356 Jean-Marie De Koninck
63 927 525 375 (= 33 · 53 · 77 · 23)
most likely the number n at which Q2(n) := min(λ(n), λ(n + 1)) reaches its
maximum value, namely 2.65551: we have
n = 63 927 525 375 =
33
·
53
·
77
· 23 with λ(n) 3.19419
n + 1 = 63 927 525 376 =
213
·
114
· 13 · 41 with λ(n + 1) 2.65551
66 433 720 320 (=
213
·
33
· 5 · 11 · 43 · 127)
the ninth 4-perfect number (see the number 30 240).
68 719 476 736
the largest number n whose sum of digits is equal to
6

n (see the number
34 012 224).
68 899 596 497
the fourth number having two representations as the sum of two co-prime fourth
powers:
68 899 596 497 =
5024
+
2714
=
4974
+
2984
(see the number 635 318 657).
70 525 124 609
the smallest prime factor of the Fermat number F19 =
2219
+ 1; the table below
reveals the status of the factorizations204 of the Fermat numbers Fn =
22n
+ 1
for 0 n 24 as of May 2009.
204Several mathematicians believe that the largest Fermat prime is F4 = 65 537. Hardy & Wright
use in their book [107] a probabilistic argument to show that the number of Fermat primes is
probably finite. Here are the highlights: it follows from the Prime Number Theorem that the
probability that a number m is prime is approximately 1/ log m, implying that the expectation for
the number of Fermat primes is

n=1
1
log(22n
+ 1)

n=1
1
2n
log 2
=
1
log 2
= O(1),
that is a finite number.
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