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 1 for which

p|n

p =

d|n

d;

• 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;

• 3 569 485 920, the number n at which the expression

Ω(n)ω(n)

n

reaches its max-

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,