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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 20 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, 2018).

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, 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).

Qui trovate il minimo valore di n tale che q(n, m) – n!m sia composto, per m da 6 a 1000 (M. Fiorentini, 2019).

 

Ho proposto una variante della congettura di Buss, che sembra stranamente sfuggita sinora all’attenzione dei matematici: se q’(n, m) è il massimo numero primo minore di n!m – 1, n!mq’(n, m) è primo per m < 7 e m = 8.

 

La tabella seguente riporta i valori di n!mq’(n, m) per n fino a 20 e m da 1 a 6 e uguale a 8 (M. Fiorentini, 2019).

n

n! – q’(n, 1)

n!2q’(n, 2)

n!3q’(n, 3)

n!4q’(n, 4)

n!5q’(n, 5)

n!6q’(n, 6)

n!8q’(n, 8)

2

-

2

3

3

3

3

5

3

3

5

5

5

17

7

7

4

5

5

17

7

17

5

17

5

7

11

11

7

19

19

31

6

11

11

23

7

47

37

137

7

17

11

23

13

61

179

191

8

31

17

17

23

17

31

53

9

13

11

13

19

13

17

31

10

11

53

17

71

13

251

19

11

13

17

29

71

23

23

73

12

13

47

41

43

19

271

137

13

23

53

73

97

199

191

17

14

17

19

227

61

37

83

491

15

47

23

73

181

367

97

701

16

53

17

239

151

107

149

193

17

59

163

47

43

71

263

19

18

41

131

307

67

53

31

353

19

101

23

107

89

151

349

131

20

31

173

41

167

31

337

223

 

La tabella seguente riporta i valori di n minori di 100 tali che n!mq’(n, m) sia composto, per m da 7 a 100 (M. Fiorentini, 2019).

