Carol-Kynea prime

Definitions

In the context of the Carol/Kynea prime search, a Carol number is a number of the form [math]\displaystyle{ (b^n-1)^2-2 }[/math] and a Kynea number is a number of the form [math]\displaystyle{ (b^n+1)^2-2 }[/math]. A Carol/Kynea prime is a prime which has one of the above forms. A prime of these forms must satisfy the following criteria:

b must be even, since if it is odd then [math]\displaystyle{ (b^n±1)^2-2 }[/math] is always even, and thus can’t be prime.
n must be greater than or equal to 1. For any b, if n is 0 then (bⁿ±1)² is equal to 1, and thus yields -1 when 2 is subtracted from it. By definition -1 is not prime. If n is negative then (bⁿ±1)² is not necessarily an integer.
b may be a perfect power of another integer. However these form a subset of another base’s primes (ex. Base 4 Carol/Kynea primes are Base 2 Carol/Kynea primes where [math]\displaystyle{ n \bmod 2 \equiv 0 }[/math]). So it is not necessary to search these bases separately.

Due to the form of these numbers, they are also classified as near-square numbers (numbers of the form n²-k).

History

Carol and Kynea numbers were first studied by Cletus Emmanuel in 1995^[1], who named them after personal acquaintances^[2]. He searched these forms for primes up to the limit of 15000.

Starting in 2004, Steven Harvey maintained a search for this form. At this time Multisieve and cksieve were used to sieve these forms and PFGW was used to test for primality. The search went dormant in 2011 and was resurrected in 2015 by Mark Rodenkirch. Initially Multisieve was used, but then later on he wrote cksieve which would later become part of the Mtsieve framework.

On 2015-12-26 Mark opened a thread^[3] for a coordinated search of Carol/Kynea numbers on MersenneForum, which continues to this day (although now Gary Barnes, maintainer of NPLB and CRUS, maintains the search).

Top 5 Carol primes

Prime	Digits	Found by	Date
(290¹²⁴¹¹⁶-1)²-2	611246	Karsten Bonath	2019-03-01
(968⁷⁹⁵¹²-1)²-2	474826	Karsten Bonath	2021-01-13
(2⁶⁹⁵⁶³¹-1)²-2	418812	Mark Rodenkirch	2016-07-16
(2⁶⁸⁸⁰⁴²-1)²-2	414243	Mark Rodenkirch	2016-07-05
(178⁸⁷⁵²⁵-1)²-2	393937	Serge Batalov	2016-05-21

Top 5 Kynea primes

Prime	Digits	Found by	Date
(362¹³³⁶⁴⁷+1)²-2	683928	Karsten Bonath	2019-06-17
(2⁸⁵²⁷⁷⁰+1)²-2	513419	Ryan Propper	2019-07-14
(30¹⁵⁷⁹⁵⁰+1)²-2	466623	Serge Batalov	2016-05-22
(2⁶⁶¹⁴⁷⁸+1)²-2	398250	Mark Rodenkirch	2016-06-18
(1968⁵⁸⁵³³+1)²-2	385619	Clint Stillman	2017-11-30

OEIS sequences

These are available OEIS sequences:

Base	Carol	Kynea
2	A091515	A091513
6	A100901	A100902
10	A100903	A100904
14	A100905	A100906
22	A100907	A100908

Data

All bases

All bases with their own page are listed here: There are 382 sequences.

Carol-Kynea table

Bases which are a power of

There are 22 sequences.

Category

Bases without a Carol prime

There are 62 sequences.

Category

Bases without a Kynea prime

There are 61 sequences.

Category

Bases without a Carol and Kynea prime

There are 1 sequences.

Category

Remaining data

All bases <= 3000 not listed yet on their own page are tested to n=2000 and listed in a file.

The data file can be found here.

