164 Jean-Marie De Koninck
4 871
the smallest prime number p such that
83p−1
1 (mod
p2):
the only prime
numbers p
232
satisfying this congruence are 4 871, 13 691 and 315 746 063
(see Ribenboim [169], p. 347);
the second component of the fourth Wieferich prime pair {p, q}, here with
p = 83 and q = 4 871 (see the number 2 903).
4 900
the only perfect square
n2
1 such that
n2
=
12
+
22
+ . . . +
k2
for a certain number k: here k = 24 (see the number 70).
4 913 (=
173)
the second number n whose sum of digits is equal to
3

n (see the number 512).
4 933
the number of digits in the decimal expansion of the Fermat number
2214
+ 1.
4 991
the fifth Lucas-Carmichael number (see the number 399).
5 002
the sixth solution of σ(φ(n)) = σ(n) (see the number 87).
5 040
the value of 7! ;
the 19th highly composite number (see the number 180).
5 041
the smallest number n such that, if the Riemann Hypothesis (according to which
all complex zeros of the Riemann Zeta function have their real part equal to
1
2
)
is true, inequality σ(m)/m

log log m (where γ is Euler’s constant) holds
for all m n (G. Robin
[176])148;
the fifth powerful number which can be written as the sum of two co-prime
3-powerful numbers = 1: 5 041 = 128 + 4 913, that is
712
=
27
+
173
(see the
number 841).
148In
that paper, Robin proves that the Riemann Hypothesis is equivalent to the fact that inequality
σ(n) n log log n is true for all n 5 041.
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