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.