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Carol generalizzati (numeri di)

Teoria dei numeri 

Si chiamano “numeri di Carol generalizzati” gli interi della forma (bn – 1)2 – 2; sono una generalizzazione dei numeri di Carol, ottenuti sostituendo una base b a 2 nella definizione.

 

In base b maggiore di 2 il numero di Carol generalizzato (bn – 1)2 – 2 si rappresenta con n – 1 cifre b – 1, seguite da una cifra b – 3 e n cifre b – 1. Per esempio, per b = 6 e n = 4 abbiamo il numero di Carol generalizzato 1677023 = 555355556.

 

Un numero di Carol generalizzato in base bn è anche un numero di Carol generalizzato in base b.

 

Le tabelle seguenti mostrano i numeri di Carol generalizzati per b fino a 20 e n fino a 20.

b \ n

1

2

3

4

5

2

–1

7

47

223

959

3

2

62

674

6398

58562

4

7

223

3967

65023

1046527

5

14

574

15374

389374

9759374

6

23

1223

46223

1677023

60450623

7

34

2302

116962

5759998

282441634

8

47

3967

261119

16769023

1073676287

9

62

6398

529982

43033598

3486666302

10

79

9799

997999

99979999

9999799999

11

98

14398

1768898

214329598

25937102498

12

119

20447

2982527

429940223

61916866559

13

142

28222

4822414

815673598

137857749262

14

167

38023

7524047

1475712223

289253579327

15

194

50174

11383874

2562789374

576648871874

16

223

65023

16769023

4294836223

1099509530623

17

254

82942

24127742

6975590398

2015991060734

18

287

104327

34000559

11019750623

3570463447487

19

322

129598

47032162

16983302398

6131061305602

20

359

159199

63983999

25599679999

10239993599999

b \ n

6

7

8

2

3967

16127

65023

3

529982

4778594

43033598

4

16769023

268402687

4294836223

5

244109374

6103359374

152587109374

6

2176689023

78363604223

2821106548223

7

13841051902

678221425762

33232919039998

8

68718952447

4398042316799

281474943156223

9

282428473598

22876782889022

1853020102758398

10

999997999999

99999979999999

9999999799999999

11

3138424833598

379749794608898

45949729434854398

12

8916094476287

1283918392885247

184884258035073023

13

23298075468862

3937376260202254

665416607551718398

14

56693897316223

11112006614731007

2177953334857793023

15

129746315109374

29192925683671874

6568408350587109374

16

281474943156223

72057593501057023

18446744065119617023

17

582622188954622

168377825738723582

48661191861715353598

18

1156831313401727

374813366357640959

121439531074554330623

19

2213314824974398

799006683995140642

288441413533654041598

20

4095999871999999

1638399997439999999

655359999948799999999

b \ n

9

10

11

2

261119

1046527

4190207

3

387381122

3486666302

31380705314

4

68718952447

1099509530623

17592177655807

5

3814693359374

95367412109374

2384185693359374

6

101559936513023

3656158319130623

131621703116673023

7

1628413517203234

79792265732661502

3909821044628334562

8

18014398241046527

1152921502459363327

73786976277658337279

9

150094634522158142

12157665452083359998

984770902120849113662

10

999999997999999999

99999999979999999999

9999999999799999999999

11

5559917308776336098

672749994880685159998

81402749386269137772098

12

26623333270565683199

3833759992323640393727

552061438910950400851967

13

112455406930748394382

19004963774605082455102

3211838877951270784369294

14

426878854169314649087

83668255424706292250623

16398978063347721975533567

15

1477891879958513671874

332525673006811787109374

74818276426774845263671874

16

4722366482732206260223

1208925819612430151450623

309485009821309884352692223

17

14063084451830549238014

4064231406643540534600702

1174562876521079915181447422

18

39346408074899818994687

12748236216388937239984127

4130428534112200791697551359

19

104127350297265866137282

37589973457533696060839998

13569980418173857927283575522

20

262143999998975999999999

104857599999979519999999999

41943039999999590399999999999

b \ n

12

13

2

16769023

67092479

3

282428473598

2541862639682

4

281474943156223

4503599493152767

5

59604644287109374

1490116116943359374

6

4738381333968052223

170581728153456820223

7

191581231352883839998

9387480337453976284834

8

4722366482732206260223

302231454902557782048767

9

79766443076307650790398

6461081889221589567275582

10

999999999997999999999999

99999999999979999999999999

11

9849732675801334237958398

1191817653772651897035844898

12

79496847203373011932545023

11447545997288067568804823039

13

542800770374323916601350398

91733330193268010908186431502

14

3214199700417627548926337023

629983141281875636173666664447

15

16834112196027973081787109374

3787675244106348436864013671874

16

79228162514263774643590529023

20282409603651661416747996545023

17

339448671314610739399029657598

98100666009922820632816624036094

18

1338258845052392388776975130623

433595865796975841944545375141887

19

4898762930960842391086457145598

1768453418076865617089615670815362

20

16777215999999991807999999999999

6710886399999999836159999999999999

b \ n

14

15

2

268402687

1073676287

3

22876782889022

205891103396834

4

72057593501057023

