22 Jean-Marie De Koninck

66

• the denominator of the Bernoulli number B10 =

5

66

;

• the smallest solution of σ2(n) = σ2(n + 11): the second one is 130;

• the smallest solution of σ(n) = σ(n + 28); the sequence of numbers satisfying

this equation begins as follows: 66, 159, 267, 282, 295, 328, 357, 580, 979, 1111,

1265, 2157, 2233, 2599, 2686, 2698, 2990, 3580, 3799, 3882, 4066, 4070, 4317,

4782, 5518, 7003, 7021, 7339, 7475, . . .

67

• the third prime number with at least two digits and whose digits are consecu-

tive (ascending or descending): the sequence of prime numbers satisfying this

property begins as follows: 23, 43, 67, 89, 101, 787, 4567, 12101, 12323, 12343,

32321, 32323, 34543, 54323, 56543, 56767, 76543, 78787, 78989, 210101, 212123,

234323, 234343, 432121, 432323, 432343, 434323, 454543, 456767, 654323,

654343, 678767, 678989, 876787, 878789, 878987, 878989, 898787, 898987,

1012321, 1210123, 1212121, 1234543, 3210101, 3210121, 32112101, 3212123,

3212323, . . . ;

• the eighth Heegner number (see the number 163);

• the largest known prime number p such that

2p

+ 3 is prime; the other known

prime numbers p for which

2p

+ 3 is prime are 2, 3 and 7.

68

• the smallest number n such that

∑

m≤n

τ (m) is a multiple of 100 (here the sum

is equal to 300).

69

• the only number such that when we gather the digits formed by its square and

its cube, we obtain all the digits from 0 to 9 once and only once: 692 = 4 761

and

693

= 328 509.

70

• the only number whose square corresponds to the number of cannonballs one

can pile up in a pyramidale shape

(702

=

12

+

22

+ . . . +

242)

(Lucas, 1873):

for an elementary proof, see W.S. Anglin

[5]30;

30One

can also solve this problem using elliptic curves. Here is the general idea. First it is clear

that we need to solve the diophantine equation

y2

=

x(x + 1)(2x + 1)

6

=

x3

3

+

x2

2

+

x

6

,