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,
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