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Kynea generalizzati (numeri di)

Teoria dei numeri 

t

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

 

In base b maggiore di 2 il numero di Kynea generalizzato (bn + 1)2 – 2 si rappresenta con una cifra 1, seguita da n – 1 cifre b – 1, un altro 1 e n cifre b – 1. Per esempio, per b = 6 e n = 4 abbiamo il numero di Kynea generalizzato 1682207 = 1000155556.

 

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

 

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

b \ n

1

2

3

4

5

2

7

23

79

287

1087

3

14

98

782

6722

59534

4

23

287

4223

66047

1050623

5

34

674

15874

391874

9771874

6

47

1367

47087

1682207

60481727

7

62

2498

118334

5769602

282508862

8

79

4223

263167

16785407

1073807359

9

98

6722

532898

43059842

3486902498

10

119

10199

1001999

100019999

10000199999

11

142

14882

1774222

214388162

25937746702

12

167

21023

2989439

430023167

61917861887

13

194

28898

4831202

815787842

137859234434

14

223

38807

7535023

1475865887

289255730623

15

254

51074

11397374

2562991874

576651909374

16

287

66047

16785407

4295098367

1099513724927

17

322

84098

24147394

6975924482

2015996740162

18

359

105623

34023887

11020170527

3570471005759

19

398

131042

47059598

16983823682

6131071209998

20

439

160799

64015999

25600319999

10240006399999

b \ n

6

7

8

2

4223

16639

66047

3

532898

4787342

43059842

4

16785407

268468223

4295098367

5

244171874

6103671874

152588671874

6

2176875647

78364723967

2821113266687

7

13841522498

678224719934

33232942099202

8

68720001023

4398050705407

281475010265087

9

282430599362

22876802020898

1853020274945282

10

1000001999999

100000019999999

10000000199999999

11

3138431919842

379749872557582

45949730292289922

12

8916106420223

1283918536212479

184884259754999807

13

23298094776098

3937376511196322

665416610814641282

14

56693927434367

11112007036385023

2177953340760949247

15

129746360671874

29192926367109374

6568408360838671874

16

281475010265087

72057594574798847

18446744082299486207

17

582622285504898

168377827380078274

48661191889618383362

18

1156831449450623

374813368806521087

121439531118634172927

19

2213315013157922

799006687570627598

288441413601588293762

20

4096000127999999

1638400002559999999

655360000051199999999

b \ n

9

10

11

2

263167

1050623

4198399

3

387459854

3486902498

31381413902

4

68720001023

1099513724927

17592194433023

5

3814701171874

95367451171874

2384185888671874

6

101559976823807

3656158560995327

131621704567861247

7

1628413678617662

79792266862562498

3909821052537641534

8

18014398777917439

1152921506754330623

73786976312018075647

9

150094636071840098

12157665466030497602

984770902246373352098

10

1000000001999999999

100000000019999999999

10000000000199999999999

11

5559917318208126862

672749994984434858402

81402749387410384454542

12

26623333291204804607

3833759992571309850623

552061438913922434334719

13

112455406973166391874

19004963775156516422498

3211838877958439425945442

14

426878854251958836223

83668255425863310870527

16398978063363920236212223

15

1477891880112287109374

332525673009118388671874

74818276426809444287109374

16

4722366483007084167167

1208925819616828197961727

309485009821380253096869887

17

14063084452304900744002

4064231406651604510202498

1174562876521217002766677954

18

39346408075693256156159

12748236216403219108890623

4130428534112457865337868287

19

104127350298556616928398

37589973457558220325871202

13569980418174323888319168398

20

262144000001023999999999

104857600000020479999999999

41943040000000409599999999999

b \ n

12

13

2

16785407

67125247

3

282430599362

2541869016974

4

281475010265087

4503599761588223

5

59604645263671874

1490116121826171874

6

4738381342675181567

170581728205699596287

7

191581231408248988802

9387480337841532326462

8

4722366483007084167167

302231454904756805304319

9

79766443077437368936322

6461081889231757030588898

10

1000000000001999999999999

100000000000019999999999999

11

9849732675813887951465282

1191817653772789987884420622

12

79496847203408676334338047

11447545997288495541626339327

13

542800770374417108941840322

91733330193269222408612800514

14

3214199700417854324575838207

629983141281878811032759681023

15

16834112196028492067138671874

3787675244106356221644287109374

16

79228162514264900543497371647

20282409603651679431146506027007

17

339448671314613069887978576642

98100666009922860251128755659842

18

1338258845052397016102500835327

433595865796975925236404837826559

19

4898762930960851244346133410242

1768453418076865785301549519843598

20

16777216000000008191999999999999

6710886400000000163839999999999999

b \ n

14

15

2

268468223

1073807359

3

22876802020898

205891160792462

4

72057594574798847

1152921506754330623

5

37252902996826171874

