146 Jean-Marie De Koninck
2 378
the fifth solution y of the Fermat-Pell equation x2 2y2 = 1: here (x, y) =
(3363, 2378) (see the number 99).
2 383
the smallest prime factor of the Mersenne number
2397
1, whose complete
factorization is given by
2397
1 = 2383 · 6353 · 50023 · 53993 · 202471 · 5877983
·814132872808522587940886856743
·1234904213576000272542841146073
·6597485910270326519900042655193;
it is the smallest Mersenne number with exactly nine prime factors (see the
number 223 for the list of the smallest Mersenne numbers with a given number
of prime factors).
2 400
the smallest number n such that P (n)
4

n and P (n + 1)
4

n + 1: here
P (2400) = P (25 · 3 · 52) = 5 6.99 . . . =
4

2400 and P (2401) = P (74) =
7 =
4

2401; the sequence of numbers satisfying this property begins as fol-
lows: 2400, 4374, 123200, 165375, 194480, 228095, 282624, 328509, 336140,
601425, . . . : one can prove that this sequence is
infinite139;
if nk stands for the
smallest number n such that max(P (n), P (n + 1)) (n +
1)1/k,
then n2 = 8,
n3 = n4 = 2 400, n5 = 5 909 560 and n6 = 1 611 308 699.
2 401
(=74)
the smallest number n 1 whose sum of digits is equal to
4

n: the others are
234 256, 390 625, 614 656 and 1 679 616;
the third perfect square which is also a Smith number (see the number 22):
2401 = 74 and 2 + 4 + 1 = 7; the smallest two are 361 and 1 600.
2 418
the smallest number n such that
Li2(x)
ex
log x
Li
x
e
for all x n (see B.C.
Berndt [21] as well as the number 38 358 837 677);
the seventh number n such that σ(φ(n)) = n (see the number 744).
139Indeed, A. Balog & Z. Ruzsa [14] have proved that, for each ε 0, the set {n : max(P (n), P (n+
1))
nε}
is of positive density, implying in particular that the set of numbers n such that P (n)
4

n and P (n + 1) 4

n + 1 is of positive density.
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