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.