116 Jean-Marie De Koninck

997

• the largest three digit prime number;

• the largest known value of ρ(k) whose last digit is 7 (see the number 389).

1 001

• the

14th

number k such that

k|(10k+1

− 1) (see the number 303).

1 008

• the smallest number n such that π(n) = n/6 (see the number 330).

1 009

• the smallest four digit prime number;

• the smallest prime number which can be written in the following ten manners:

x2 + y2, x2 + 2y2, . . . , x2 + 10y2 (Problem stated by Gregory Wulcszyn and

solved by A.M. Vaidya in Amer. Math. Monthly 75 (1968), p. 193); indeed,

1 009 =

152

+

282

=

192

+ 2 ·

182

=

312

+ 3 ·

42

=

152

+ 4 ·

142

=

172

+ 5 ·

122

=

252

+ 6 ·

82

=

12

+ 7 ·

122

=

192

+ 8 ·

92

=

282

+ 9 ·

52

=

32

+ 10 ·

102;

• the smallest number which can be written as the sum of three distinct cubes

in two distinct ways: 1 009 =

13

+

23

+

103

=

43

+

63

+

93;

if we denote by nk

the smallest number which can be written as the sum of three distinct cubes

in k distinct ways, then

n2 = 1 009 = 13 + 23 + 103 = 43 + 63 + 93,

n3 = 5 104 = 13 + 123 + 153 = 23 + 103 + 163 = 93 + 103 + 153,

n4 = 13 896 = 13 + 123 + 233 = 23 + 43 + 243

= 43 + 183 + 203 = 93 + 103 + 233,

n5 = 161 568 = 23 + 163 + 543 = 93 + 153 + 543 = 173 + 393 + 463

= 183 + 193 + 533 = 263 + 363 + 463,

n6 = 1 296 378 = 33 + 763 + 953 = 93 + 333 + 1083

= 213 + 773 + 943 = 313 + 593 + 1023

=

333

+

813

+

903

=

603

+

753

+

873,

n7 = 2 016 496 =

63

+

723

+

1183

=

103

+

663

+

1203

=

193

+

213

+

1263

=

473

+

973

+

1003

=

543

+

603

+

1183

=

663

+

903

+

1003

=

833

+

853

+

943,

n8 = 2 562 624 =

83

+

363

+

1363

=

83

+

643

+

1323

=

123

+

1003

+

1163

=

173

+

463

+

1353

=

303

+

1033

+

1133

=

363

+

603

+

1323

=

513

+

853

+

1223

=

693

+

723

+

1233,