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
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