Those Fascinating Numbers 5

14

• the smallest solution5 of σ(n) = σ(n + 1); the sequence of numbers satisfying

this equation begins as follows: 14, 206, 957, 1334, 1364, 1634, 2685, 2974,

4364, 14841, 18873, 19358, 20145, 24957, 33998, 36566, 42818, 56564, 64665,

74918, 79826, 79833, 84134, 92685, . . . ;

• the fourth Catalan number: Catalan

numbers6

are the numbers of the form

1

n + 1

2n

n

.

15

• the third smallest solution of φ(n) = φ(n + 1); the sequence of numbers satis-

fying this equation begins as follows: 1, 3, 15, 104, 164, 194, 255, 495, 584, 975,

2204, 2625, 2834, 3255, 3705, 5186, 5187, 10604, 11715, 13365, 18315, 22935,

25545, 32864, 38804, 39524, 46215, 48704, 49215, 49335, 56864, 57584, 57645,

64004, 65535, 73124, . . . : R. Baillie [10] found 391 solutions n 2 · 108 ;7

• one of the three numbers n such that the polynomial x5 −x±n can be factored:

the other two are n = 22 440 and n = 2 759 640: here we have x5 − x ± 15 =

(x2

± x +

3)(x3

∓

x2

− 2x ± 5); see the number 22 440;

• the value of the sum of the elements of a diagonal, a row or a column of a

3 × 3 magic square: for a k × k magic square with k ≥ 3, the common value

is

k(k2

+ 1)/2, which gives place to the sequence whose first terms are 15, 34,

65, 111, 175, 260, 369, 505, 671, 870, 1105, 1379, 1695, . . . (see Sierpinski [185],

p. 434).

16

• the only number n for which there exist two distinct integers a and b such that

n =

ab

=

ba:

here a = 2, b = 4;

• the smallest perfect square for which there exists another perfect square with

the same sum of divisors: σ(16) = σ(25) = 31.

5This

sequence of numbers is probably infinite, but no one has yet proved it.

6Catalan

numbers appear when one wants to find in how many ways it is possible to partition a

convex polygon in triangles by drawing some of its diagonals.

7P.

Erd˝ os, C. Pomerance & A. S´ ark¨ ozy [79] provide a heuristic argument which suggests that, for

each fixed ε 0, equation φ(n) = φ(n + 1) has at least

x1−ε

solutions n ≤ x. However, A. Schinzel

[180] believes that it may be possible that equation φ(n) = φ(n + 1) has only a finite number of

solutions, but he conjectures that for each even integer k ≥ 2, equation φ(n) = φ(n+k) has infinitely

many solutions. Let us add that equation φ(n) = φ(n + k) has very few solutions when k is odd

and divisible by 3; thus by letting Ek be the set of solutions n 108 of φ(n) = φ(n + k), we have

E3 = {3, 5}, E9 = {9, 15}, E15 = {13, 15, 17, 21}, E21 = {21, 35} and E27 = {27, 45, 55}, while the

cardinality of each of the other sets Ek , 1 ≤ k ≤ 32, is at least 12.