931322574676513671874

6

6140942214621543825407

221073919721673727868927

7

459986536546096407122498

22539340290701753210883134

8

19342813113842862888321023

1237940039285450643643301887

9

523347633027406290798421442

42391158275216615296558622498

10

10000000000000199999999999999

1000000000000001999999999999999

11

144209936106499993537343230562

17449402268886415673055142585102

12

1648446623609515111787972788223

237376313799769837142993440604159

13

15502932802662404090022306504098

2619995643649945062752337461014562

14

123476695691247958050243432972287

24201432355484595733077228359450623

15

852226929923929332468035888671874

191751059232884087544279144287109374

16

5192296858534827772645684405075967

1329227995784915875209650069494038527

17

28351092476867701224485760484864898

8193465725814765562278847131811850434

18

140485060518220187032940669665050623

45517159607903340369286996011242815487

19

638411683925748519729618688479710882

230466617897195215075871773465682889998

20

2684354560000000003276799999999999999

1073741824000000000065535999999999999999

b \ n

16

17

2

4295098367

17180131327

3

1853020274945282

16677181957946894

4

18446744082299486207

295147905213712564223

5

23283064365692138671874

582076609136199951171874

6

7958661109952043104206847

286511799958104285156999167

7

1104427674243987112166438402

54116956037952576929987635262

8

79228162514264900543497371647

5070602400912922109586440191999

9

3433683820292516190698226792962

278128389443693544611649175564898

10

100000000000000019999999999999999

10000000000000000199999999999999999

11

2111377674535255377445074981354242

255476698618765890561913502757987982

12

34182189187166852481137359756197887

4922235242952026708474335456602882047

13

442779263776840699635146410515144962

74829695578286078030729761311907388354

14

4743480741674980707056349975408672767

929722225368296217790269580217488769023

15

43143988327398919513547373504638671874

9707397373664756887789427529144287109374

16

340282366920938463500268095579187314687

87112285931760246647214195312891367784447

17

2367911594760467245941428681072284984322

684326450885775034050601200449527584015682

18

14747559712960682275520042650358467657727

4778209346999261057194172825685027839016959

19

83198449060887472632005819332677161253122

30034640110980377619956806852216202333683598

20

429496729600000000001310719999999999999999

171798691840000000000026214399999999999999999

b \ n

18

2

68720001023

3

150094636071840098

4

4722366483007084167167

5

14551915228374481201171874

6

10314424798490738666085285887

7

2651730845859656728606219202498

8

324518553658426762811953039540223

9

22528399544939174712029418468770882

10

1000000000000000001999999999999999999

11

30912680532870672646793187563871916322

12

708801874985091845434590973571339649023

13

12646218552730347184494399894865371196898

14

182225556172186058675793987513151243419647

15

2184164409074570299711240221416473388671874

16

22300745198530623141545163005614100796407807

17

197770344305988984840173728227447304580820098

18

1548139828427760582529574217647388937563930623

19

10842505080063916320800658689039324237764596962

20

68719476736000000000000524287999999999999999999

b \ n

19

2

274878955519

3

1350851719997515022

4

75557863726464079233023

5

363797880709209442138671874

6

371319292745660498381670187007

7

129934811447123042914962516444734

8

20769187434139310802352361468592127

9

1824800363140073130060755413202568098

10

100000000000000000019999999999999999999

11

3740434344477351389038793886260210949262

12

102067469997853225735552540208119205068799

13

2137210935411428674144467495263237025619682

14

35716209009748467500300237649644971935924223

15

491436992041778317434408335229091644287109374

16

5708990770823839524233294993525432374177824767

17

57155629504430816618802557139790346181461899714

18

501597304410594428739557966516011934289717362687

19

3914144333903073791808966563635592278543811040398

20

27487790694400000000000010485759999999999999999999

b \ n

20

2

1099513724927

3

12157665466030497602

4

1208925819616828197961727

5

9094947017729473114013671874

6

13367494538843741380155726102527

7

6366805760909028145325967734448002

8

1329227995784915875209650069494038527

9

147808829414345923340398541124497155202

10

10000000000000000000199999999999999999999

11

452592555681759518060239060338834324676802

12

14697715679690864505835223070135321077219327

13

361188648084531445929958887568889918143195202

14

7000376965910699630056671204689357094600572927

15

110573323209400121422732564707701396942138671874

16

1461501637330902918203687250567922248914281955327

17

16517976926780506002833808957994397324121772166402

18

162517526629032594911616347711156509243210494640127

19

1413006104539009638843035576233404340899820824782402

20

10995116277760000000000000209715199999999999999999999

 

