Frank Buss propose una congettura analoga alla congettura di Fortune: se q(n, m) è il minimo numero primo maggiore di n!m + 1, q(n, m) – n!m è primo per m < 6. La congettura di Fortune corrisponde al caso m = 1.
La tabella seguente riporta i valori di q(n, m) – n!m per n fino a 1000 e m da 1 a 5 (M. Fiorentini, 2018).
|
n |
q(n, 1) – n! |
q(n, 2) – n!2 |
q(n, 3) – n!3 |
q(n, 4) – n!4 |
q(n, 5) – n!5 |
|
1 |
2 |
2 |
2 |
2 |
2 |
|
2 |
3 |
3 |
3 |
3 |
5 |
|
3 |
5 |
5 |
7 |
5 |
13 |
|
4 |
5 |
11 |
5 |
5 |
13 |
|
5 |
7 |
7 |
17 |
7 |
7 |
|
6 |
7 |
11 |
11 |
29 |
7 |
|
7 |
11 |
11 |
17 |
19 |
11 |
|
8 |
23 |
13 |
23 |
29 |
71 |
|
9 |
17 |
23 |
23 |
181 |
23 |
|
10 |
11 |
17 |
103 |
19 |
19 |
|
11 |
17 |
13 |
59 |
31 |
197 |
|
12 |
29 |
59 |
17 |
173 |
17 |
|
13 |
67 |
23 |
29 |
79 |
101 |
|
14 |
19 |
31 |
79 |
43 |
53 |
|
15 |
43 |
23 |
59 |
379 |
17 |
|
16 |
23 |
41 |
23 |
61 |
47 |
|
17 |
31 |
59 |
347 |
101 |
73 |
|
18 |
37 |
67 |
307 |
127 |
97 |
|
19 |
89 |
29 |
53 |
101 |
53 |
|
20 |
29 |
31 |
227 |
83 |
433 |
La tabella seguente riporta i valori di n minori di 100 tali che q(n, m) – n!m sia composto, per m da 6 a 100 (M. Fiorentini, 20187).
|
m |
n |
|
6 |
28 |
|
7 |
6, 10 |
|
8 |
5, 10, 17 |
|
9 |
2, 3, 8, 24, 59 |
|
10 |
8, 9, 28 |
|
11 |
3, 8, 22, 51 |
|
12 |
3, 15, 65 |
|
13 |
3, 16, 17, 18, 22, 44, 45, 53, 65 |
|
14 |
2, 16, 20, 27, 43 |
|
15 |
3, 21, 35, 40, 46 |
|
16 |
36, 58 |
|
17 |
3, 4, 6, 16 |
|
18 |
4, 6, 11, 12, 17, 27, 39 |
|
19 |
2, 4, 6, 7, 22, 84 |
|
20 |
10, 11, 29, 60, 65, 73, 82, 89 |
|
21 |
4, 7, 16, 51, 52, 63 |
|
22 |
2, 5, 28, 42, 59, 65 |
|
23 |
2, 13, 22, 35, 37, 57, 60, 77 |
|
24 |
4, 7, 9, 13, 29, 56 |
|
25 |
2, 9, 18, 28, 41, 48, 58, 63, 74, 80, 81, 88 |
|
26 |
2, 8, 9, 26, 30, 42, 64, 82, 98 |
|
27 |
5, 15, 35, 41, 42, 46, 55, 57 |
|
28 |
3, 4, 9, 11, 22, 27, 39, 40, 42, 45, 48, 56, 93 |
|
29 |
10, 14, 15, 18, 28, 34, 36, 39, 46, 53, 88 |
|
30 |
6, 10, 11, 14, 17, 22, 34, 40, 50, 52, 61, 66 |
|
31 |
6, 8, 9, 10, 13, 26, 27, 48, 51, 58, 86 |
|
32 |
2, 5, 8, 10, 11, 12, 13, 17, 22, 39, 48, 87, 96 |
|
33 |
4, 6, 8, 11, 18, 24, 25, 31, 43, 49, 55, 64, 76, 80, 98 |
|
34 |
2, 3, 11, 13, 14, 31, 40, 50, 54, 72, 76 |
|
35 |
3, 10, 13, 18, 23, 28, 33, 40, 48, 51, 88, 96 |
|
36 |
7, 8, 12, 14, 15, 19, 32, 36, 38, 42, 55 |
|
37 |
2, 9, 17, 24, 45, 88 |
|
38 |
10, 13, 16, 26, 32, 44, 45, 53, 60, 78, 86, 96 |
|
39 |
12, 36, 56, 60, 97 |
|
40 |
