Talk:Regular prime

Untitled
Alright, I can't follow exactly what a regular prime is, but I'm pretty sure that 2 is either regular or irregular. Thus, it should appear on one of the lists of the first few fooregular primes. LizardWizard 04:06, Feb 14, 2005 (UTC)


 * That is a little like asking for a two-sided polygon and trying to distinguish it from a straight line. --Henrygb 16:01, 15 July 2005 (UTC)


 * 2 would be regular, FWIW. Charles Matthews 16:30, 15 July 2005 (UTC)


 * But it is not considered either way, in part because of the statement in the article "Historically, regular primes were considered by Kummer since he was able to prove that Fermat's last theorem holds true for regular prime exponents (and consequently for all exponents that were multiples of regular primes)." --Henrygb 14:11, 16 July 2005 (UTC)


 * Sloppy, I think. But let's have odd primes only as regular, then. Charles Matthews 20:50, 16 July 2005 (UTC)
 * By the definition we give 2 should be trivially regular; the field in question is gotten by adjoining -1 to the rationals (i.e., is the rationals); and its class number is 1. But odd prime is probably best. Septentrionalis 13:55, 14 July 2006 (UTC)

Kummer
Isn't Kummer's proof for the first case of FLT, where p doesn't divide any of the three bases, a, b, c? Septentrionalis 13:55, 14 July 2006 (UTC)

Jensen
The infinitude of irregular primes wasn't proven by Johan Jensen, but by K L Jensen; see or search Zentrallblatt for the reference, which is: K. L. Jensen, "Om talteoretiske Egenskaber ved de {\it Bernoulliske} Tal" (Danish), published in Nyt Tidsskr. for Math. 26, pages 73--83 (1915). Throwawayhack 19:38, 11 April 2007 (UTC)
 * About this Jensen, I found a genealogy file online with the following content (in Danish, but linkified by me): Kaj Løchte Jensen, født Hjørring 27.2.1893, døbt (Catharinæ) 7.5.; død Risskov 21.12.1932, begr. Hjørring 29.12. Student 1914, cand. phil. 1915. Syg fra 1916 til sin død. I am certain this is the correct person. It says he was ill from 1916 to his death. I speculate that this was mental illness; most likely he was a patient at Jydske Asyl most of his adult life, and died there. /80.71.142.78 (talk) 13:27, 22 January 2021 (UTC)

Way too hard to follow
This article is simply way too hard to follow. It would be much better if someone actually made an effort to explain the subject instead of just relying on an intense burst of jargon in the opening sentence that is utterly impossible for the layman to understand. —Preceding unsigned comment added by 141.161.109.134 (talk) 16:39, 10 March 2008 (UTC)


 * That is how sources define regular primes. I think it would be too complicated to try to explain the technical linked terms here, but I have moved the simpler looking Bernoulli number criterion to the first paragraph: . Is that better? PrimeHunter (talk) 21:51, 10 March 2008 (UTC)

This is not an article for laymen. This is a technical article for readers who have some familiarity with algebraic number theory. — Preceding unsigned comment added by 204.99.170.188 (talk) 04:59, 17 June 2011 (UTC)

"... way too hard to follow." Agreed. However, since "prime" is easily understood, are we not only one step away (on the "complexity axis", if you will) to begin to wonder about "regular primes" and "irregular primes"? Wikipedia has importance as a bridge for those who do not know, but desire to do so. For that reason alone, "...not for laymen..." seems a sad flag of surrender given the depth of other expository writing herein, cf. Graham's Number, Abelian Algebra, Fermat's Conjecture. — Preceding unsigned comment added by 67.183.183.75 (talk) 03:05, 16 June 2013 (UTC)

List of irregular pairs sorted by p
These are irregular pairs with odd prime p <= 2069.

