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
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