1152921502459363327

5

37252902972412109374

931322574554443359374

6

6140942214308087169023

221073919719792987930623

7

459986536543383514831102

22539340290682762964843362

8

19342813113825270702276607

1237940039285309906154946559

9

523347633027314783628601598

42391158275215791732030243902

10

9999999999999799999999999999

999999999999997999999999999999

11

144209936106498474538008897598

17449402268886398964062464922498

12

1648446623609509976114114592767

237376313799769775514907142258687

13

15502932802662388340516763706942

2619995643649944858008765404651534

14

123476695691247913602216130740223

24201432355484595110804846128201727

15

852226929923929215696331787109374

191751059232884085792703582763671874

16

5192296858534827484415308253364223

1329227995784915870597964051066650623

17

28351092476867700550974454247261182

8193465725814765550829154925772587262

18

140485060518220185533687199336726527

45517159607903340342300433545332981759

19

638411683925748516533591945348174398

230466617897195215015147265346183696802

20

2684354559999999996723199999999999999

1073741823999999999934463999999999999999

b \ n

16

17

2

4294836223

17179607039

3

1853020102758398

16677181441386242

4

18446744065119617023

295147905144993087487

5

23283064365081787109374

582076609133148193359374

6

7958661109940758664577023

286511799958036578519220223

7

1104427674243854180444159998

54116956037951646407931686434

8

79228162514263774643590529023

5070602400912913102387185451007

9

3433683820292508778617471385598

278128389443693477902922376898622

10

99999999999999979999999999999999

9999999999999999799999999999999999

11

2111377674535255193646155527065598

255476698618765888540125388760812898

12

34182189187166851741600324176052223

4922235242952026699599891029641134079

13

442779263776840696973479973782425598

74829695578286077996128097634382036622

14

4743480741674980698344536624171188223

929722225368296217668304193300163985407

15

43143988327398919487273740081787109374

9707397373664756887395323027801513671874

16

340282366920938463426481119284349108223

87112285931760246646033603692173956481023

17

2367911594760467245746783913569617510398

684326450885775034047292239401982236958974

18

14747559712960682275034284525972090650623

4778209346999261057185429179446073052889087

19

83198449060887472630852053678406676582398

30034640110980377619934885304785063124939842

20

429496729599999999998689279999999999999999

171798691839999999999973785599999999999999999

b \ n

18

2

68718952447

3

150094634522158142

4

4722366482732206260223

5

14551915228359222412109374

6

10314424798490332426258612223

7

2651730845859650214951827560702

8

324518553658426690754359001612287

9

22528399544939174111650877280774398

10

999999999999999997999999999999999999

11

30912680532870672624553518309902990398

12

708801874985091845328097640447798673407

13

12646218552730347184044578267057541624382

14

182225556172186058674086472096308696449023

15

2184164409074570299705328653896331787109374

16

22300745198530623141526273539682622215553023

17

197770344305988984840117475889639033680856062

18

1548139828427760582529416832015087751413628927

19

10842505080063916320800242179638132592798465598

20

68719476735999999999999475711999999999999999999

b \ n

19

2

274876858367

3

1350851715348469154

4

75557863725364567605247

5

363797880709133148193359374

6

371319292745658060942710145023

7

129934811447122997319381774952162

8

20769187434139310225891609165168639

9

1824800363140073124657348542510599742

10

99999999999999999979999999999999999999

11

3740434344477351388794157524466552764098

12

102067469997853225734274620210636713361407

13

2137210935411428674138619814101735241176974

14

35716209009748467500276332433809176278335487

15

491436992041778317434319661716289520263671874

16

5708990770823839524232992762070528716884148223

17

57155629504430816618801600850047605576162511102

18

501597304410594428739555133574630512939011932159

19

3914144333903073791808958649956969637289454544482

20

27487790694399999999999989514239999999999999999999

b \ n

20

2

1099509530623

3

12157665452083359998

4

1208925819612430151450623

5

9094947017729091644287109374

6

13367494538843726755521965850623

7

6366805760909027826156902543999998

8

1329227995784915870597964051066650623

9

147808829414345923291767879288269439998

10

9999999999999999999799999999999999999999

11

452592555681759518057548060359104084639998

12

14697715679690864505819888030165531176730623

13

361188648084531445929882867713790394945439998

14

7000376965910699630056336531667655955394330623

15

110573323209400121422731234605009365081787109374

16

1461501637330902918203682414864643790397583130623

17

16517976926780506002833792701068770733831682559998

18

162517526629032594911616296718211643658897796890623

19

1413006104539009638843035425873510510715988051359998

20

10995116277759999999999999790284799999999999999999999

 

I numeri di Carol generalizzati sono stati studiati principalmente cercando tra essi i numeri primi, che sono primi vicini a potenze.

