6 Jean-Marie De Koninck

17

• the third Fermat prime (17 =

222

+1), the first two being 3 and 5: a number of

the form

22k

+ 1, where k is a non negative integer, is called a Fermat number

and is often denoted by Fk (see the number 70 525 124 609); if such a number

is prime, we say that it is a Fermat prime;

• the only prime number which is the sum of four consecutive prime numbers:

17 = 2 + 3 + 5 + 7;

• the exponent of the sixth Mersenne prime (131 071 =

217

− 1) (Cataldi, 1588);

• the smallest Stern number (see the number 137).

18

• the largest known number x for which there exist numbers n ≥ 3, y and q ≥ 2

such that (xn − 1)/(x − 1) = yq ; the only known solutions of this last equation

are given by

35

− 1

3 − 1

=

112,

74

− 1

7 − 1

=

202,

183

− 1

18 − 1

=

73

(see Y. Bugeaud, M. Mignotte & Y. Roy [26]);

• the fifth number n such that τ (n) = φ(n) (see the number 8).

19

• the smallest number r which has the property that each number can be written

as x1 4 +x2 4 +. . .+xr 4, where the xi’s are non negative integers (see the number 4);

• one of the nine known numbers k such that 11 . . . 1

k

is prime: the others8 are 2,

23, 317, 1 031, 49 081, 86 453, 109 297 and 270 343;

• the largest known prime pk such that ν(pk) :=

p≤pk

p + 1

p − 1

is an integer: here,

ν(p8) = ν(19) = 21;

• the exponent of the seventh Mersenne prime (524 287 =

219

−1) (Cataldi, 1588).

20

• the smallest solution of σ(n) = σ(n + 6); the sequence of numbers satisfying

this equation begins as follows: 20, 155, 182, 184, 203, 264, 621, 650, 702, 852,

893, 944, 1343, 1357, 2024, 2544, 2990, 4130, 4183, 4450, 5428, 5835, 6149,

6313, 6572, 8177, 8695, . . .

8Such a number k must be a prime, for if it was not, then we would have k = ab with 1 a ≤

b k, in which case

10ab−1

9

=

10ab−1

10b−1

·

10b−1

9

, the product of two numbers 1.