Those Fascinating Numbers http://dx.doi.org/10.1090/mbk/064 Those Fascinating Numbers Jean-Marie De Koninck Translated by Jean-Marie De Koninck Providence, Rhode Island This work was originally published in French by Ellipses under the title: Ces nombres qui nous fascinent, c 2008 Edition´ Marketing S.A. The present translation was created under license for the American Mathematical Society and is published by permission. Translated by Jean-Marie De Koninck. Cover image by Jean-S´ebastien B´erub´e. 2000 Mathematics Subject Classification. Primary 11–00, 11A05, 11A25, 11A41, 11A51, 11K65, 11N05, 11N25, 11N37, 11N56. For additional information and updates on this book, visit www.ams.org/bookpages/mbk-64 Library of Congress Cataloging-in-Publication Data Koninck, J.-M. de, 1948- [Ces nombres qui nous fascinent. English] Those fascinating numbers / Jean-Marie De Koninck ; translated by Jean-Marie De Koninck. p. cm. Includes bibliographical references and index. 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Visit the AMS home page at http://www.ams.org/ 10987654321 141312111009 To my mother who showed me the way Contents Preface ..........................................................................ix Notations.......................................................................xiii Themainfunctions..............................................................xv Frequentlyusedtheoremsandconjectures .......................................xvii Thosefascinatingnumbers ........................................................1 Appendix: The prime numbers < 10 000 ........................................ 409 Bibliography . ..................................................................413 Index ..........................................................................425 vii Preface One day, in 1918, G.H. Hardy, the great English mathematician, took what he thought was an ordinary cab ride to go visit his young prot´eg´e at the hospital, the Indian mathematician S. Ramanujan. To break the ice, Hardy mentioned that the number 1729 on his taxicab was a rather dull number. Ramanujan immediately replied that, on the contrary, it was a very fascinating number since it was the smallest positive integer which could be written as the sum of two cubes in two distinct ways: 1729 = 123 +13 =103 +93. This anecdote certainly shows the genius of Ramanujan, but it also stirs our imagination. In some sense, it challenges us to find the remarkable characteristics of other numbers. This is precisely the task we undertake in this project. The reader will find here “famous” numbers such as 1729, Mersenne prime numbers (those prime numbers of the form 2p − 1, where p is itself a prime number) and perfect numbers (those numbers equal to the sum of their proper divisors); also “less famous” numbers, but no less fascinating, such as the following ones: • 37, the median value of the second prime factor of an integer; thus, the prob- ability that the second prime factor of an integer chosen at random is smaller 1 than 37 is approximately 2 ; • 277, the smallest prime number p which allows the sum 1 1 1 1 1 1 + + + + + ...+ 2 3 5 7 11 p (where the sum is running over all the prime numbers ≤ p) to exceed 2; • 378, the smallest prime number which is not a cube, but which can be written as the sum of the cubes of its prime factors: indeed, 378 = 2·33 ·7=23 +33 +73; • 480, possibly the largest number n such that n(n +1)...(n + 5) has exactly the same distinct prime factors as (n +1)(n +2)...(n + 6); indeed, 480 · 481 · ...· 485 = 28 · 32 · 52 · 7 · 112 · 13 · 23 · 37 · 97 · 241, 481 · 482 · ...