Those Fascinating Numbers 83
the smallest Lucas-Carmichael number: such numbers are square-free compos-
ite numbers n for which p|n =⇒ p + 1|n + 1: the sequence of these numbers
begins as follows: 399, 935, 2015, 2915, 4991, 5719, 7055, 8855, 12719, 18095,
20705, 20999, 22847, 29315, 31535, 46079, 51359, 60059, 63503, 67199, 73535,
76751, 80189, 81719, 88559, 90287, . . . ; it is interesting to generalize the ques-
tion and search, for each positive integer k, for the square-free composite num-
bers n such that p|n =⇒ p + k|n + k: thus, if we denote by nk the smallest
number n such that p|n =⇒ p + k|n + k, we have the following table86:
k nk
1 399
2 598
3 165
4 1886
5 715
6 148219
7 273
8 343027
9 231
10 935
k nk
11 3445
12 4681203
13 1547
14 2821
15 1105
16 21098
17 43183
18 258482
19 27071
20 2117473
k nk
21 3059
22 10373
23 2737
24 690501
25 1595
26 23959
27 42653
28 1621007
29 9331
30 4169135
k nk
31 35465
32 543973
33 14839
34 36941
35 1885
36 4405589
37 11063
38 55537
39 19721
40 207581
k nk
41 133127
42 1548318
43 14645
44 13981
45 370691
46 96209
47 7285
48 3318267
49 4991
50 45136855
k nk
51 55709
52 143318
53 33337
54 568514
55 2465
56 2649157
57 16583
58 183251
59 684229
60 3398223
k nk
61 48323
62 240157
63 4018237
64 8495423
65 32767
66 13188109
67 66263
68 7792402
69 109411
70 12690146
k nk
71 228241
72 13698613
73 39785
74 407851
75 323765
76 109649
77 27379
78 31533842
79 305345
80 2311867
k nk
81 948119
82 318643
83 190957
84 38714713
85 42251
86 595249
87 481777
88 374402
89 454951
90 81430490
k nk
91 1078181
92 61680733
93 21507
94 1661027
95 89053
96 12580829
97 147323
98 1514602
99 24211
100 6022007
k nk
101 298699
102 66621453
103 200777
104 2592061
105 6038039
106 1960523
107 528283
108 427333057
109 282131
110 10193281
k nk
111 455729
112 831290
113 242767
114 512386761
115 34661
116 800299
117 385323
118 3891581
119 22321
120 ? ?
86Running
through this table, one will notice that for the prime numbers k, the corresponding
number nk is relatively small, while if k is composite, the corresponding number nk is generally
large. A legitimate question is to ask if the number nk exists for each number k: we may at least
confirm that there are no numbers n 5 ·
109
such that p|n =⇒ p + 120|n + 120.
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