108 Jean-Marie De Koninck

and112 most likely,

n8 = 1 770 019 255 373 287 038 727 484 868 192 109 228 823,

with f(n8 + i) = 6 for i = 1, 2, . . . , 8.

844

• the smallest number n such that n, n + 1, n + 2, n + 3, n + 4 are all divisible by

a square 1: here 844 =

22

· 211, 845 = 5 ·

132,

846 = 2 ·

32

· 47, 847 = 7 ·

112,

848 =

24

· 53 (see the number 242).

854

• the largest number n such that if A and B stand respectively for the set of digits

of n and of

n2,

then A∪B = {1, 2, 3, . . . , 9} and A∩B = ∅: here

8542

= 729 316;

the only other number satisfying this property is 567.

857

• the smallest prime number q such that

1

11

+

1

13

+

1

17

+ . . . +

1

q

1 (see the

number

347)113.

858

• the second Giuga number (see the number 30).

860

• the smallest solution of σ2(n) = σ2(n + 8): the list of numbers satisfying this

equation begins as follows: 860, 4316, 3790076, 5448956, 8921084, . . .

114

112J.M.

De Koninck & F. Luca [58] proved that the number nk exists for each k ≥ 2 while also

providing lower and upper bounds for the number nk.

113Given

a large prime pk, one can estimate the size of the smallest prime qk such that

pk ≤p≤qk

1

p

is the nearest to 1. To do so, we use the formula

∑

p≤x

1

p

= log log x + c + O

1

log2

x

, yielding

pk ≤p≤qk

1

p

= log log qk − log log pk + O

1

log2

pk

≈ 1,

which occurs when log

(

log qk

log pk

)

≈ log e, that is log qk ≈ log

pe

k

, meaning that qk ≈

pe

k

.

114One

can prove that if Hypothesis H is true (see its statement on page xvii), then this equation

has infinitely many solutions.