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

than 37 is approximately

1

2

;

• 277, the smallest prime number p which allows the sum

1

2

+

1

3

+

1

5

+

1

7

+

1

11

+ . . . +

1

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