Base	Carol primes	Kynea primes
366	15, 43, 61, 339	32, 149, 828, 947
370	2, 23, 24, 194, 244	1, 3, 86, 164, 590, 791, 1613
372	1, 2, 7, 51	6, 7, 31, 73, 103, 409, 1408
374	735	22, 48
376	10, 1545	2, 3, 90
378	4, 10, 21, 48, 132, 763, 1277	3, 4, 7, 18, 173, 427
380	4	2, 3, 6, 44, 707, 1220, 1278, 1347, 1962
382	5, 1491	75, 588, 696
384	321	13
386	2, 3, 8, 9, 12, 105, 399	1, 13, 254
388	1, 9, 50, 63	282
390	3	1, 13, 79, 200, 441, 1645
392	1, 5, 15, 223	2, 32, 203
396	3	3, 13, 69, 130, 147
398	512	1, 2, 4, 749
402	6, 21, 135	6, 136
404	460	1, 44, 74, 230
406	1, 21, 49, 495, 1531	5, 6, 9, 79
408	72, 168, 1581, 1660	3, 275
410	3, 5	3, 58, 655
412	2, 3, 15, 33, 280, 503, 544, 663	32
414	2, 220	1, 121, 297
416	1, 2, 12	4, 5, 287
418	3, 4, 5, 6	270
420	8, 345	1, 645, 775
422	1, 6, 31, 42	647
424	9	1, 4, 48, 220, 700, 828
428	20, 200	1, 2, 7, 8, 292, 1466
430	1, 2, 3, 18, 56	4, 30
434	2, 15, 141	1
436	1, 6, 162	2, 6, 8, 1944
438	2, 8, 201, 1731, 1826	85, 106
440	2, 231	1, 2, 28, 35, 50, 98, 233
442	1, 97, 236, 397	1, 3, 28, 232
444	1, 2, 3, 8, 109	21, 82, 657, 1630
446	5, 172, 300	1, 2, 14, 44, 642, 769
450	1, 78	45, 120, 783
452	78	72, 88, 124
454	2, 11, 16, 22, 31	264, 1547
456	7, 34	3
458	9, 33, 114	12, 77
460	3, 5, 11, 1133, 1253, 1636	85, 288, 1687
462	5, 44, 113, 135, 421, 703, 1254	15, 911
464	133, 630	2, 665, 1646
466	11, 68	1, 3, 6, 135
468	1, 6, 8, 400, 603, 998	1, 2, 5, 56, 61, 62, 1774
470	1, 2, 3, 15, 128, 451	2, 106, 162, 288, 586
474	33, 231	3, 57
476	2, 8, 48	8, 12, 1030, 1692
478	3, 45, 900	18, 140, 1611
480	2, 50, 239	1, 3, 10, 13, 64, 153, 315, 535
482	1	2, 42, 129, 132
486	45, 138, 170	4, 15, 203, 1432
488	12, 111	1, 13, 85
490	1, 4, 13, 84, 210, 1885	1, 2, 7, 183, 545, 1359, 1895
492	1, 370, 1518	8, 555, 774
494	323	1, 16, 303
496	1, 440, 690	1, 4, 6, 26, 116, 134, 751
498	1, 5, 24, 92, 99, 1123	4, 9, 11, 216, 347
502	3	6
504	2, 10, 117, 821	1, 7, 34, 220, 490
506	1, 6, 136, 445	140, 404
508	2, 102, 816	30
510	4, 9, 254, 305	216, 231, 1546
514	1, 30, 117	3
516	100	2
518	3, 19, 177	10, 23, 55, 723, 819
520	3, 6	134, 1371
522	2, 1059	1, 3, 517
524	1, 8, 109, 156, 166, 244, 471	1, 3
526	1, 39	2, 3, 5, 154
528	3, 1059	3, 4, 9, 1985
530	5, 84, 1049	1, 93
532	1, 17, 128, 716, 1982	2
534	13, 510	2, 12, 170, 1476
536	3, 357, 653, 1472	7, 10, 70
538	319	2, 6, 120, 868, 1374
540	6, 55, 547, 752, 827, 1877	1, 2, 212
544	24, 84, 94	1, 38, 90, 230
546	1, 3, 51, 196, 426, 1196	11, 21
548	24, 1218	6, 9, 33, 138, 162, 200, 444, 1496
550	30, 81, 168, 330	4, 181, 319
554	3, 72, 645	6, 138, 1887
556	3, 595	405
558	162	2, 314
