Those Fascinating Numbers 35
107
the exponent of the
11th
Mersenne prime
2107
1 (Fauquembergue, 1913);
the largest known number k such that the decimal expansion of
2k
does not
contain the digit 4; indeed, using a computer, one can verify that
2107
= 162259276829213363391578010288128
does not contain the digit 4, while each number
2k,
for k = 108, 109, . . . , 3000,
contains it (see the number 71).
108
the largest known number k such that the decimal expansion of
2k
does not
contain the digit 9; indeed, using a computer, one can verify that
2108
= 324518553658426726783156020576256
does not contain the digit 9, while each number 2k, for k = 109, 110, . . . , 3000,
contains it (see the number 71);
the smallest number which can be written as the sum of a cube and a square
in two distinct ways: 108 = 23 + 102 = 33 + 92;
the third solution of φ(n) = γ(n)2: the only solutions41 of this equation are 1,
8, 108, 250, 6 174 and 41 154 (see also the number 1 782).
109
the sixth number n such that (12n 1)/11 is prime: the sequence of numbers
satisfying this property begins as follows: 2, 3, 5, 19, 97, 109, 317, 353, 701,
9739, . . . (H. Dubner [71])42 ;
the smallest three digit number = 100, 200, 300, and whose square contains only
two distinct digits: the numbers satisfying this property are 109, 173, 212, 235
and 264 (for instance,
1092
= 11881): Sin Hitotumatu has asked if, besides the
numbers
10n,
2 ·
10n
and 3 ·
10n,
there exist only a finite number of squares
containing only two distinct digits (see the numbers 3 114 and 81 619).
41The
proof of this statement is the object of Problem #745 in the book of J.M. De Koninck &
A. Mercier [62].
42It
is clear that such a number n must be prime, since otherwise n = ab, with 1 a b n, in
which case
12n 1
11
=
12ab 1
11
=
12a 1
11
(
12a(b−1)
+
12a(b−2)
+ . . . +
12a
+ 1
)
,
the product of two numbers 1.
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