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
,
Previous Page Next Page