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 =

·
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.
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