560	2, 4, 8, 706	1, 2, 7, 20, 33, 68, 135
562	1, 4, 567, 1053	330
564	110, 1016, 1050	1, 76, 756
566	1, 77	56, 636, 643
568	11, 15, 72, 212, 515, 1075, 1208	1, 11, 12, 844
570	1, 2	36, 1061
572	15, 18, 293	1, 535
574	1, 3, 166, 651	1, 12, 144, 212, 298, 1190, 1353, 1951
578	9, 539	19
580	2, 3, 14, 53, 131, 167, 365, 559	2, 19, 68
582	46	1, 152, 301
584	1	9, 406, 591
586	21, 465	1, 2, 3, 12
588	1, 5, 789	11, 13, 28
590	235, 937	23, 155, 762
592	393, 582	12, 1855
594	3	1, 55, 107, 237, 280, 924, 1585
596	1, 2, 18, 25, 66, 185, 414	79, 95, 1434
598	2, 20	57, 129, 845
602	5, 79, 221, 454	640
604	13, 15, 48	38, 159, 1356
606	2, 23, 32, 60, 288	1, 100, 1129
608	1, 3, 13, 26, 326	1, 2, 3, 8, 315, 513
610	1, 9, 10, 78, 1165, 1252	19, 456
612	2, 6, 42	66, 288, 359
614	838	1, 3, 31, 135, 252, 276
616	1, 19, 1208	3, 78, 89
618	4, 16, 24, 618	3
620	11, 21, 23	1, 3, 766
622	1, 2, 4, 16, 323	118, 312, 348
624	2, 56, 83, 102, 282, 285, 286	13, 24, 32, 1589
626	8, 54, 146	69, 120, 154, 534
628	293	2, 120, 288
630	2, 5, 19, 63, 117	234, 807
636	52, 202	1, 483
638	1, 4, 1483	3, 1288
642	3, 4, 60, 1787	13, 20, 440
644	9, 393	1, 2, 5, 54, 146, 399
646	1, 4, 1884	201
648	3, 225, 759, 840	709
650	2, 6, 7, 1042, 1603	29, 100, 287
652	40, 107, 140	1, 3, 46
656	5, 6, 81, 136, 228	2, 6, 8, 30
658	60, 442, 1272	8, 20, 1015
660	43, 972	1476
662	2, 3, 137, 147, 438	1, 3
664	1, 2, 4, 50, 1010, 1600	23, 29, 44, 45, 169, 434, 483, 712, 839
666	120	1, 2, 17, 25, 142, 742
668	1, 3	16, 125, 226
670	54, 1524	1, 2, 19, 117, 660
672	1, 46, 92, 133, 941	37, 64
674	3, 52	2, 722, 1182
678	1, 21, 37, 39, 50, 280	143, 193
680	9, 12, 15, 574	2, 4, 7, 44, 521, 569
682	30	5, 9, 21
684	10, 15, 63	2
686	2, 29, 44, 172, 1170, 1260	2, 3, 800, 1866
690	413	30, 36, 681, 1162
692	57, 586, 1787	91
694	2, 6, 62, 256	2, 318, 591
696	3, 87, 1675	3, 897
700	3, 16, 60, 70, 306	29, 64, 151, 1047, 1410
704	2, 3, 5, 380	6, 7, 49
706	2, 3, 6, 1955	3, 4, 5, 10, 49, 53
708	10, 14, 21	2, 7, 162, 215, 282, 428, 538, 742
712	9, 24	1, 7, 15, 37, 42, 375, 591
714	1, 4, 84	1, 19, 81, 155
716	1, 210	2, 18, 39, 81, 1475
718	2, 48, 440	1, 3, 6, 13, 408, 1708
722	6, 86, 222	4
724	7, 216, 595	275, 393, 948
726	4, 669	1, 2, 13, 494, 1005
728	1, 4, 88, 364, 907	2, 1054, 1751
730	3, 15, 69, 91, 97, 160	27, 105
732	5, 288, 464	1, 4, 15, 27, 61, 91, 273
734	1, 4, 52, 151, 304, 620, 676	8
736	2, 52	2, 671
740	18, 47, 503	3, 50, 291
742	3, 6, 979	1, 4, 96, 585, 596, 618, 680, 1210
744	1, 49	3, 6, 14
746	2, 3, 516	1, 535
748	1, 124, 135, 179, 320	2, 14, 126
750	3, 149	1, 2, 6, 71, 381
752	1, 14, 804	60, 328
754	6	127, 372, 817