 

Un numero di Carol generalizzato può essere primo solo se la base è pari; la tabella seguente mostra quelli noti (Ray Chandler, Cletus Emmanuel, Steven Harvey, Pab Ter, Eric W. Weisstein, The Online Encyclopedia of Integer Sequences http://oeis.org, M. Fiorentini, 2018).

b

Valori di n che producono primi

Limite della ricerca esaustiva

2

2, 3, 4, 6, 7, 10, 12, 15, 18, 19, 21, 25, 27, 55, 129, 132, 159, 171, 175, 315, 324, 358, 393, 435, 786, 1459, 1707, 2923, 6462, 14289, 39012, 51637, 100224, 108127, 110953, 175749, 185580, 226749, 248949, 253987, 520363, 653490

653490

4

1, 2, 3, 5, 6, 9, 66, 162, 179, 393, 3231, 19506, 50112, 92790, 326745

326745

6

1, 2, 6, 7, 20, 47, 255, 274, 279, 308, 1162, 2128, 3791, 9028, 9629, 10029, 13202, 38660, 46631, 48257, 117991

117991

8

1, 2, 4, 5, 6, 7, 9, 43, 44, 53, 57, 105, 108, 131, 145, 262, 569, 2154, 4763, 13004, 33408, 58583, 61860, 75583, 82983, 217830

217830

10

1, 8, 21, 123, 4299, 6128, 11760, 18884, 40293

70000

12

3, 29, 51, 7824, 15456, 22614, 28312, 47014

47014

14

1, 6, 13, 45, 74, 240, 553, 12348, 13659

40000

16

1, 3, 33, 81, 9753, 25056, 46395

163372

18

2, 8, 30, 98, 110, 185, 912, 2514, 4074, 10208, 15123, 19395

20000

20

1, 2, 53, 183, 1281, 1300, 8041, 29936

29936

22

1, 8, 35, 88, 503, 8642, 8743, 14475, 72820

72820

24

2, 27, 4950, 20047

20047

26

159, 879, 4744, 5602, 74387

74387

28

1, 22, 127, 165, 2520, 6492, 6577, 22960, 25528

25528

30

1, 6, 19, 30, 166, 495, 769, 826, 1648, 3993

20000

32

2, 3, 5, 11, 35, 63, 87, 37116, 130698

130698

34

1, 4, 258

2000

36

1, 3, 10, 137, 154, 581, 1064, 4514, 6601, 19330

58995

38

1, 2, 13, 560

2000

40

4, 15, 39, 138,

2000

42

3, 6, 14, 15, 29, 78, 195, 255, 272, 713,

2000

44

1, 7, 30, 90, 1288, 1947

2000

46

12, 269, 1304

2000

48

1, 2, 4, 6, 12, 13

2000

50

1, 3, 4, 9, 31, 66, 115, 430, 1233

2000

52

2, 14, 24, 85

2000

54

9, 17, 65, 963

2000

56

1, 2, 3, 11, 177, 1698

2000

58

88, 1720

2000

60

2, 5, 155

2000

62

1, 454, 1573

2000

64

1, 2, 3, 22, 54, 131, 1077, 6502, 16704, 30930, 108915

108915

66

12, 475, 1902

2000

68

4, 59, 779

2000

70

1, 5, 9, 18, 24, 277, 337, 431, 516

2000

72

1, 9, 187

2000

74

183

2000

76

1, 2, 32, 37, 51,

2000

78

1, 2, 85

2000

80

320, 363

2000

82

24, 1074, 1212

2000

84

4, 66, 82, 140, 1253, 1922

2000

86

3, 39, 75, 1120

2000

88

2, 5, 9, 134, 594

2000

90

1, 5, 43, 1105

2000

92

3, 4, 6, 60, 78, 116, 638

2000

94

1, 2, 666

2000

96

5, 106, 448, 561, 609

2000

98

2, 21, 130, 378, 896

2000

100

4, 3064, 5880, 9442

35000

 

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