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,
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