152 Jean-Marie De Koninck

2 903

• the first component of the largest known Wieferich pair {p, q}, here with

p = 2 903 and q = 18 787; we say that two prime numbers p and q form a

Wieferich prime pair if

pq−1

≡ 1 (mod

q2)

and

qp−1

≡ 1 (mod

p2);

if equa-

tion

xp

−

yq

= ±1 has a non trivial solution in integers x, y and in prime

numbers p, q larger than 3, then (p, q) satisfies the above pair of congruences,

a result established by P. Mihailescu en 2000; the only known Wieferich pairs

are {2, 1 093}, {3, 1 006 003}, {5, 1 645 333 507}, {83, 4 871}, {911, 318 917} and

{2 903, 18 787}.

2 915

• the fourth Lucas-Carmichael number (see the number 399).

2 970

• the fifth number which is not perfect or multi-perfect, but whose harmonic

mean is an integer (see the number 140).

2 971

• the rank of the largest known prime Fibonacci number (that is F2971) (see the

number 89; see also R.K. Guy [101], A3).

2 974

• the eighth solution of σ(n) = σ(n + 1) (see the number 206).

2 988

• the smallest abundant number n such that n + 2, n + 4, n + 6 and n + 8 are

also abundant (see the number 348).

3 001

• the second prime number of the form k · 10k + 1; the sequence of numbers

satisfying this property begins as follows: 11, 3 001, 9 000 000 001,

21 000 000 000 000 000 000 001, . . . (see the numbers 201 and 363).

3 024

• the largest number of the form n(n + 1)(n + 2) . . . (n + k − 1) for which there

are no prime numbers p

3

2

k dividing n(n + 1)(n + 2) . . . (n + k − 1); here

3 024 = 6 · 7 · 8 · 9 (see D. Hanson [106]);

• the largest solution n

109

of γ(n + 1) − γ(n) = 13: the others are 18 and 152

(see the number 98).