m

n

7

2, 19

8

-

9

9

10

4, 65

11

2, 8, 12, 29, 38, 64

12

4, 35, 81

13

100

14

5, 29, 51, 64, 97, 98

15

9, 12, 24, 28, 44, 60

16

2, 5, 24, 29, 40, 42

17

2, 3, 10, 12, 35

18

12, 55, 74

19

6, 23, 37

20

5, 7, 9, 12, 16, 18, 36, 71

21

2, 5, 6, 8, 22, 28, 29

22

3, 4, 29, 31, 38, 56, 60, 86

23

2, 4, 15, 37, 43, 52, 55, 65, 69

24

4, 20, 30, 69

25

2, 8, 16, 33, 55, 62

26

3, 9, 16, 22, 36, 55

27

2, 6, 9, 13, 17, 18, 27, 51, 60, 64, 86, 99

28

2, 3, 5, 15, 23, 34, 35, 37, 42, 58, 59, 81, 90, 96

29

6, 8, 10, 25, 26, 33, 36, 37, 45, 47, 55, 66, 75

30

2, 7, 16, 27, 29, 30, 31, 32, 38, 46, 47, 80

31

4, 20, 21, 22, 30, 39, 44, 47, 70, 99

32

4, 5, 6, 7, 16, 18, 19, 34, 38, 41, 51, 59, 75, 90

33

2, 3, 5, 10, 15, 16, 35, 36, 45, 49, 53, 55, 57, 58, 70, 76, 84, 85

34

12, 23, 26, 38, 55, 58, 85, 86

35

14, 16, 17, 22, 26, 35, 38, 67, 84

36

3, 18, 25, 27, 28, 49, 62, 66, 70, 73, 82, 97

37

2, 4, 6, 10, 14, 16, 24, 34, 40, 51, 65

38

2, 4, 10, 12, 14, 17, 25, 41, 64, 97, 100

39

3, 4, 7, 25, 49, 64, 83

40

2, 3, 6, 9, 11, 13, 18, 21, 47, 60, 68, 87, 95, 96

41

2, 12, 17, 20, 30, 46, 49, 65, 78, 86, 87, 97

42

4, 8, 9, 29, 34, 44, 68, 78, 97

43

2, 4, 5, 9, 10, 11, 18, 20, 25, 39, 41, 44

44

4, 11, 12, 13, 27, 33, 39, 43, 45, 52, 60, 66, 90, 98, 99

45

2, 5, 6, 8, 9, 30, 39, 51, 57, 64, 68, 69, 75, 77

46

2, 8, 10, 11, 20, 24, 25, 27, 39, 43, 53, 54, 59, 69, 85, 86

47

2, 7, 9, 18, 19, 21, 34, 36, 39, 41, 56, 69, 78, 98

48

6, 9, 12, 14, 16, 18, 21, 25, 26, 27, 32, 33, 35, 47, 55, 58, 78, 81, 86, 91

49

2, 4, 5, 10, 12, 40, 46, 59, 66, 75, 92

50

2, 3, 10, 13, 14, 18, 19, 39, 44, 50, 64, 67, 77, 100

51

2, 5, 7, 18, 26, 36, 49, 54, 60, 63, 78, 90, 98

52

7, 8, 18, 22, 26, 28, 36, 39, 40, 60, 66, 72, 88, 93, 99

53

2, 8, 12, 13, 15, 18, 25, 27, 31, 67, 68, 84, 88

54

2, 3, 13, 16, 17, 24, 25, 30, 33, 34, 48, 72, 73, 92, 95, 96, 99

55

2, 5, 7, 12, 13, 22, 24, 28, 29, 31, 36, 41, 42, 46, 64, 65, 69, 73, 77, 93, 97

56

10, 12, 13, 23, 30, 33, 35, 38, 40, 41, 52, 66, 71, 81, 85, 90, 91

57

3, 11, 17, 18, 22, 25, 30, 34, 37, 44, 51, 95

58

2, 4, 5, 7, 12, 13, 16, 17, 23, 29, 31, 32, 35, 41, 44, 59, 74

59

2, 6, 9, 18, 23, 24, 27, 31, 34, 39, 57, 60, 68, 77, 86, 91, 98, 100

60

2, 4, 9, 28, 32, 37, 40, 69, 78, 83, 89

61

7, 12, 14, 20, 22, 28, 31, 34, 35, 55, 61, 67, 87, 95, 98

62

2, 3, 9, 10, 19, 25, 35, 42, 51, 58, 59, 82, 89

63

2, 6, 11, 16, 20, 22, 23, 24, 25, 27, 29, 30, 41, 56, 67, 93

64

5, 7, 13, 14, 16, 17, 21, 22, 27, 28, 32, 38, 43, 44, 69, 77

65

2, 4, 5, 6, 12, 15, 18, 22, 25, 26, 32, 37, 40, 46, 57, 58, 71, 79, 81, 83, 88

66

4, 6, 9, 13, 21, 29, 34, 36, 50, 51, 53, 76, 84, 89, 93

67

3, 7, 10, 14, 15, 24, 25, 27, 33, 35, 42, 43, 44, 49, 53, 56, 58, 61, 63, 67, 69, 70, 71, 76, 80, 82

68

3, 4, 7, 14, 15, 20, 44, 46, 64, 73

69

4, 5, 9, 10, 11, 12, 15, 22, 23, 25, 26, 34, 36, 38, 42, 57, 64, 69, 70, 76, 80, 85, 91, 95, 96

70

2, 4, 8, 11, 12, 18, 22, 24, 29, 30, 34, 37, 44, 48, 50, 55, 82, 95, 99

71

2, 3, 12, 14, 17, 18, 21, 28, 34, 36, 37, 39, 50, 52, 55, 65, 74, 97

72

2, 6, 10, 13, 20, 21, 23, 24, 28, 31, 33, 35, 42, 43, 58, 60, 69, 84, 86, 88, 95

73

2, 7, 8, 9, 10, 12, 14, 15, 20, 21, 22, 23, 25, 27, 37, 41, 44, 45, 46, 48, 56, 59, 60, 65, 72, 97

74

2, 5, 7, 13, 15, 20, 21, 26, 34, 42, 44, 57, 58, 63, 79, 100

75

3, 4, 6, 13, 17, 25, 26, 35, 38, 39, 41, 42, 45, 55, 64, 73, 74, 95, 96

76

2, 4, 6, 7, 8, 17, 19, 38, 39, 59, 61, 62, 64, 73, 87, 90, 96

77

2, 4, 10, 12, 14, 15, 28, 33, 34, 35, 46, 63, 69, 86, 88, 94, 97, 100

78

3, 4, 7, 9, 10, 14, 19, 20, 21, 22, 24, 27, 31, 35, 37, 43, 44, 48, 50, 55, 63, 66, 77, 81, 88, 90, 92, 100

