Those Fascinating Numbers 9

• the second number n (and possibly the largest) such that n3 + 1 is a powerful

number (a number is said to be powerful (or squarefull) if p|n implies that p2|n);

the smallest number satisfying this property10 is n = 2;

• one of the nine known numbers k such that 11 . . . 1

k

is prime (see the number 19);

• the largest number which cannot be written as the sum of two non square-free

numbers (see the number 933 for a more general problem).

24

• the only number n 1 such that

12 +22

+. .

.+n2

is a perfect square (E. Lucas,

1873) (see the number 70);

• the smallest number m such that equation σ(x) = m

has11

exactly three solu-

tions, namely 14, 15 and 23;

• the sixth number n such that τ (n) = φ(n) (see the number 8);

• the smallest solution of σ2(n) = σ2(n + 2) (see the number 1 079);

• the smallest number with at least two digits, having all its digits different from

1 and 0, and whose sum of digits, as well as the product of its digits, divides

n: the sequence of numbers satisfying this property begins as follows: 24, 36,

224, 432, 624, 735, 2232, 3276, 4224, 6624, 23328, 32832, 33264, 34272, 34992,

42336, 42624, 43632, 73332, 82944, 83232, 92232, 93744, . . .

25

• the only odd perfect square = 1 which is not the sum of three perfect squares

= 0 (see E. Grosswald [99], Chapter 3);

• the only perfect square which when increased by 2 yields a cube:

52

+ 2 =

33;

• the number of prime numbers 100.

10One

can easily prove that if the abc Conjecture is true, then there is only a finite number of

numbers satisfying this property.

11K. Ford & S. Konyagin [82] proved a conjecture of Sierpinski according to which, for each k ≥ 2,

there exists a number m such that equation σ(x) = m has exactly k solutions x. Later, K. Ford

[83] proved that this result is also valid for the Euler φ function; moreover, this time, the proof

also reveals that for each k ≥ 2, there exist infinitely many m’s such that φ(x) = m has exactly k

solutions in x.