Problems & Puzzles: Conjectures

Conjecture 92. For any integer m there is at least one set of consecutive primes...

Paola Lava sent on March 2022, the following conjecture:

Conjecture

For any integer m there is at least one set of n>1, consecutive primes p_1, p_2, , p_n such that the sum of every two consecutive primes in this set is divisible by m.

E.g. m=5

The first least sets for n from 2 to 10 are:

2+3=5 and 5/5=1

3   [43, 47, 53]

43+47=90 and 90/5=18; 47+53=100 and 100/5=20.

4   [157, 163, 167, 173]

157+163=320 and 320/5=64; 163+167=330 and 330/5=66; 167+173=340 and 340/5=68.

And so on

5   [3593, 3607, 3613, 3617, 3623]

6   [6037, 6043, 6047, 6053, 6067, 6073]

7   [32633, 32647, 32653, 32687, 32693, 32707, 32713]

8   [32633, 32647, 32653, 32687, 32693, 32707, 32713, 32717]

9   [833843, 833857, 833873, 833887, 833893, 833897, 833923, 833927, 833933]

10   [833843, 833857, 833873, 833887, 833893, 833897, 833923, 833927, 833933, 833947]

Here below the list of the least sets for n from 2 to 5 for m from 1 to 50 (Giovanni Resta helped me in finding values for n = 5 and  m = 29, 31, 35, 37, 41, 43, 45, 47, 49):

m

n = 2

n = 3

n = 4

n = 5

1

2, 3

2, 3, 5

2, 3, 5, 7

2, 3, 5, 7, 11

2

3, 5

3, 5, 7

3, 5, 7, 11

3, 5, 7, 11, 13

3

5, 7

5, 7, 11

5, 7, 11, 13

5, 7, 11, 13, 17

4

3, 5

3, 5, 7

23, 29, 31, 37

47, 53, 59, 61, 67

5

2, 3

43, 47, 53

157, 163, 167, 173

3593, 3607, 3613, 3617, 3623

6

5, 7

5, 7, 11

5, 7, 11, 13

[5, 7, 11, 13, 17

7

19, 23

977, 983, 991

977, 983, 991, 997

10487, 10499, 10501, 10513, 10529

8

3, 5

53, 59, 61

53, 59, 61, 67

523, 541, 547, 557, 563]