3

5

7

11

13

17

19: 11

23

29

31: 23

37: 32

41

43: 13

47: 15

53

59: 44

61: 7

67: 27, 58

71: 29

73

79: 19

83

89

97

101: 63, 68

103: 24

107

109

113

127

131: 22

137: 43

139: 129

149: 130, 147

151

157: 62, 110

163

167

173

179

181

191

193: 75

197

199

211

223: 133

227

229

233: 84

239

241: 211, 239

251: 127

257: 164

263: 100, 213

269

271: 84

277: 9

281

283: 20

293: 156

307: 88, 91, 137

311: 87, 193, 292

313

317

331

337

347: 280

349: 19, 257

353: 71, 186, 300

359: 125

367

373: 163

379: 100, 174, 317

383

389: 200

397

401: 382

409: 126

419: 159

421: 240

431

433: 215, 366

439

443

449

457

461: 196, 427

463: 130, 229

467: 94, 194

479

487

491: 292, 336, 338, 429

499

503

509: 141

521

523: 400

541: 86, 465

547: 270, 486

557: 222

563: 175, 261

569

571: 389

577: 52, 209, 427

587: 45, 90, 92

593: 22

599

601

607: 592

613: 522

617: 20, 174, 338

619: 371, 428, 543

631: 80, 226

641

643

647: 236, 242, 554

653: 48

659: 224

661

673: 408, 502

677: 529, 628

683: 32

691: 12, 200, 549

701

709: 493

719

727: 378

733

739: 495

743

751: 290, 297, 711

757: 514

761: 105, 260

769: 247

773: 499, 732

787

797: 220

809: 330, 628

811: 544, 727

821: 623, 744

823

827: 102

829

839: 66

853

857

859

863

877: 287, 868

881: 162

883

887: 418, 561

907: 319, 819

911

919

929: 520, 723, 820

937

941: 687, 805

947

953: 156

967: 13

971: 166, 825

977

983: 557

991

997

1009

1013: 411

1019: 89, 289, 501

1021

1031: 279

1033

1039: 293

1049: 343

1051: 361

1061: 474

1063

1069: 545, 613

1087

1091: 888

1093

1097

1103

1109

1117: 794

1123

1129: 348

1151: 115, 534, 784, 968

1153: 802

1163: 871

1171

1181

1187: 167, 335

1193: 262

1201: 676

1213

1217: 784, 866, 1118

1223: 365

1229: 784, 931

1231: 767

1237: 874

1249

1259

1277: 481

1279: 509, 518

1283: 510, 1029

1289

1291: 206, 675, 824

1297: 202, 220

1301: 176

1303

1307: 382, 852, 1071

1319: 304, 1187

1321

1327: 466

1361: 441

1367: 234

1373

1381: 266, 609

1399: 1115

1409: 358, 363

1423: 653

1427: 1315, 1411

1429: 627, 996

1433

1439: 574, 1193

1447: 1081

1451

1453: 323

1459

1471

1481

1483: 224

1487

1489

1493

1499: 94

1511

1523: 265, 1310

1531: 473, 849

1543

1549

1553

1559: 862, 1403

1567

1571

1579

1583: 439

1597: 842

1601: 53

1607

1609: 1356

1613: 172

1619: 560

1621: 783, 980

1627

1637: 591, 718

1657

1663: 270, 1508, 1627

1667

1669: 388, 1086

1693: 1601

1697: 607

1699

1709

1721: 30

1723: 593, 1167

1733: 483, 810, 942

1741

1747

1753: 712

1759: 1003, 1520

1777: 1192

1783

1787: 397, 963, 1606

1789: 848, 1442

1801: 869

1811: 550, 698, 1520

1823

1831: 349, 1274

1847: 954, 1016, 1558

1861

1867: 263

1871: 1794

1873: 1705

1877: 925, 1026

1879: 199, 423, 1260

1889: 242, 1613

1901: 1479, 1722

1907: 369

1913

1931: 1763

1933: 1058, 1320, 1801

1949

1951: 257, 1656

1973

1979: 148

1987: 510, 933

1993: 179, 912

1997: 772, 1731, 1888

1999

2003: 60, 600

2011: 983, 1601

2017: 1204

2027

2029

2039: 69, 853, 1300, 1699

2053: 1932

2063: 1977

2069: 505

— Preceding unsigned comment added by 49.215.7.19 (talk) 14:53, 19 August 2015 (UTC)

List of irregular pairs sorted by n
These are irregular pairs with n <= 199.