· 486 = 24 · 36 · 5 · 7 · 112 · 13 · 23 · 37 · 97 · 241; ix x Preface • 736, the only three digit number abc such that abc = a+bc; indeed, 736 = 7+36; • 1 782, possibly the only integer n>1forwhich p = d; p|n d|n • 548 834, the only number > 1 which can be written as the sum of the sixth powers of its digits: indeed, 548 834 = 56 +46 +86 +86 +36 +46; • 11 859 210, the smallest number n for which P (n)4|n and P (n +1)4|(n +1), where P (n) stands for the largest prime factor of n (here P (n) = 11 and P (n + 1) = 19); the second smallest known number n satisfying this property is n = 632 127 050 601 113 666 430 (here P (n) = 2131 and P (n + 1) = 3691); • 89 460 294, the smallest number n (and the only one known) for which β(n)= β(n+1) = β(n+2), where β(n) stands for the sum of the distinct prime factors of n; • 305 635 357, the smallest composite number n for which σ(n +4)=σ(n)+4, where σ(n) stands for the sum of the divisors of n; √ • 612 220 032, the smallest number n>1 whose sum of digits is equal to 7 n; • 3 262 811 042, possibly the only number which can be written as the sum of the fourth powers of two prime numbers in two distinct ways: 3 262 811 042 = 74 + 2394 = 1574 + 2274; Ω(n)ω(n) • 3 569 485 920, the number n at which the expression reaches its max- n imum value, namely 2.97088..., where ω(n) stands for the number of distinct prime factors of n and Ω(n) stands for the number of prime factors of n counting their multiplicity. Various numbers also raise interesting issues. For instance, does there exist a number which is not the square of a prime number but which can be written as the sum of the squares of its prime factors ? Given an arbitrary integer k ≥ 2, does there exist a number n such that P (n)k|n and P (n +1)k|(n + 1) ? For each integer k ≥ 2 which is not a multiple of 3, can one always find a prime number whose sum of digits is equal to k ? These are some of the numerous open problems stated in this book, each of them standing for an enigma that will certainly feed the curiosity of the reader. Actually my hope for this book is to encourage many to explore more thoroughly some of the questions raised all along this book. There are currently several books whose main purpose is to exhibit interesting properties of numbers. This book is along the lines of these works but offers more features. For instance, one will find – mainly in the footnotes – short proofs of key results as well as statements of many new open problems. Finally, I would like to acknowledge all those who contributed to this manus- cript. With their precious input, suggestions and ideas, this project was expansive but enjoyable. Thanks to Jean-Lou De Carufel, Charles Cassidy, Zita De Kon- inck, Eric´ Doddridge, Nicolas Doyon, Eric´ Drolet, David Gr´egoire, Bernard Hodgson, Preface xi Imre K´atai, Patrick Letendre, Claude Levesque, Florian Luca, Michael Murphy, Erik Pronovost and J´erˆome Soucy. This edition is a translation of my French book Ces nombres qui nous fascinent published by Ellipses in 2008. Anyone enjoying this book is welcome to send me suggestions and ideas which could improve and enlighten this project. Jean-Marie De Koninck D´epartement de math´ematiques et de statistique Universit´eLaval Qu´ebec G1V 0A6 CANADA [email protected] Notations • In this book, unless indicated otherwise, by “number” we mean a “positive integer”. • The sequence p1,p2,p3,... stands for the sequence of prime numbers 2, 3, 5, th ...Thus pk stands for the k prime number. • Unless indicated otherwise, the letters p and q stand for prime numbers. • By a|b,wemeanthata divides b.Bya |b,wemeanthata does not divide b. Given a positive integer k,bypkn,wemeanthatpk|n but that pk+1 | n. • When we write f(p), we mean the infinite sum f(2) + f(3) + f(5) + f(7) + p ...+ f(p)+.... Similarly we write f(p) to indicate that the summation p≤x runs over all primes p ≤ x. • The expressions f(p)and f(p) are analogue to the ones mentioned just p p≤x above, except that this time they stand for products and not summations. • By f(d), we mean that the summation runs on all divisors d of n;by d|n f(p), we mean that the summation runs over all prime factors p of n.We p|n use the corresponding notations for the products, that is f(d)and f(p). d|n p|n • We denote by γ the Euler constant, which is defined by N 1 γ = lim − log N =0.5772156649 .... N→∞ n n=1 • Given an integer b ≥ 2andanumbern whose digits in base b are d1,d2,...,dr, we sometimes use the notation n =[d1,d2,...,dr]b. If the base is not men- tioned, it should be understood that we are working in base 10.