46 Jean-Marie De Koninck

144 (= 24 · 32)

• the only Fibonacci number 1 which is a perfect square (see J.H.E. Cohn

[34])58;

• the smallest number whose fifth power can be written as the sum of four non

zero fifth powers: 1445 = 275 + 845 + 1105 + 1335 (R.K. Guy [101], D1); for a

similar question with fourth powers, see the number 422 481;

• the only solution n 1012 of σ(n) = 2n + 115;

• the smallest number n having at least two distinct prime factors and such

that B1(n) = β(n)2: here 24 + 32 = (2 + 3)2; the sequence of numbers sa-

tisfying this property begins as follows: 144, 1 568, 3 159, 5 346, 8 064, 56 000,

123 008, 380 000, 536 544, 570 752, 584 064, 729 088, 2 267 136, 8 258 048 . . . ;

it is clear that if there exist infinitely many Mersenne primes, then equation

B1(n) = β(n)2 has infinitely many solutions59; nevertheless, J.M. De Koninck

& F. Luca [59] proved that, without any conditions, this sequence is infinite60;

• the smallest number 2 which is equal to the product of the factorials of its

digits in base 5: 144 = [1, 0, 3, 4]5 = 1! · 0! · 3! · 4!; the only known numbers

satisfying this property are 1, 2, 144, 1 728, 47 775 744 and

27 134 923 845 424 074 797 548 044 288 (see the number 17 280 for the table of

the smallest numbers with this property in a given base).

145

• the smallest number 2 which is equal to the sum of the factorials of its digits

(145 = 1!+4!+5!); the only other number 2 satisfying this property is 40 585

(L. Janes, 1964)61.

147

• the number of solutions 2 ≤ x1 ≤ x2 ≤ x3 ≤ x4 ≤ x5 of

5

i=1

1

xi

= 1 (see

R.K. Guy [101], D11);

58Recently

(in 2006), Y. Bugeaud, M. Mignotte & S. Siksek [27] proved that 1, 8 and 144 are the

only Fibonacci numbers which are powers.

59Indeed,

one can easily check that if p =

2α−2

− 1 is prime, then n =

2α

·

p2

is a solution of

B1(n) = β(n)2.

60In

fact they proved more, namely that if A(x) stands for the number of n ≤ x such that equation

B1(n) = β(n)2 is verified, then the following bounds hold as x → ∞:

x

exp (2 34/3 + o(1)) log x log log x

≤ A(x) ≤

x

exp (

1

√

2

+ o(1)) log x log log x

.

61If n = [d1, d2, . . . , dr ] is a number satisfying this property, then 10r−1 ≤ n ≤ r · 9!, in which

case we must have r ≤ 8, and this is why it is easy to verify using a computer that 145 and 40 585

are the only two numbers with this property.