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).
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