0

1

2

3

4

5

6

7: 61

8

9: 277

10

11: 19, 2659

12: 691

13: 43, 967

14

15: 47, 4241723

16: 3617

17: 228135437

18: 43867

19: 79, 349, 87224971

20: 283, 617

21: 41737, 354957173

22: 131, 593

23: 31, 1567103, 1427513357

24: 103, 2294797

25: 2137, 111691689741601

26: 657931

27: 67, 61001082228255580483

28: 9349, 362903

29: 71, 30211, 2717447, 77980901

30: 1721, 1001259881

31: 15669721, 28178159218598921101

32: 37, 683, 305065927

33: 930157, 42737921, 52536026741617

34: 151628697551

35: 4153, 8429689, 2305820097576334676593

36: 26315271553053477373

37: 9257, 73026287, 25355088490684770871

38: 154210205991661

39: 23489580527043108252017828576198947741

40: 137616929, 1897170067619

41: 763601, 52778129, 359513962188687126618793

42: 1520097643918070802691

43: 137, 5563, 13599529127564174819549339030619651971

44: 59, 8089, 2947939, 1798482437

45: 587, 32027, 9728167327, 36408069989737, 238716161191111

46: 383799511, 67568238839737

47: 285528427091, 1229030085617829967076190070873124909

48: 653, 56039, 153289748932447906241

49: 5516994249383296071214195242422482492286460673697

50: 417202699, 47464429777438199

51: 5639, 1508047, 10546435076057211497, 67494515552598479622918721

52: 577, 58741, 401029177, 4534045619429

53: 1601, 2144617, 537569557577904730817, 429083282746263743638619

54: 39409, 660183281, 1120412849144121779

55: 2749, 3886651, 78383747632327, 209560784826737564385795230911608079

56: 113161, 163979, 19088082706840550550313

57: 5303, 7256152441, 52327916441, 2551319957161, 12646529075062293075738167

58: 67, 186707, 6235242049, 37349583369104129

59: 1459879476771247347961031445001033, 8645932388694028255845384768828577

60: 2003, 5549927, 109317926249509865753025015237911

61: 6821509, 14922423647156041, 190924415797997235233811858285255904935247

62: 157, 266689, 329447317, 28765594733083851481

63: 101, 6863, 418739, 1042901, 91696392173931715546458327937225591842756597414460291393

64: 1226592271, 87057315354522179184989699791727

65: 25349, 85297, 12989360531548972327803547656767339375006258039696642617507398739

66: 839, 159562251828620181390358590156239282938769

67: 105075119, 508679461, 155312172341, 155737429414728656346088798821794971221082287203779

68: 101, 123143, 1822329343, 5525473366510930028227481

69: 2039, 66041, 29487071944189, 15138431327918641, 484510273389546188488228650507868434878928667

70: 688531, 20210499584198062453, 3090850068576441179447

71: 353, 2586437056036336027701234101159, 312210239910371909857727050224078527206101218811162523

72: 3112655297839, 1872341908760688976794226499636304357567811

73: 2341, 4014623, 24259423, 30601587075439337, 482132394333433671681711454588230154366429871388577

74: 923038305114085622008920911661422572613197507651

75: 193, 34629826749613, 4207222848740394629, 22060457167870794468746201, 2084356623048603581413664959497121

76: 58231, 22284285930116236430122855560372707885169924709

77: 145007, 3460859370585503071, 581662827280863723239564386159, 2046494332840854220697501265093364699008503

78: 787388008575397, 33364652939596337, 1214698595111676682009391

79: 2740019561103910291228417123054994825316979387, 2653485331720644497330964662311698866076250195175420143

80: 631, 10589, 5009593, 141795949, 969983603247099340617362338794263364709

81: 7701306020743, 3572363603188902175396213, 38846764704262590259300934027789308313372462321468975007497723

82: 4003, 38189, 267564809427749238542649199594159701256952090203379

83: 4395659, P98

84: 233, 271, 68767, 167304204004064919523, 2786903827245650053311240128451928279

85: 4397, 739762335239015186706527735192795520726707, P62

86: 541, 21563, 1317161453956258384019814501134446230216181176462038507

87: 311, 390751, 46053168570671, P92

88: 307, 2682679, P60

89: 1019, 588528876550967927, 16292380848703930709213, P72

90: 587, 1758317910439, P57

91: 307, 1964309984670433843694580256152588601980583986713006597, P64

92: 587, 108023, P63

93: 7096363493, 7308346963823, 120476813565517, P85

94: 467, 1499, 2459153, 4217126617741589575995641, 3577922013827274976860631840900289

95: 53089, 20609829625906839913745698187, 180986288780569828566819992453, P66

96: 7823741903, 4155593423131, 10017952436526113, 96454277809515481, 6735480167773644873691271

97: 835823, 2233081, 1951860271597317997069749059, 9416370608392625586845089085196635167, 15038064837301437151577874662131773543986118119

98: 2857, 3221, 1671211, 9215789693276607167, 9778263152874996218584617307180549616435599