2, 4, 10, 11, 17, 23, 33, 41, 54, 58, 59, 83 |
|
41 |
2, 3, 6, 8, 9, 13, 34, 47, 66, 86 |
|
42 |
2, 4, 9, 10, 12, 15, 20, 36, 44, 58, 72, 75, 85 |
|
43 |
3, 6, 9, 12, 18, 26, 28, 34, 35, 36, 39, 50, 55, 59, 64, 75, 88 |
|
44 |
3, 4, 10, 13, 15, 20, 21, 24, 26, 30, 34, 46, 50, 54, 58, 64, 76, 86, 88, 92 |
|
45 |
4, 6, 7, 10, 13, 16, 17, 19, 28, 29, 36, 38, 45, 49, 63, 66, 81, 97 |
|
46 |
2, 3, 10, 14, 25, 34, 48, 64, 65, 66, 70, 73, 81, 94, 100 |
|
47 |
3, 5, 9, 12, 15, 16, 24, 26, 29, 31, 34, 42, 82 |
|
48 |
2, 5, 11, 18, 23, 31, 35, 42, 43, 44, 67, 99, 100 |
|
49 |
2, 3, 6, 21, 35, 37, 51, 52, 76, 77, 82 |
|
50 |
2, 3, 4, 7, 10, 22, 23, 32, 34, 40, 42, 48, 51, 52, 64, 81, 99 |
|
51 |
2, 3, 4, 5, 6, 7, 14, 15, 16, 17, 20, 21, 22, 23, 30, 36, 41, 59, 63, 75, 96 |
|
52 |
2, 3, 7, 8, 13, 14, 16, 17, 24, 26, 27, 33, 52, 54, 59, 62, 68, 87 |
|
53 |
8, 16, 18, 25, 26, 27, 38, 42, 51, 66 |
|
54 |
2, 8, 11, 14, 22, 37, 41, 46, 47, 50, 77, 93, 98 |
|
55 |
5, 6, 8, 10, 14, 19, 33, 40, 41, 47, 55, 59, 61, 73, 93, 96, 99 |
|
56 |
2, 3, 7, 11, 27, 30, 37, 40, 57, 64, 69, 84 |
|
57 |
2, 3, 6, 7, 9, 10, 11, 12, 24, 32, 33, 34, 44, 47, 49, 53, 55, 63, 65, 85 |
|
58 |
2, 6, 10, 21, 23, 56, 71, 85, 92, 95, 98 |
|
59 |
12, 16, 20, 28, 31, 35, 41, 42, 58, 66, 68, 97 |
|
60 |
2, 3, 6, 24, 26, 35, 56, 69 |
|
61 |
2, 9, 13, 14, 36, 38, 47, 48, 62, 85, 86, 88, 93, 96 |
|
62 |
2, 7, 9, 10, 13, 20, 27, 33, 41, 45, 60, 78, 84 |
|
63 |
4, 9, 10, 19, 21, 31, 34, 38, 55, 69, 75, 77, 78, 80 |
|
64 |
5, 10, 13, 14, 31, 42, 52, 70, 86, 90, 92, 93 |
|
65 |
3, 5, 6, 9, 12, 14, 17, 18, 19, 27, 33, 36, 38, 53, 56, 60, 61, 70, 78, 86, 87, 98 |
|
66 |
2, 3, 4, 5, 11, 17, 18, 23, 25, 34, 35, 36, 41, 48, 49, 57, 60, 68, 73, 98, 99 |
|
67 |
3, 6, 8, 10, 18, 19, 22, 23, 24, 31, 34, 40, 43, 47, 62, 74, 76, 91 |
|
68 |
2, 3, 5, 8, 27, 40, 52, 59, 83, 86, 89 |
|
69 |
5, 6, 9, 10, 15, 20, 21, 23, 27, 30, 31, 36, 45, 50, 79 |
|
70 |
2, 5, 6, 8, 9, 13, 23, 24, 35, 43, 46, 47, 50, 51, 56, 61, 65, 69 |
|
71 |
3, 4, 6, 7, 8, 9, 11, 14, 15, 23, 24, 26, 30, 31, 34, 48, 50, 56, 57, 58, 63, 66, 72 |
|
72 |
2, 4, 9, 13, 15, 16, 21, 33, 34, 43, 47, 48, 76, 78, 88, 92 |
|
73 |
3, 6, 7, 13, 15, 17, 18, 23, 25, 27, 35, 41, 43, 50, 52, 55, 57, 67, 69, 70, 78, 81, 82 |
|
74 |
3, 4, 6, 7, 9, 11, 14, 15, 16, 17, 20, 26, 33, 36, 39, 51, 52, 61, 67, 68, 69, 77, 86 |
|
75 |
2, 6, 11, 17, 20, 22, 23, 29, 34, 37, 39, 41, 46, 51, 75, 79, 80, 84, 87, 93, 98 |
|
