34 Jean-Marie De Koninck
102
the smallest number n such that
n64
+1 is prime; if we denote by nk the smallest
number n 2 such that
n2k
+ 1
is40
prime, then n1 = n2 = n3 = n4 = 2,
n5 = 30, n6 = 102, n7 = 130, n8 = 278, n9 = 46, n10 = 824 and n11 = 150;
the smallest solution of σ2(n) = σ2(n + 17).
103
the smallest prime number p such that ω(p + 1) = 2 and ω(p + 2) = 3 (see also
the number 64);
the largest prime number p
232
such that
43p−1
1 (mod
p2):
the only
other prime number p
232
satisfying this congruence is p = 5 (see Ribenboim
[169], p. 347).
104
the smallest composite number n such that σ(n + 6) = σ(n) + 6: the only
numbers n
109
satisfying this equation are 104, 147, 596, 1 415, 4 850, 5 337,
370 047, 1 630 622, 35 020 303 and 120 221 396;
the fourth solution of φ(n) = φ(n + 1) (see the number 15).
105
the smallest number n such that φ(n) φ(n + 1) φ(n + 2): here 48
52 106; n = 1484 is the smallest number n such that φ(n) φ(n + 1)
φ(n + 2) φ(n + 3); Nicolas Doyon claims (private communication) that it is
possible to prove that, for all integers k 2, if a1, a2, . . . , ak is any permutation
of the integers 1, 2, . . . , k, then there exist infinitely many integers n such that
φ(n + a1) φ(n + a2) . . . φ(n + ak);
the largest number n such that all odd numbers n which are co-prime with
n are prime, namely the numbers 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53,
59, 61, 67, 71, 73, 79, 83, 89, 97, 101 and 103 (see L. Cseh [40]);
the largest known number n for which n
2k
is prime for each number k such
that
2k
n: in the case n = 105, we indeed have that 103, 101, 97, 89, 73 and
41 are prime numbers;
the number of Carmichael numbers
107
(see the number 646).
40It
is not known if the sequence (nk)k≥1 is well defined: indeed, it is not obvious that for each
number k, there exists n such that
n2k
+ 1 is prime.
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