756	2, 57	1, 7, 1808
758	1, 538	2, 75, 548
760	2, 5, 38, 89	1, 3, 40, 61, 354
762	1, 12, 1341	1, 5, 39, 243, 1569
764	1, 146	115
766	93, 414, 420	95
768	42, 154, 305	2, 27, 43
770	3, 14, 77, 240, 513	2, 31, 131, 771, 1680
774	26	1, 28, 45, 81
776	1, 32, 236, 610, 808, 1537	2, 9, 11, 20, 71, 449, 973, 1432
778	8, 17	1, 2, 20
780	1, 6, 288, 936, 1200	15, 22
782	3, 40, 760	48, 171, 645
786	1	3, 8, 191, 677
788	447	1, 3, 700, 759
790	1, 2, 40, 602	4, 213
792	3, 10, 12, 112, 698	6, 51, 162, 696
794	3, 126, 290, 1232	4
796	207	1
798	1, 3, 1048	2, 689, 944
800	13, 30	5, 1506
802	2	27, 48
804	6, 257, 550	2, 4, 1660
806	2, 4, 10, 45, 51, 171, 1477, 1978	1, 8, 657, 774, 909
808	1, 3, 6, 20, 314	4, 5, 52, 214
810	5, 9, 321	1
812	1, 11, 276, 310, 376	3, 20, 43, 56, 79, 107, 760
814	45, 69, 130	530
816	6, 1910	1, 345, 1149
820	5, 255, 901	174
822	30	3, 22, 454
824	1215	1, 195
826	1, 11, 14, 32, 190, 483, 852	10, 18
828	3	465
830	375	487
834	21	45, 57, 110, 233, 730, 1294, 1626
838	33, 78	925
840	11, 54, 1595	1, 117
842	1, 19, 75, 1402	21, 33, 72, 201
844	21	33
846	5, 27, 224	3, 5, 10, 41, 116, 181
848	38	11
850	2, 12, 60, 78	6, 1815
852	3, 4, 15, 54	2, 462
854	5, 211, 522	2, 12, 152, 613
856	10, 70, 306	44
858	5	88
860	10, 64, 1052	3, 31, 72, 269
862	42, 95	1, 454
864	1, 2, 104, 993	15, 42, 1049
868	7, 357, 1166	105, 449, 517
870	385	20, 795
872	2, 5, 8, 17, 1415	3, 4, 564
874	2, 123	1, 5, 200
876	1, 21, 60, 105	4, 5
878	2, 7	180, 244
880	723, 971	1, 3, 9, 80, 121, 386
882	1, 493	842
884	3, 4, 61, 75, 252, 1213	198
886	5, 27, 33, 147	10, 84
888	11, 1505	5, 29, 124, 1063, 1568
890	2, 24	25
892	200	11, 112
896	11	1, 2, 3, 12, 376, 978
898	1, 4	6, 18
902	1	2, 449, 625, 1313, 1313
904	3, 7	14, 42, 78
906	3, 6	558
910	2, 10, 24, 49, 51, 54, 90, 200	110, 179
912	60, 208, 1638	12, 50, 1467
914	278	46, 69
916	9, 27, 57	2, 3
918	3, 21, 26, 808	8, 18
920	14, 780	221, 250
922	190, 315	3, 8
924	2, 4, 16, 288	3, 9, 18, 422
930	9, 843	1, 293, 712, 942
932	1, 10, 16, 98	46, 55, 79, 250, 260
934	3, 20, 501, 1585	65, 77
936	58, 365	3, 26, 33, 103
940	10, 55, 1788	1692
944	1, 4, 9, 12, 14, 1302	1, 3, 112, 685, 1433
946	1, 6, 7, 64, 146, 1022	10, 15, 49, 126, 140, 385
948	6, 9, 13, 21, 1923	5, 168, 1036
952	1, 245, 784	3, 17, 83, 160, 855
954	10, 120, 247, 579, 1719	84, 147
956	5, 630	6, 529
958	15, 800	1, 2, 5, 14, 21, 110
960	1, 3, 5, 252	4, 706
962	51, 282, 732	1901
972	46	1, 6, 9, 151, 225
974	1, 10, 294, 1175	2, 90
976	3	40
978	42, 53, 202	1, 1063
980	1, 2, 21, 31, 175, 634, 1354	129, 589
984	5, 14, 527	3, 10, 31
986	2	1, 2, 216, 310, 565
988	1, 6, 61	29, 275, 930
