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The Little Book of Bigger Primes Second Edition Paulo Ribenboim The Little Book of Bigger Primes Second Edition PauloRibenboim DepartmentofMathematicsandStatistics Queen’sUniversity Kingston,ONK7L3N6 Canada MathematicsSubjectClassification(2000):11A41,11B39,11A51 LibraryofCongressCataloging-in-PublicationData Ribenboim,Paulo. Thelittlebookofbiggerprimes/PauloRibenboim. p.cm. Includesbibliographicalreferencesandindex. ISBN0-387-20169-6(alk.paper) 1.Numbers,Prime. I.Title. QA246.R4732004 512.7′23—dc22 2003066220 ISBN0-387-20169-6 Printedonacid-freepaper. Seefirstedition1991PauloRibenboim. 2004Springer-VerlagNewYork,Inc. All rights reserved. This work may not be translated or copied in whole or in part without the writtenpermissionofthepublisher(Springer-VerlagNewYork,Inc.,175FifthAvenue,NewYork, NY10010,USA),exceptforbriefexcerptsinconnectionwithreviewsorscholarlyanalysis.Use inconnection withany formof informationstorageand retrieval,electronic adaptation,computer software,orbysimilarordissimilarmethodologynowknownorhereafterdevelopedisforbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if theyarenotidentifiedassuch,isnottobetakenasanexpressionofopinionastowhetherornot theyaresubjecttoproprietaryrights. PrintedintheUnitedStatesofAmerica. (EB) 9 8 7 6 5 4 3 2 1 SPIN10940969 Springer-VerlagisapartofSpringerScience+BusinessMedia springeronline.com Nel mezzo del cammin di nostra vita mi ritrovai per una selva oscura che la diritta via era smarrita Dante Alighieri, L’Inferno This page intentionally left blank Preface This book could have been called “Selections from the New Book of Prime Number Records.” However, I preferred the title which pro- pelled you on the first place to open it, and perhaps (so I hope) to buy it! But the book is not very different from its parent. Like a bonsai, which has all the main characteristics of the full-sized tree, this pa- perback should exert the same fatal attraction. I wish it to be as dangerous as the other one. I wish 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. This second edition is still a little book, but the primes have “grown bigger”. An irrepressible activity of computation special- ists has pushed records to levels previously unthinkable. These en- deavours generated—or were possible by—new algorithms and great advances in programming techniques and hardware developments. A fruitful interplay for the intended aim, to produce large, awesome numbers. Theseupdatedrecordsarereported;theyarelikeasnapshottaken May 2003. However, only limited progress was made in the theoret- ical results. They are explained in the appropriate place. The old viii Preface classicalproblemsremainopenandcontinuedefyingourgreatminds. Withaninnersmile:“Ifyousolveme,you’llbecomeidle”.Notknow- ing that we, mathematicians, invent more problems than we can solve. Idle, we shall not be. Paulo Ribenboim Acknowledgements First and foremost, I wish to express my gratitude to Wilfrid Keller. He spent uncountable hours working on this book, informing me of thenewestrecords,discussingmytexttogreatdepths,withjudicious comments.Healsotookupthearduoustaskofpreparingthecamera- ready copy. Like the proud Buenos Aires tailor who was not happy until the jacket fitted to perfection. I have also obtained great support from many colleagues who explained patiently their results. As a consequence, their names are included in the text. Finally, Chris Caldwell maintains a rich, well selected, informative website on prime numbers, which I consulted often with great profit.