99: 376003429, 5160267661, 4363907262506552373343, P94

100: 263, 379, 28717943, 65677171692755556482181133, P45

101: 37425288022730945391029688959184999895624961872309545117125516009, P69

102: 827, 17833331, 86023144558386407, 299116358909830276447443337, 8417841532399822926231891659

103: 8647, 198943119321654388058500384086517043195558620394228397755851, P79

104: 776253902057299, 6644689804135385589700423, P45

105: 761, 2477, P138

106: 3967, 37217, 77272435237709, P65

107: 4858416191, 98985829942673, 1150887066548393492521971151372616707, P88

108: 656884664663, 23657486502844933, P69

109: 1462621, 8445961, 4675063901, 142310099610444540136513337624011455219491905450233709110803, P66

110: 157, 76493, 150235116317549231, 36944818874116823428357691, P44

111: 509053, 116904299, 134912677, 748079839770433, P120

112: 887569, 8065483, P86

113: 8185757, 617575481323, 1522046069820268709, 265053146030428876430329, P94

114: P97

115: 1151, 5290253211544727, 22557103319451713, 2565948669867461313318215567,, 118972684453835135392634192556273454718187595705343, P52

116: 7559, 7438099, 6795944986967, P77

117: 1098948437923935829829, 17698520871521406115634951924463689, 11661906593316353058846911847709511061777523, P69

118: P100

119: 86611938909696635972683149781, 14465489614111569999691521198240690587831, P102

120: 6495690221, 8070196213, P93

121: 3783751296217426258321287795930437607814627399445703026216549471614268601506896712447, P85

122: 1545314586433142560447, 1545923474257037240728199709913, P54

123: 6997, 5571781, 1526627072504771936814787447304051999806158513499861179250074093751, P103

124: 74747, 162263, 14066893, 8262971607841, 3498285428145163, 16743250272239551, 559028822384715164688625676524544680328026657

125: 359, 6043, 26111, 118463, 3322981, 545893110893363273374339, 340434085979481287216483227078798002216360327742620827466139, P77

126: 409, 216363744721, P102

127: 251, 8647, 2941927, 51082969901, 44305294819613, 167237174851562092201, P128

128: 35089, 5953097, 12349588663, 13349390911530343, 6996505560116602097773394576621473, P46

129: 139, 70663, 238229, 91486803609919, 33397018471037747, 38280927951817207, 1823694188853227904949904627, 252181896718842913832793991507441358249, P64

130: 149, 463, 2264267, 3581984682522167, P92

131: 7753, 1476089794776829083186088935097008657059172428532830012025534581117, P125

132: 804889, 10462099, P112

133: 223, 29851802334168169974816953851717491292941047351257691, P142

134: 42859, 338420464438865099, 6005440277888093849051345046242759, P65

135: 4463, 6255577, 321639994822891, 214074317717282326017498018953, P148

136: 10995389191, 29835096585483934621, P98

137: 307, P200

138: 2957, 9733, 1373021071, 554744941981, 756906736720877, 9959596661942153266426403135574603847379, P48

139: 1652869, 2401621, 2152717661, 12254459673349, 34356165690119899, P157

140: 17681, 6251263, 1914841969, 44124706530665069, 49919098955213994432243162077, P68

141: 509, 24379, 232439, P199

142: 4003, 111781954908479484383981, P105

143: 2978734769, 8557612247, P197

144: 6500309593, P135

145: 157823, 72378952903, 978576085558923501179, 170513218370189155958048891371, P149

146: 1377371, 22639970526343, 6726159702783854797, 37996324998547740539691528067877, 1754821172656266926966923716442469, 4036138055144761320534304068715607

147: 149, 3343, 48711863, P213

148: 1979, 30817, 172331, 4975417507662031677157, 1248863436460860523032749, P84

149: 37663, 1392851, 238661068231279, 9792965881638773573903833054687345233935055294576309732623214361308544381554891261555176895512167, P105

150: 153427, 2517869, 5810708205829, 21664796739499531040947, 2409795082015672566733218756037, P72

151: 649724429, 13621373428254587, 111381973999260228282238167431335585433059, 141000785435055584990802770623376828639701620449, 1193722701951839165150952421110134799375110059438006543, P67

152: 9743, 230165249, 3720341037827029338655181363717044961, 37835716074058426890725596550304118196498159, P52

153: 25149009833, 18051556174129735359181, 3957666449530267510589053, 438321334095183824658294709367, 1767165620447279332603545521778737, P115

