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Buss (congettura di) (II)

Congetture  Teoria dei numeri 

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).

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