990	457	5, 6, 47
994	1, 9, 358	1, 2, 19, 35, 100
1002	2, 3	1, 13
1004	1, 27	258
1006	3, 16, 18, 786	140, 152, 254, 1111
1008	4, 77, 110, 1248, 1667	4, 74, 115
1012	14	1, 4, 22, 697, 1821, 1954
1014	1, 4, 7, 19, 1014	15, 954
1016	12, 299, 1375, 1426	11, 32, 42
1018	2, 3, 9, 15, 597, 1531	3, 35, 132, 760, 1481
1020	574, 624, 970	1, 33, 205, 450
1022	1, 17, 62, 429, 1553	1, 3, 28, 37, 1393
1026	18, 33	7, 9, 102
1028	4, 166	1, 2, 18, 53, 75, 201
1030	1, 4, 6	2, 288
1034	1440	1, 2, 6, 37, 1842
1036	1, 9, 12, 20, 240, 377, 1197	2
1040	20	10, 45
1042	10, 273, 1511	13, 141, 303, 1872
1044	2	1, 261, 1966
1046	1, 14	9, 1632
1048	108	2
1050	25, 267	1, 57
1052	1, 6, 42, 520	62, 312, 359, 627
1054	135	3
1056	18, 118, 578	1, 369, 476, 556
1058	1, 2, 27, 35, 195	5, 12, 140
1060	63, 396, 1880	70, 1362
1062	5	1814
1064	3, 5, 18, 66, 130, 1995	3, 11, 29, 52
1066	6, 22	2, 72, 128, 147
1070	1, 15, 382, 685	1
1072	1	1, 2, 122, 206
1074	1, 2, 44	12
1076	21	2, 12
1078	2, 5, 261	2, 902
1080	19, 48, 537, 1008, 1609	77, 302
1082	65, 71	144, 279
1084	4, 32, 40, 57, 408	1, 11, 381, 923
1086	1, 1201	13, 72, 203, 967
1088	9, 813	41, 1235
1090	3	1, 21
1092	1, 5, 821	2, 1150, 1316
1094	13	21, 210
1096	5, 230	10, 106, 136, 171
1098	3, 37	82, 100, 105
1100	2, 4, 229	1, 2, 3, 5, 24, 90, 182
1102	1, 36, 114, 127	5
1104	6, 15	3, 15, 25, 56
1106	2, 11, 39, 182, 672	3, 50, 218, 604
1110	36, 179	4, 52, 64, 379
1112	10	1, 3, 5, 6, 280
1114	1, 3, 4, 319, 434, 826, 1341	15, 21, 75, 81
1118	5, 63, 88	1, 57, 126
1120	1, 7, 21, 67	2, 3, 8, 525, 1112
1122	153, 1416	3, 696
1124	117	1, 216, 264, 324
1128	27, 56, 279	40, 108
1130	2, 18	15
1134	2, 13, 14, 1838	42, 154, 694
1138	14	10, 12, 426
1140	16	1, 74
1142	1, 200, 318	1
1144	1, 37	3
1146	66	1, 3, 91, 229, 459, 795, 878
1148	1, 8, 10	1, 4, 10, 21, 58, 88, 1667
1154	1, 7	2, 3, 348
1156	2, 129	1, 7, 73
1158	1, 8, 43, 66	5, 792
1160	3, 9	1, 25, 141, 1922
1162	1, 70, 125, 126, 740, 1687	3, 115, 126, 1273
1164	12, 15	9, 17
1168	36	6, 108, 322
1170	75, 121, 381, 420	12, 37, 661
1172	2, 3, 6, 14	17, 34
1174	120, 837	1
1176	1, 2, 7, 112, 212, 1672	1, 11, 12, 66, 180
1178	1, 4, 117, 426	2, 1131
1180	62, 72, 597	16, 21, 70, 1827
1182	7	2, 5, 14, 23, 27, 70, 731
1184	4, 15, 55, 56, 149, 410	2, 4, 6, 7, 16, 26, 31, 279, 286, 505
1186	3, 21, 54, 104, 254, 1272, 1899	627
1190	352, 1196	2, 6, 7, 15, 141, 256, 259
1192	34	3, 5, 8, 1239
1194	117, 120, 711	63, 70
1196	4, 20, 27	1, 180, 901
1198	1, 6, 59	4, 25, 307, 633
1200	3	3, 11, 75, 136
1202	832	2, 7, 18, 37, 50, 1266
1204	17	442
1206	3, 61, 274	3, 5, 561, 609
1208	3, 9, 548	6, 13
1212	1	3, 98
