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)