10 Jean-Marie De Koninck
26
the smallest number which is not a palindrome, but whose square is a palin-
drome; a palindrome is a number which reads the same way from the left as
from the right; the first ten numbers n satisfying this property are listed below:
n n2
26 676
264 69696
307 94249
836 698896
2285 5221225
n n2
2636 6948496
22865 522808225
24846 617323716
30693 942060249
798644 637832238736
the smallest solution of σ(n) = σ(n + 15); it is mentioned in R.K. Guy [101],
B13, that Mientka & Vogt could only find two solutions to this equation,
namely 26 and 62: there are at least seven others, namely 20 840 574, 25 741 470,
60 765 690, 102 435 795, 277 471 467, 361 466 454 and 464 465 910.
27
the smallest number n such that n and n + 1 each have exactly three prime
factors counting their multiplicity: 27 = 33 and 28 = 22 · 7 (see the number 135
for the general problem with k prime factors instead of only three).
28
the only even perfect number of the form
an
+
bn,
with n 2 and (a, b) = 1:
in fact, 28 = 13 + 33 (T.N. Sinha [187]).
29
the smallest prime number p 2 such that
41p−1
1 (mod
p2):
the only
prime numbers p 232 satisfying this congruence are 2, 29, 1 025 273 and
138 200 401 (see Ribenboim [169], p. 347).
30
the smallest Giuga number: we say that a composite number n is a Giuga
number if
p|n
1
p

p|n
1
p
is a positive integer: if we could find a number n which
is both a Giuga number and a Carmichael number (which is most unlikely!),
we would then have found a composite number n satisfying the congruence
1n−1
+
2n−1
+ . . . + (n
1)n−1
−1 (mod n)
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