1214	397	12, 1408
1220	30, 76	2, 15, 249, 252, 722
1224	1, 6, 13, 24, 25, 64, 188, 197, 270, 453	2, 672, 684
1226	12, 94, 185, 251, 534	3, 16, 416
1228	129	3, 767
1230	65	1, 7, 67, 355
1232	1, 2, 7, 8, 118	1, 9, 125, 202
1234	1, 94, 1683	99, 1406
1238	2, 876	1, 464, 1796, 1900
1240	1, 111, 175, 518	1, 165, 176, 1010
1242	1, 896	4, 347
1244	12, 436	6, 9, 158
1246	3, 28	5, 93, 202, 1707
1248	3, 4	786
1250	2, 57, 348	1, 352, 379, 579
1252	1, 9, 25, 32	1, 12, 1058
1254	1, 1218, 1686	5, 14, 175, 1646
1256	2, 3, 21, 228	4, 5, 23
1258	39, 155, 245, 749, 843, 1260	3, 14, 859
1260	5, 20, 121, 165	1, 5, 10
1262	1, 165	11, 26, 66, 113
1264	41, 56, 425	1, 72
1266	1, 3, 4, 54, 152, 813	1, 2, 446
1268	1, 132, 651, 1530	5, 37
1270	444, 680	1253
1272	30, 45	20, 1034
1274	342	15, 26
1276	63, 118, 1077	6, 111
1278	1145	6, 7, 783
1280	2, 3, 43, 176, 272, 400	2, 5, 1546
1282	358, 404	3
1284	3, 224	201, 1895
1288	1, 2, 160	1, 2, 6, 7, 14, 142, 1713
1292	11, 41	37
1294	3, 53, 98	375
1296	5, 77, 532, 2257, 9665	1, 3, 14, 94, 108, 1942, 8392, 12700
1298	1, 120, 312	6, 15, 108
1300	4, 10, 65	3, 7, 12, 36, 56, 183
1302	231	2, 6, 12
1304	4, 75, 195, 297, 889	14, 60, 132
1306	2, 9, 42, 236	1, 6, 19, 682
1308	1, 9	6
1310	66, 78, 391, 459	1, 98
1312	1, 2, 295, 799	100, 189, 810
1314	3, 29	2, 12, 108
1316	110	131, 235, 871
1320	168, 255, 1548	1, 60, 426
1322	1, 21, 621, 1150	435
1324	2, 3, 4	6, 17
1328	46	2, 3, 30, 49
1330	12, 20, 27, 119	1, 2, 5, 52, 128, 427
1332	1, 16, 43	182, 1304
1336	59, 387, 516	756
1338	2, 547	5, 138, 179, 334, 893, 1419
1340	13, 18	5, 192, 700
1342	4, 143, 334, 654	1, 2
1344	1	12, 15, 354, 534, 780, 1033
1348	741	3
1350	2, 10, 422, 488	3, 46, 700, 1224
1352	3, 4, 39, 549	9, 53, 93, 163
1354	39, 54, 57	4, 40
1358	1, 405	1, 4, 14, 16, 204, 265, 614
1362	65	1, 1083
1364	1, 109, 315	1, 25, 990
1366	1, 8, 39, 1087, 1436	1, 3, 15, 127
1368	1, 69, 876	279
1370	3	24
1372	3, 4, 1740	1248, 1574
1374	160, 456, 618	99, 1706
1376	2, 21, 147, 1098	9, 58
1378	19, 43, 227, 619	1
1380	1, 1466	15, 100
1382	355	664
1386	4, 64, 604, 725	5, 57, 103
1390	102, 272	1, 10, 25
1392	1, 4, 6, 597	1
1394	1, 43, 59, 67	7, 9, 408, 880, 1738
1396	840	168, 522
1398	5, 6, 52	6, 18, 25, 90, 1369
1400	7	5
1402	10, 90, 519, 1105, 1915	6
1406	2, 3	17, 50
1408	2, 31, 93, 312, 338, 1166	1, 2, 15, 32, 36, 72
1410	1, 2, 235, 915, 1021	29
1412	310	1176
1414	154	1, 8
1416	1, 3, 30, 139	6
1418	9, 768	7, 513
1420	6, 30, 781, 1965	210
1422	2, 21, 236	7, 47, 353
1424	24, 60, 87	946, 1396
1426	5, 10	39, 117, 560, 1392, 1452
1428	22, 29	1, 495
1430	1	2, 39
1434	4, 15, 56, 59, 184, 462, 748, 830, 1714	1, 127, 207, 442
