 11:03:06 AM October 4, 2016 8:36 Prime Numbers, Friends Who Give Problems 9in x 6in b2394-fm page xi Contents Preface vii Acknowledgments? Acknowledgments! ix 1. What are Prime Numbers? 1 2. Division is Harder than Multiplication 5 3. Another Paulo! Is a Dialogue of Three Possible? 17 4. How Natural Numbers are Made Out of Primes 21 5. Tell Me: Which is the Largest Prime? 29 6. Trying Hard to Find Primes 33 7. A Formula, A Formula, Please 39 8. Paulo Came with a Lasso 47 9. Beautiful Old Elementary Arithmetic 51 xi October 4, 2016 8:36 Prime Numbers, Friends Who Give Problems 9in x 6in b2394-fm page xii xii Prime Numbers, Friends Who Give Problems 10. The Old Man Still Knows 61 11. Can You Tell Me All About Congruences? 71 12. Homework Checked 85 13. Testing for Primality and Factorization 89 14. Fermat Numbers are Friendly. Are They Primes? 93 15. This World is Perfect 105 16. Unfriendly Numbers from a Friend of Fermat’s 111 17. Paying My Debt 123 18. Money and Primes 159 19. Secret Messages 175 20. New Numbers and Functions 185 21. Princeps Gauss 195 22. Gathering Forces 207 23. The After “Math” of Gauss 225 24. Primes After Dinner: Bad Dreams? 239 25. Primes in Arithmetic Progression 245 October 4, 2016 8:36 Prime Numbers, Friends Who Give Problems 9in x 6in b2394-fm page xiii Contents xiii 26. Selling Primes 261 27. The Great Prime Mysteries 273 28. Mysteries in Sequences: More But Not All 291 29. The End and the Beginning 311 Name Index 317 Subject Index 319 October 4, 2016 8:33 Prime Numbers, Friends Who Give Problems 9in x 6in b2394-ch01 page 1 1 What are Prime Numbers? Eric. Papa Paulo, I’ve heard about prime numbers. Some smart people talk about these numbers with voices full of mystery. I know nothing about them. Can you explain prime numbers to me? Papa Paulo (the friendly Papa Paulo). I will be happy to do it. I tell you, if I explain everything slowly, if you pay attention and understand all that I say, you will learn about prime numbers. Then it will be your turn to impress the people who appear to know it all. Let us agree that you’ll not be shy and that you will ask all the questions which come to your mind. If you have a question, my attitude is that you want to know the answer and I will provide it with patience and clarity. Eric. At least with you I don’t have to hide my ignorance. Papa Paulo. Ignorance calls for learning, so it is a privileged situation. Eric. After you learn, you are no longer ignorant, and the privileged position is lost. Why learn? Papa Paulo. It is a paradox. The more you learn, the more questions and problems appear requiring answers and solutions. If we continue our dialogue for a while, you’ll see how questions arise with any piece of knowledge acquired. Eric. Your game between ignorance and knowledge is nothing so new. What I want to know is: What is a prime number? 1 October 4, 2016 8:33 Prime Numbers, Friends Who Give Problems 9in x 6in b2394-ch01 page 2 2 Prime Numbers, Friends Who Give Problems Papa Paulo took an almost solemn air and said in an irrefutable voice: Papa Paulo. A prime number is a natural number greater than 1 whose only factors are 1 and the number itself. Eric. You promised to explain everything. What is a natural number? Papa Paulo. The natural numbers are 0, 1, 2, 3, 4,...,1093,..., 301258,.... Eric. And what are the factors of a number? Papa Paulo. I will give an example and you will understand right away: The factors of 10 are 1, 2, 5 and 10. These factors are also called divisors. Another example: the factors or divisors of 12 are 1, 2, 3, 4, 6 and 12. You see that 12 divided by 5 is more than 2 and less than 3, so 12 divided by 5 is not an integer. Therefore 5 is not a divisor (= factor) of 12. Eric. I understand what you said. I can see that every natural number n greater than 1 has the divisors 1 and n, and sometimes has other divisors. Papa Paulo. You are right — 1 and n are called the trivial divisors (= factors) of n.AsIsaid,thenumbern greater than 1 is called a prime number if it has only the trivial divisors. Eric. If one thinks just a little, making simple mental calculations, it is easy to see that 2, 3, 5, 7, 11, 13, 17, 19 are prime numbers. Papa Paulo. Yes, Eric. Let us see how one has to proceed to find out if 19 is a prime number. 19 19 • Does 2 divide 19? No, because 9 < 2 < 10, so 2 is not an integer. 19 • Does 3 divide 19? No, because 6 < 3 < 7.