154: 2213, 125929, 1569473, 384785986561, 83697900175217338619182484215561594711, P85

155: 6779, 11232821, 139668927262709710013, 18526916368663653639476639296503123344751098807, P163

156: 293, 953, 167604149935534865064907, 94884267483295622200143616179947, P101

157: 1088239, 11102051, 100153607, 130223537, 227071134239, P198

158: 5309, 10463, 42487, 50929, 10481243, 1749855366374444668341589937589990596230011702982943022995765557, P66

159: 419, 6127384099, 5519160811451003, 21846610457557327344557544721254743124424589840548059277069, P162

160: 6807624661, 40094692599177383, 12830086712891890983430059948563, 1744826505423362390046833266050403703791289, P62

161: 2543, 74567, 1397180350713344753917243720749772943, 577769769631936427594208754309870324583454884649989987999, P148

162: 881, 1356077, 22767953612964575737798380133664917, 6529339197711546201002267709627054755633, P82

163: 373, 174175655449, 9110273370757630942069196741476911790511415633167, 227268167662143428963168079206940398921235185460026379651599777158266191, P122

164: 257, 1434031, 104386532651, 2903061743891, 9898920431428993, 716563254696398958818280936436469476402929223, P72

165: 2591, 805781401, 65492318651, 63488848774356502730543060633, 35148563470374881695890678645018818080373286214180248173, P154

166: 971, 85754183, 4877261843, 311318618909, 37074748512889, 60519068332988964084651891032717, 117092287618059239620235259605532189619, P52

167: 1187, 3343, 2106833, 1793322371, 38720170561, 50150236900098278077, C214

168: 28211, 19254163575306510187, 10094494587919631151637, 1104790013606614517447652064159916593151013237167098468511, P71

169: 558587, 861433987, 86771436435012390277, 16669946091045819953746739504935989816954757, P187

170: 2504129, 751612064207, P154

171: 70727223023077, 1034326231547973051559, P239

172: 1613, P174

173: 208001, 743155422133, 2840083403239, 84005508665545362459530332377303058295981062837, P196

174: 379, 617, 8419, 264899, 6659961564676431900928667503, 93193525172231316499819296116439042677911, P96

175: 563, 1199714371, C269

176: 1301, 333026571343, 110783038328477, 124813394943812621, 161682280601750017807051565594123306297091203059258385807467, P79

177: 8629, 708473, C273

178: 9433, 129180506448277, 182363423482601296739326836920802601519, P129

179: 1993, 459293, 463761292322706383024896643864485966179727861, P235

180: 403783, 972607943, 249829228470043, 2076252436787489535833, 4241477436592626145879, P127

181: 6923483330327017, 2551820624140592595425268234790493384082614077333545481, P215

182: 73107144475261423, 311089841618633327, 3627027615648746666477, 2122174114227419648093461601, 8327616545832330042958707170640293981592673849, P59

183: 2141, 6263, 42451, C287

184: 389621, 21983088204089362967, P169

185: 2804389579706797633, C284

186: 353, 1301919607, 922966808867, 9161904079472101, 107856487459065437158480612197729025253133196481, P109

187: 241679, 36999818357, 22658461432253, 54342802734882461, 1086110887390889008410968159777, C229

188: 53377, 18974159366624817405627752670504479132613571595050983959444958694223874973021, P117

189: C310

190: 5101, 60860762760882373, 174262092707971020104538709609, 131410417049682678695361379910908937724385222976450357113181662889, P87

191: 559570609330768709, 6386014734599369410586902768943, C265

192: 40833790860803270336710504624737304862569304959957, P163

193: 311, 1469840300183, 6895766514961118059, 1269672106384218692615790692911, 3136426284780626707900456422430291796488693834153, 527970984341964329713816165139855340458712234958050634766242533991, P134

194: 467, 1649059, 95812875598016433532365219084195658008281, 2505663816946125800334764295277843127504620817, P111

195: 34110029, 28024555486506389, 2436437750204310804841, P278

196: 461, 825337, 58273617156601282072242637946609, 16936665062202361732611820380328005721, 1077894157071847644151421507667924461777695091, P91

197: 26034939865747697437451558982836040663625026070193, C274

198: 34470847, 723357738211, P201

199: 1879, 4339, 585653083, 2507798651531, 49639305210453901009432031, 105555375856176303898432906280701535874458049677, C230

— Preceding unsigned comment added by 49.215.7.19 (talk) 14:56, 19 August 2015 (UTC)

Assessment comment
Substituted at 02:33, 5 May 2016 (UTC)