76 |
2, 3, 4, 6, 9, 10, 13, 37, 44, 57, 67, 81, 83 100 |
|
77 |
7, 14, 19, 21, 28, 40, 44, 48, 53, 55, 56, 61, 66, 78, 81, 89, 95, |
|
78 |
3, 4, 6, 7, 11, 12, 14, 24, 35, 36, 57, 64, 72, 91 |
|
79 |
10, 11, 14, 16, 30, 32, 34, 43, 46, 47, 52, 63, 73, 85, 93, |
|
80 |
3, 6, 8, 13, 17, 21, 23, 24, 26, 28, 29, 30, 38, 57, 61, 67, 69, 76, 77, 80, 88, 92, 98 |
|
81 |
3, 7, 8, 10, 14, 16, 18, 19, 20, 21, 23, 27, 30, 37, 44, 45, 51, 57, 64, 67, 70, 71, 76, 81, 83, 85, 87 |
|
82 |
2, 5, 6, 8, 12, 13, 14, 19, 23, 27, 36, 37, 41, 57, 61, 65, 68, 69, 73, 83, 90, 96 |
|
83 |
2, 3, 5, 13, 15, 16, 47, 69, 77, 83, 94 |
|
84 |
5, 6, 8, 10, 12, 17, 20, 22, 26, 27, 30, 37, 41, 49, 54, 59, 78, 80 |
|
85 |
2, 3, 5, 6, 8, 12, 13, 15, 18, 20, 27, 29, 32, 37, 38, 40, 46, 50, 54, 55, 57, 60, 64, 68, 71, 83, 98 |
|
86 |
2, 3, 4, 6, 11, 13, 19, 20, 22, 28, 35, 46, 51, 55, 58, 64, 71, 73, 75, 83, 84 |
|
87 |
2, 4, 5, 8, 11, 15, 16, 20, 25, 29, 33, 34, 87;2, 4, 5, 8, 11, 15, 16, 20, 25, 29, 33, 34, 36, 42, 52, 54, 56, 57, 65, 78, 96 |
|
88 |
5, 6, 10, 11, 14, 16, 20, 23, 25, 27, 29, 30, 36, 37, 42, 44, 51, 65, 67, 76, 100 |
|
89 |
5, 6, 10, 13, 14, 22, 26, 27, 29, 30, 49, 50, 61, 63, 67, 71, 89, 98 |
|
90 |
2, 3, 5, 6, 8, 9, 10, 25, 27, 32, 35, 39, 41, 43, 45, 49, 51, 52, 62, 68, 70, 73, 77, 79, 80, 85 |
|
91 |
7, 13, 16, 25, 28, 44, 47, 48, 51, 54, 57, 60, 63, 65, 73, 86, 88, 97, 98 |
|
92 |
2, 4, 5, 10, 14, 15, 17, 20, 21, 22, 32, 35, 39, 45, 47, 62, 68, 69, 73, 76, 78, 79, 81, 86, 98 |
|
93 |
2, 3, 5, 7, 8, 14, 20, 21, 23, 28, 30, 32, 36, 38, 49, 51, 56, 66, 69, 77 |
|
94 |
2, 7, 8, 9, 10, 13, 18, 37, 38, 41, 63, 65, 66, 67, 74, 75, 76, 77, 81, 84, 85, 99 |
|
95 |
2, 3, 8, 9, 11, 12, 15, 19, 20, 21, 24, 27, 31, 35, 38, 39, 48, 51, 56, 58, 71, 85, 87, 94, 98 |
|
96 |
6, 8, 10, 14, 21, 22, 39, 41, 42, 47, 52, 96;42, 47, 52, 60, 62, 64, 67, 83, 96, 98 |
|
97 |
2, 4, 6, 9, 11, 18, 28, 30, 32, 49, 65, 82 |
|
98 |
4, 8, 11, 15, 17, 24, 27, 29, 35, 40, 43, 47, 49, 51, 52, 60, 63, 70, 71, 97 |
|
99 |
2, 3, 5, 7, 9, 14, 15, 18, 19, 23, 25, 26, 33, 34, 35, 38, 46, 59, 72, 78, 92, 93, 99, 100 |
|
100 |
4, 5, 7, 9, 10, 11, 17, 24, 25, 26, 27, 29, 37, 45, 47, 58, 65, 72, 76, 85, 90, 93 |
L’evidenza sperimentale indica che è probabile che esistano casi del genere per tutti i valori di m maggiori di 5.
Non si conosce però alcun controesempio con m minore di< 6; se esistono, n è maggiore di 1000 (M. Fiorentini, 2018).