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.