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Those Fascinating

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 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. ISBN 978-0-8218-4807-4 (alk. paper) 1. theory. I. Title. QA241.K686 2009 512.7–dc22 2009012806

Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made by e-mail to [email protected]. c 2009 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. 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 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, numbers (those prime numbers of the form 2p − 1, where p is itself a ) and perfect numbers (those numbers equal to the sum of their proper ); 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 , 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 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 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 . • 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.

xiii xiv Notations

• The factorization of a number usually appears in the form α α α 1 · 2 · · r n = q1 q2 ... qr ,

where q1

in this case, the expression P136 stands for a (known) 136 digit prime number which there is no need to write at length, since it can be obtained explicitly by 7 simply dividing 122 + 1 by 257. Another possible situation could occur, as for instance: 28 12 + 1 = 8253953 · 295278642689 · C258;

in this case, the expression C258 stands for a composite 258 digit number for which no non trivial factorization is known. • In order to compare the size of certain expressions in the neighborhood of infin- ity, we use various notations, some of which have been introduced by Landau, namely O(...)ando(...). Hence, given two functions f and g defined on [a, ∞) (where a ≥ 0), we write:

(i) f(x)=O(g(x)) if there exist two constants M>0andx0 for which |f(x)|

(ii) f(x)=o(g(x)) if, for each ε>0, there exists a constant x0 = x0(ε)such that |f(x)| <εg(x) for all x ≥ x0;thuswehave 1 = o(1), sin x = o(x), log x = o(x),x4 = o(ex); x

(iii) f(x)=Ω(g(x)) if there exist two constants M>0andx0 such that |f(x)| >M|g(x)| for all x ≥ x0; instead of writing f(x)=Ω(g(x)), we sometimes write f(x) g(x); thus we have √ √ x =Ω( x), x =Ω(logx),ex =Ω(x4),xex ex;

f(x) (iv) f(x) ∼ g(x) to mean that lim =1;thus,asx →∞,wehave x→∞ g(x) 1 sin 1/x ∼ 0, ∼ 1,x2 + x ∼ x2. x 1/x

(v) f(x) ≈ g(x)tomeanthatwehavebothf(x)  g(x)andg(x)  f(x). The Main Functions

[x], the largest integer ≤ x B(n)= αp, the sum of the prime factors of n with multiplicity pαn α B1(n)= p , the sum of the largest prime powers dividing n pαn P (n)=max{p : p|n}, the largest prime factor of the number n ≥ 2 p(n)=min{p : p|n}, the smallest prime factor of the number n ≥ 2 β(n)= p, the sum of the distinct prime factors of n p|n β∗(n)= p = β(n) − P (n), the sum of the prime factors of n except for the p|n p

xv xvi The Main Functions

σ(n)= d, the sum of the divisors of n d|n σ∗(n)= d, the sum of the unitary divisors of n d|n, (d,n/d)=1 k th σk(n)= d , the sum of the k powers of the divisors of n d|n σI (n)= d, the sum of the odd divisors of n d|n,d odd τ(n)= 1, the number of divisors of n d|n ω(n)= 1, the number of distinct prime factors of n p|n Ω(n)= α, the number of prime factors of n counting their multiplicity α p n πk(x)= 1, the number of numbers n ≤ x such that ω(n)=k n≤x ω(n)=k µ(n), the Moebius function defined by ⎧ ⎨ 1ifn =1, p2|n p, µ(n)=⎩ 0iffor a certain prime (−1)ω(n) otherwise

Ω(n) λ0(n)=(−1) , the Liouville function log n λ(n)= , the index of composition of the number n ≥ 2 log γ(n) ι(n)= min |n − m|, the index of isolation of the number n ≥ 2, that is the dis- 1≤m= n P (m)≤P (n) tance to the nearest integer whose largest prime factor does not exceed that of n

n 1 ξ(n)= gcd(i, n) i=1 Frequently used Theorems and Conjectures

Fermat’s Little Theorem Let p be a prime number. Given any positive integer a co-prime with p, then ap−1 ≡ 1(modp).

The Prime Number Theorem As x →∞, π(x)=(1+o(1))Li(x), with

x x x x x Li(x)= + +2! + ...+(r − 1)! r + O r , log x log2 x log3 x log x log +1 x where r is any given fixed integer.

The Chinese Remainder Theorem

Let m1,m2,...,mr be co-prime integers, and let a1,a2,...,ar be arbi- trary integers. Then the system of congruences ⎧ ⎪ n ≡ a1 (mod m1) ⎨⎪ n ≡ a2 (mod m2) . ⎪ . ⎩⎪ . n ≡ ar (mod mr) has a solution given by r m n0 = biai, mi i=1

where m = m1m2 ...mr and where each bi is the solution of the congru- ence (m/mi)bi ≡ 1(modmi).

xvii xviii Frequently used Theorems and Conjectures

Hypothesis H (or Schinzel Hypothesis)

Let  ≥ 1 and f1(x),...,f(x) be irreducible with integer coefficients and positive leading coefficient. Assume that there are no integers > 1 dividing the product f1(n) ...f(n) for all positive integers n. Then there exist infinitely many positive integers m such that all numbers f1(m),...,f(m) are primes. This conjecture was first stated in 1958 by A. Schinzel and W. Sierpinski [181].

The Let ε>0. There exists a positive constant M = M(ε) such that, given any co-prime integers a, b, c verifying the conditions 0

An equivalent statement is the following: Let ε>0. There exists a positive constant M = M(ε) such that, given any co-prime integers a, b, c satisfying a + b = c,wehave

⎛ ⎞1+ε max{|a|, |b|, |c|}

The abc Conjecture was first stated in 1985 by D.W. Masser and J. Oesterl´e.

Those Fascinating Numbers 413

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425 426 Jean-Marie De Koninck

Ramanujan, 1 729 , 2047 Ruth-Aaron, 714 Sub-factorial function, 148 349 Sastry, 183 Syracuse Conjecture, 41 self contained, 293 self described, 6 210 001 000 von Sterneck Conjecture, 7 725 038 629 self replicating, 954 Sierpinski, 78 557 Waring Problem, 4 Smith, 22 pair, 2903 S-perfect, 126 squarefull, 23 square pyramidal, 208 335 star, 121 Stern, 137 sublime, 12 superabundant, 110 880 super-prime, 73 939 133 symmetric, 35853 tetrahedral, 10 triangular, 3 trimorphic, 491 tri-perfect, 120 vampire, 1260 voracious, 1807 unitary hyperperfect, 288 unitary perfect, 6 Wieferich, 16 547 533 489 305 Wilson, 5 971 Woodall, 115

Palindrome (see palindrome number) Prime , 1905 Fermat, 17 Fibonacci pseudoprime, 323 irregular, 59 Lucas, 613 Mersenne, 3 regular (see irregular prime) twin, 35 Wieferich, 1093 Wilson, 5 Wolstenholme, 16 843 Prime Number Theorem, page xvii Pseudoprime in base a,91

Riemann Hypothesis, 5041 , 177

Schinzel Hypothesis (see Hypothesis H)