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

 

Un numero di Kynea generalizzato può essere primo solo se la base è pari; la tabella seguente mostra quelli noti (Ray Chandler, Cletus Emmanuel, Steven Harvey, Mark Rodenkirch, 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

1, 2, 3, 5, 8, 9, 12, 15, 17, 18, 21, 23, 27, 32, 51, 65, 87, 180, 242, 467, 491, 501, 507, 555, 591, 680, 800, 1070, 1650, 2813, 3281, 4217, 5153, 6287, 6365, 10088, 10367, 37035, 45873, 69312, 102435, 106380, 108888, 110615, 281621, 369581, 376050, 442052, 621443, 661478

661478

4

1, 4, 6, 9, 16, 90, 121, 340, 400, 535, 825, 5044, 34656, 53190, 54444, 188025, 221026, 330739

330739

6

1, 2, 3, 4, 9, 12, 30, 49, 56, 115, 118, 376, 432, 1045, 1310, 6529, 7768, 8430, 21942, 26930, 33568, 50800

50800

8

1, 3, 4, 5, 6, 7, 9, 17, 29, 60, 167, 169, 185, 197, 550, 12345, 15291, 23104, 34145, 35460, 36296, 125350

220492

10

22, 351, 1061

70000

12

1, 2, 8, 60, 513, 1047, 7021, 7506

20000

14

1, 5, 60, 72, 118, 181, 245, 310, 498, 820, 962, 2212, 3928, 5844, 5937

20000

16

2, 3, 8, 45, 170, 200, 2522, 17328, 26595, 27222, 110513

165369

18

1, 10, 21, 25, 31, 1083

20000

20

1, 15, 44, 77, 141, 208, 304, 1169, 3359, 5050, 22431, 34935

34935

22

3, 166, 814, 1851, 2197, 3172, 3865, 19791

20000

24

24, 321, 971, 984

20000

26

1, 2, 8, 78, 79, 111, 5276, 8226, 19545, 215057

215057

28

1, 2, 11, 15, 586, 993, 5048, 24990

24990

30

2, 3, 57, 129, 171, 9837, 30359, 157950

157950

32

1, 3, 13, 36, 111, 136, 160, 214, 330, 1273, 7407, 20487, 21276, 22123, 75210

132295

34

1, 2, 14, 29, 61, 146

2000

36

1, 2, 6, 15, 28, 59, 188, 216, 655, 3884, 4215, 10971, 13465, 16784, 25400

25400

38

6, 279

2000

40

2, 49, 144, 825

2000

42

1, 3, 4, 81, 119

2000

44

3, 195, 1482

2000

46

1, 54

2000

48

1, 207, 329, 1153

2000

50

4, 38, 93, 120

2000

52

3, 5, 166, 456

2000

54

1, 1910

2000

56

8, 14, 73, 122, 136, 706

2000

58

2, 21, 35, 213, 296, 1734

2000

60

1, 9, 21, 49, 1717

2000

62

1, 2, 212, 761

2000

64

2, 3, 30, 275, 11552, 17730, 18148, 62675

110246

66

172, 375, 945, 1751

2000

68

1, 3, 6, 32

2000

70

1, 7, 11

2000

72

354, 1920

2000

74

1, 3, 694

2000

76

1, 2, 3, 22, 29, 126, 173, 284

2000

78

3, 5, 13, 79

2000

80

29, 34, 45, 81, 148, 562, 1445

2000

82

3, 9, 595, 1866

2000

84

423

2000

86

8, 18, 36, 260

2000

88

1, 40, 324, 1072

2000

90

11, 504, 520, 676

2000

92

1, 2, 900, 1795

2000

94

5

2000

96

2, 13, 15, 54

2000

98

4, 12, 35, 459

2000

100

11

35000

 

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