1436	1, 10, 11, 705	4, 64, 103, 294
1438	1500	18
1440	5	13, 51, 102
1444	1, 280	3, 1745
1446	5, 20	1, 6
1448	1, 18, 25, 240, 1116	1195
1450	126, 154, 156, 646	15, 52
1454	6	38, 555, 1435
1456	19	4, 32
1458	541	3, 5
1460	65	1, 21, 37, 72, 184, 522
1462	1, 5, 27, 34	1, 26, 43
1464	1, 2	7, 9, 23, 120, 142, 150, 1314
1470	9	85, 732
1474	2	10, 439
1476	7, 363	17, 53, 410, 1098, 1370
1478	2, 20	17
1480	111	5, 36, 45, 177, 375
1486	3, 150	1046
1488	14, 548	90
1490	8, 168, 236	1, 2
1492	1, 146, 383	10, 50
1494	9, 20	5, 6, 156, 1402
1496	15, 27, 382, 1097	1128
1498	4	1, 27, 420, 682, 1700, 1792
1500	1, 190	2, 39
1502	2, 14, 905	87, 100, 1566
1504	1303, 1308, 1743	3, 120, 1307
1506	4, 196	2, 3
1508	24, 38, 1332	3, 6, 65, 106, 903
1510	4, 6, 34, 125	26, 92, 540
1512	6, 29, 145, 493	28, 621
1514	3, 37, 520, 1363, 1623, 1980	1176
1518	1, 3, 34, 514, 1010	3, 6, 36, 94, 198, 948, 1624
1520	5, 19, 26, 52	1, 130, 944
1522	1	5, 10, 101
1526	1, 2, 6, 13, 106, 334	2, 15, 36, 227, 694
1528	3, 4, 42, 51	365
1530	36, 257	265, 657
1532	3, 119, 166, 781	5, 11, 14
1536	1633	12
1538	9	42, 79, 183, 1128
1544	2, 62	1, 87, 433
1546	1, 2, 3, 5, 104, 116	14, 30, 234, 599
1548	31, 388	1, 14, 32, 101
1550	1, 216	11, 30, 125
1552	6, 996	32, 302
1554	26, 52, 216	564, 904
1556	4, 31	240, 263, 330
1558	51, 270	9, 16
1560	4, 27, 40, 53	1
1562	1, 53	1, 34, 84, 331
1564	1, 8, 1527	3, 288, 777
1568	1, 4	5, 16
1572	11, 62	1083
1574	2, 26	4, 35, 40, 114
1576	3, 1100	907
1578	18, 763	3, 4, 47, 1913
1580	3, 34, 233	2, 25, 1278
1582	2, 4, 8, 555	7, 45, 1493
1584	9, 19, 72	42
1588	13, 867	7, 103, 124
1590	5, 155, 355, 369	10, 48, 86
1594	46	1119
1596	305	1, 2, 4, 6, 149, 1012
1598	1, 10, 16	122, 1688
1600	2, 69, 2042	1, 72, 1428, 1498, 2583, 3912, 4696, 20389
1602	1, 49, 354, 1038	1, 6, 26
1604	1, 10, 17, 18, 32, 186, 1614	55, 88
1606	237	11, 42, 168, 198
1608	45	13, 60, 138
1610	2, 15, 70, 383	1, 315, 730, 1484
1612	1	95, 854
1616	3, 44, 960, 1059, 1231	1, 5, 80, 815
1618	1, 618, 1273, 1843	1, 13, 89, 390, 1217
1620	1, 2, 74	3, 693
1622	540	6, 13, 1916
1624	3	2, 32
1626	81, 334, 1639	114, 126, 140, 168
1628	50	3, 522, 810
1630	61, 139	2, 26, 1222
1632	23, 37, 1460	37, 707, 1131
1634	3, 21, 37, 606	4, 6, 22, 147, 148, 1665
1636	491	18, 96
1638	2, 9, 25, 160, 317, 660, 1233	1, 192
1640	1, 186, 229, 1723	80, 108, 324
1642	2, 32	15, 94
1644	5, 99	21, 67, 949
1646	70, 432	2, 3, 20, 26, 288
1648	2, 660	5, 6
1652	14, 49, 442, 878, 1257	1
1654	1, 294, 510	5, 36
1658	2, 25, 142, 982, 1053	3, 10, 18, 40, 737, 896
1660	12, 101, 167	2, 6, 26, 120, 182, 400
1662	7, 1257	6, 89
