30 Jean-Marie De Koninck

19 134 702 400 093 278 081 449 423 917 . . . .; if we let Fn stand for the nth Fi-

bonacci number, and if Fk is prime, then one can show that k is either 4 or a

prime number: this helps in establishing that the set of numbers k 100 000

for which Fk is prime is {3, 4, 5, 7, 11, 13, 17, 23, 29, 43, 47, 83, 131, 137, 359,

431, 433, 449, 509, 569, 571, 2971, 4723, 5387, 9311, 9677, 14431, 25561, 30757,

35999, 37511, 50833, 81839}; no one has yet proved that there are infinitely

many prime Fibonacci numbers;

• the fourth prime number with at least two digits and whose digits are consec-

utive integers (see the number 67).

90

• the only number

109

which is not perfect but which is equal to the sum of

its proper deficient divisors: 90 = 1 + 2 + 3 + 5 + 9 + 10 + 15 + 45; a number n

is said to be deficient if σ(n) 2n;

• the largest solution of φ(x) = 24, the others being 35, 39, 45, 52, 56, 70, 72, 78

and 84;

• the third unitary perfect number (see the number 6);

• the only number which is equal to the sum of its digits added to the sum of the

squares of its digits.

91

• the smallest pseudoprime in base 3: given a number a 1, a composite number

n a is said to be pseudoprime in base a if

an−1

≡ 1 (mod n); the ten smallest

pseudoprimes in base 3 are 91, 121, 286, 671, 703, 949, 1105, 1541, 1729 and

1891;

• the largest known number k such that the decimal expansion of

2k

does not

contain the digit 1; indeed, using a computer, one can verify that

291

= 2475880078570760549798248448

does not contain the digit 1, while each number

2k,

for k = 92, 93, . . . , 3000,

contains it (see the number 71);

• the rank of the prime number which appears the most often as the tenth prime

factor of an integer : p91 = 467 (see the number 199).

92

• the number of integer zeros of the function M(x) :=

n≤x

µ(n) located in the

interval [1, 1000]: if we denote by mk the number of times that the function

M(x) crosses the “x” axis in the interval [1, 10k], then the sequence (mk)k≥1