Those Fascinating Numbers 15

39

• the smallest number n such that 2n − 7 is prime (a question raised by Erd˝os

in 1956): using a computer, one obtains that the other numbers n 20 000

satisfying this property are 715, 1983, 2319, 2499, 3775 and 12819.19

40

• the smallest solution of equation

σ(n)

n

=

9

4

; the sequence of numbers satisfying

this equation begins as follows: 40, 224, 174 592, 492 101 632, . . .

41

• the largest odd number which is not the sum of four non zero squares (Sierpinski

[185], p. 404);

• the largest number n such that the polynomial

x2

+ x + n is prime for each of

the numbers x = 0, 1, 2, . . . , n − 2; the other numbers n satisfying this property

are n = 1, 2, 3, 5, 11 and 17 (see D. Fendel & R.A. Mollin [80]);

• the integer part of the number γ0 = 41.677647, that is the conjectured value

of lim supn→∞

σ∞(n)

log n

, where σ∞(n) stands for the smallest number k such that

f

k(n)

= 1, where

f(n) =

⎧

⎨

⎩

1 if n = 1,

n/2 if n is even,

3n + 1 if n is odd,

f

1(n)

= f(n), f

2(n)

= f(f(n)), f

3(n)

= f(f

2(n))

and so on; according to the

Syracuse conjecture (also called the 3x + 1 problem ), this sequence inevitably

reaches the number 1 (see J.C. Lagarias & A. Weiss [120]);

• the smallest prime number of the form (x4 +y4)/2: here 41 = (34 +14)/2: there

exist only eight prime numbers 10 000 satisfying20 this property, namely 41,

313, 353, 1201, 3593, 4481, 7321 and 8521.

42

• the smallest number n 1 such that σ2(n) is a perfect square: the sequence

of numbers satisfying this property begins as follows: 42, 246, 287, 728, 1434,

1673, 1880, 4264, 6237, 9799, 9855,. . . ;

• the fifth Catalan number (see the number 14);

19While

performing this search, one can ignore all even numbers n, all numbers n ≡ 1 (mod 4),

all numbers n ≡ 7 (mod 10) as well as all numbers n ≡ 11 (mod 12), since in these four cases,

we obtain respectively that 3, 5, 11 and 13 divide 2n − 7.

20It

is interesting to mention that there exist other forms which generate infinitely many prime

numbers: it is the case, for instance, for the form x2 + y4 as it was proved by J. Friedlander &

H. Iwaniec [85], as well as for the form

x3

+

2y3

as was established by D.R. Heath-Brown [110] (see

also for that matter the number 3 391).