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].
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