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.
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