79

5, 9, 12, 16, 20, 31, 37, 46, 47, 48, 57, 64, 73, 81, 82, 91

80

2, 5, 9, 13, 14, 15, 28, 32, 34, 38, 41, 43, 45, 46, 51, 63, 88, 96, 97, 99

81

2, 3, 4, 11, 12, 14, 22, 34, 39, 40, 44, 55, 56, 61, 63, 66, 68, 70, 76, 87, 91

82

2, 3, 4, 5, 6, 7, 13, 16, 25, 27, 31, 34, 42, 61, 63, 73, 77

83

2, 3, 4, 5, 8, 9, 14, 16, 19, 27, 32, 39, 41, 45, 46, 48, 53, 56, 74, 82, 84, 89, 92

84

2, 3, 4, 5, 6, 9, 10, 12, 16, 20, 22, 24, 26, 28, 30, 34, 37, 48, 53, 61, 66, 69, 70, 75, 76, 77, 84, 86, 89, 91, 95

85

3, 4, 5, 7, 10, 11, 14, 15, 16, 19, 21, 32, 54, 58, 61, 67, 74, 85

86

2, 14, 18, 20, 27, 30, 32, 33, 40, 46, 52, 56, 57, 59, 66, 76, 79, 80, 91, 96, 98, 99

87

3, 4, 5, 7, 8, 31, 32, 33, 44, 45, 46, 56, 59, 73, 85, 88, 95, 98

88

2, 9, 10, 12, 16, 17, 20, 26, 31, 32, 34, 43, 47, 51, 57, 58, 64, 65

89

2, 4, 6, 9, 11, 15, 16, 23, 27, 34, 38, 40, 41, 55, 58, 63, 68, 81, 88, 97

90

2, 7, 10, 12, 15, 19, 20, 24, 26, 28, 29, 30, 37, 43, 47, 51, 53, 58, 62, 65, 68, 83

91

2, 3, 4, 6, 7, 8, 10, 14, 19, 24, 27, 32, 38, 41, 42, 45, 49, 53, 60, 79, 82, 96

92

3, 8, 10, 11, 12, 13, 15, 16, 25, 36, 45, 50, 56, 66, 92, 100

93

2, 3, 8, 9, 15, 17, 24, 25, 39, 49, 52, 54, 55, 58, 63, 67, 70, 72

94

6, 7, 8, 11, 19, 20, 22, 25, 26, 29, 35, 52, 58, 61, 72, 73, 84, 93, 97

95

2, 3, 7, 8, 13, 17, 25, 27, 29, 32, 36, 44, 45, 52, 53, 55, 61, 64, 66, 85

96

3, 12, 16, 20, 22, 28, 36, 40, 42, 46, 47, 48, 65, 68, 69, 92, 93, 98

97

2, 3, 6, 11, 19, 20, 23, 26, 34, 38, 39, 42, 43, 51, 55, 56, 63, 72, 74, 82, 86, 94, 99

98

2, 7, 8, 10, 13, 18, 20, 24, 28, 31, 38, 48, 54, 55, 58, 65, 72, 73, 82

99

2, 3, 10, 13, 15, 18, 25, 29, 31, 36, 48, 53, 55, 57, 63, 74, 76, 84, 96

100

2, 3, 9, 12, 14, 17, 21, 24, 29, 31, 34, 35, 40, 62, 65, 67, 82, 89

 

L’evidenza sperimentale indica che è probabile che esistano casi del genere per tutti i valori di m maggiori di 6, tranne 8.

 

Qui trovate il minimo valore di n tale che n!mq’(n, m) sia composto, per m da 6 a 1000 (M. Fiorentini, 2019).

Contattami

Potete contattarmi al seguente indirizzo bitman[at]bitman.name per suggerimenti o segnalazioni d'errori relativi a questo articolo.