Those Fascinating Numbers 3

• the only triangular number 1 whose square is also a triangular number

(W. Ljunggren, 1946): here 62 = 36 = 1 + 2 + 3 + . . . + 8.

7

• one of the two prime numbers (the other one is 5) which appears most often as

the third prime factor of an integer (1 time in 30);

• the second Mersenne prime: 7 =

23

− 1.

8

• the third number n such that τ (n) = φ(n): the only numbers satisfying this

equation are 1, 3, 8, 10, 18, 24 and 30;

• the number of twin prime pairs 100 (see the number 1 224).

9

• the only square which follows2 a power of 2: 23 + 1 = 32;

• the only perfect square which cannot be written as the sum of four squares

(Sierpinski [185], p. 405);

• the smallest number r which has the property that each number can be written

as x1 3 +x2 3 +. . .+xr, 3 where the xi’s are non negative integers (see the number 4).

10

• one of the five numbers (the others are 1, 120, 1 540 and 7 140) which are both

triangular and tetrahedral (see E.T. Avanesov [8]): a number n is said to be

tetrahedral if it can be written as n = 1

6

m(m + 1)(m + 2) for some number m:

it corresponds to the number of spheres with same radius which can be piled

up in a tetrahedron;

• the fourth number n such that τ (n) = φ(n) (see the number 8).

2Much

more is known. Indeed, according to the Catalan Conjecture (first stated

by Catalan [31] in 1844), the only consecutive numbers in the sequence of powers

1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81, 100, 121, 125, . . . are 8 and 9; this conjecture was recently proved

by Preda Mihailescu [135].