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.
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