Talk:Regular prime

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

text: Sloppy, I think. But let's have odd primes only as regular, then. Charles Matthews 20:50, 16 July 2005 (UTC)
replies: text: 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)
thread_title: Untitled

message: 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)
thread_title: Kummer

message: 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)
replies: text: 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)
thread_title: Jensen

message: 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)
replies: text: 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)
thread_title:  Way too hard to follow 

message: 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)
thread_title:  List of irregular pairs sorted by p

message: 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)
thread_title:  List of irregular pairs sorted by n 

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