9

7, 11

313, 317, 331

5171, 5179, 5189, 5197

38377, 38393, 38431, 38447, 38449

10

13, 17

43, 47, 53

157, 163, 167, 173

3593, 3607, 3613, 3617, 3623

11

97, 101

787, 797, 809

33871, 33889, 33893, 33911

1796671, 1796677, 1796693, 1796699, 1796759

12

5, 7

137, 139, 149

137, 139, 149, 151

409, 419, 421, 431, 433

13

23, 29

9587, 9601, 9613

159293, 159311, 159319, 159337

947423, 947431, 947449, 947483, 947501

14

19, 23

977, 983, 991

977, 983, 991, 997

10487, 10499, 10501, 10513, 10529

15

13, 17

2473, 2477, 2503

2969, 2971, 2999, 3001

60383, 60397, 60413, 60427, 60443

16

53, 59

541, 547, 557

541, 547, 557, 563

62501, 62507, 62533, 62539, 62549

17

31, 37

3967, 3989, 4001

406873, 406883, 406907, 406951

18164651, 18164689, 18164719, 18164723, 18164753

18

7, 11

313, 317, 331

5171, 5179, 5189, 5197

38377, 38393, 38431, 38447, 38449

19

73, 79

28979, 29009, 29017

471313, 471353, 471389, 471391

15095579, 15095611, 15095617, 15095687, 15095693

20

29, 31

947, 953, 967

6047, 6053, 6067, 6073

32633, 32647, 32653, 32687, 32693

21

19, 23

3121, 3137, 3163

166739, 166741, 166781, 166783

3272567, 3272587, 3272609, 3272629, 3272651

22

97, 101

787, 797, 809

33871, 33889, 33893, 33911

1796671, 1796677, 1796693, 1796699, 1796759

23

67, 71

72823, 72858, 72869

2112193, 2112217, 2112239, 2112263

116863451, 116863469, 116863543, 116863561, 116863589

24

11, 13

283, 293, 307

5309, 5323, 5333, 5347

67819, 67829, 67843, 67853, 67867

25

47, 53

47441, 47459, 47491

520763, 520787, 520813, 520837

65835479, 65835521, 65835529, 65835571, 65835629

26

23, 29

9587, 9601, 9613

159293, 159311, 159319, 159337

947423, 947431, 947449, 947483, 947501

27

79, 83

81463, 81509, 81517

207869, 207877, 207923, 207931

7005239, 7005277, 7005293, 7005331, 7005347

28

41, 43

4363, 4373, 4391

5443, 5449, 5471, 5477

1165217, 1165223, 1165273, 1165279, 1165301

29

347, 349

61153, 61169, 61211

2404471, 2404483, 2404529, 2404541

1154953243, 1154953249, 1154953301, 1154953307, 1154953417

30

13, 17

2473, 2477, 2503

2969, 2971, 2999, 3001

60383, 60397, 60413, 60427, 60443

31

89, 97

478001, 478039, 478063

1531487, 1531499, 1531549, 1531561

800037461, 800037521, 800037523, 800037583, 800037647

32

61, 67

21617, 21647, 21649

88919, 88937, 88951, 88969

7442557, 7442563, 7442621, 7442627, 7442653

33

97, 101

160243, 160253, 160309

2673791, 2673793, 2673857, 2673859

15442121, 15442183, 15442187, 15442249, 15442253

34

31, 37

3967, 3989, 4001

406873, 406883, 406907, 406951

18164651, 18164689, 18164719, 18164723, 18164753

35

103, 107

132763, 132817, 132833

6056569, 6056581, 6056639, 6056651

771405431, 771405559, 771405571, 771405629, 771405641

36

17, 19

8017, 8039, 8053

95737, 95747, 95773, 95783

558791, 558793, 558827, 558829, 558863

37

109, 113

227873, 227893, 227947

8480357, 8480369, 8480431, 8480443

1956833159, 1956833183, 1956833233, 1956833257, 1956833381

38

73, 79

28979, 29009, 29017

471313, 471353, 471389, 471391

15095579, 15095611, 15095617, 15095687, 15095693

39

37, 41

218279, 218287, 218357

561829, 561839, 561907, 561917

128805491, 128805499, 128805569, 128805577, 128805647

40

59, 61

12163, 12197, 12203

73477, 73483, 73517, 73523

24653231, 24653249, 24653311, 24653329, 24653351

41

199, 211

1772119, 1772167, 1772201

8756689, 8756707, 8756771, 8756789

46267575553, 46267575593, 46267575799, 46267575839, 46267575881

42

19, 23

3121, 3137, 3163

166739, 166741, 166781, 166783

3272567, 3272587, 3272609, 3272629, 3272651

43

83, 89

3070187, 3070213, 3070273

15409001, 15409013, 15409087, 15409099

9523396021, 9523396187, 9523396193, 9523396273, 9523396279

44

151, 157

57413, 57427, 57457

839087, 839117, 839131, 839161

30716599, 30716641, 30716687, 30716729, 30716731

45

43, 47

841459, 841541, 841549

2517289, 2517311, 2517379, 2517401

1036205567, 1036205623, 1036205657, 1036205713, 1036205747

46

67, 71

72823, 72859, 72869

2112193, 2112217, 2112239, 2112263

116863451, 116863469, 116863543, 116863561, 116863589

47

281, 283

6135979, 6136003, 6136073

36726643, 36726649, 36726737, 36726743

72019040869, 72019040917, 72019040963, 72019041011, 72019041151

48

71, 73

39451, 39461, 39499

60811, 60821, 60859, 60869

17426083, 17426093, 17426131, 17426141, 17426179

49

439, 443

1194059, 1194103, 1194157

67473239, 67473251, 67473337, 67473349

23035373797, 23035373863, 23035373993, 23035374059, 23035374091

50

47, 53

47441, 47459, 47491

520763, 520787, 520813, 520837

65835479, 65835521, 65835529, 65835571, 65835629

 

 


 

Records   |  Conjectures  |  Problems  |  Puzzles