1666	11, 26	20, 233, 243, 247, 1126
1670	2, 5	3
1672	2, 11, 14	1, 2, 34, 67, 689
1674	1, 15	12, 29
1676	13, 14	6, 111
1678	5, 52, 83, 516, 1380	1, 2
1680	1, 429, 864	1
1682	1, 39, 102, 120	17, 24, 44, 680
1686	4, 561, 951, 1334	1, 39, 494
1688	1, 2, 7, 12, 397	2, 158, 432
1692	174	1, 7, 25, 345, 366
1694	1, 2	1, 12
1696	1	2, 165
1698	2, 1170	4, 321
1700	2, 33, 566	1, 13, 47
1702	1, 3, 60, 95	1, 7, 31, 183, 266
1704	1, 2	9, 10, 315
1708	183, 720	9, 531, 794
1710	4, 7	26
1712	140	934
1714	859, 1627	1, 6, 198, 850
1716	1, 3, 553	1, 2
1718	1, 2, 50, 1064	184
1720	3, 366	1, 888
1722	1	2
1728	1, 17, 2608, 5152, 7538, 22945	20, 171, 349, 2502, 26286
1730	1095, 1653	6, 84, 223, 1727
1732	384, 519	1067
1734	3, 102, 1011, 1889	105
1736	8, 29, 104	1, 9, 17
1738	1, 3	9, 417
1740	2, 329	1, 16
1742	1, 16, 328	2, 5, 17, 86, 1384
1744	26	13, 128
1746	170, 806	4, 6, 30
1748	28, 684	1, 56
1750	1, 2, 123, 175, 765	1, 5
1752	1, 1182, 1522	3
1754	3, 596	64, 66, 150
1756	176	47
1758	12, 153, 1340	2, 4
1762	360	2, 8, 207
1764	3, 7, 39, 136, 1263, 2426	2, 1023, 1233, 2010, 4212, 12501
1766	4, 34, 1227	5, 11, 24, 1613
1768	17, 36, 59, 231	15, 46
1770	42, 68, 390, 780	2, 5, 9, 31, 69
1772	1445	2, 14, 756, 795
1776	23, 393	8, 12, 27
1778	2, 14, 36	2, 3, 134, 198, 1456
1780	6	2, 36
1782	8, 50, 698	1
1784	1, 2, 1845	1, 44, 119, 386
1786	1, 15, 71, 72	2, 5
1790	420, 462	2, 36, 170
1792	2, 318	5, 54
1794	9, 28, 87	11, 14, 1824
1796	2, 6, 195	1, 9
1798	1, 486, 1128	4, 9, 11, 20, 244
1800	30, 32, 264, 1280	104, 180
1802	61, 264	22, 538
1804	3, 437	175, 1743
1808	1, 9, 15, 25, 61, 1044, 1803	20, 68
1810	6, 21, 26, 39, 45, 114, 194, 290	7, 10, 15, 16, 499, 520
1812	1317	32, 96
1814	4, 16, 60, 63, 236, 1341	6, 270
1816	2, 54	534
1818	101, 156	1, 2, 7, 8
1820	1, 165, 1090	5, 665
1822	3	6, 408, 1143, 1868
1824	2, 1530	1069
1826	3, 22, 363, 968, 1816	1, 5, 11, 1508
1828	1, 304	34
1830	2	4
1832	5, 100	50, 60, 115, 187, 559
1834	2, 16, 234, 299	2, 271, 1006
1836	64, 493	14, 17, 125, 1205
1838	305	1, 808
1840	1, 427, 540	1, 4, 75
1842	1, 2, 6, 13, 61, 280	15, 64, 159, 208, 1612, 1992
1844	2, 200	141, 363
1846	10	267, 276
1848	9, 205, 1032, 1650	1, 5, 6
1850	1, 16	26, 33
1854	5, 18, 42, 45, 374, 449, 985, 1779	2, 1376
1856	2, 18, 55, 118, 1500, 1800	63, 1016
1858	374	18, 89, 90, 124, 1157
1860	76	1, 2, 3, 1873
1862	1, 4, 9, 61, 337, 732, 978	5, 25, 43, 50, 126, 310
1864	997	6, 12, 114, 506
1866	3, 65	1, 79
1868	1, 2, 7, 74, 371	8, 9, 22
1870	5, 175, 752, 1190	11
1872	3, 30, 62, 115, 824	17, 946
1874	10, 264	1, 153
1876	1, 16, 18	2, 41, 402, 659, 953
1878	3, 1228	9, 78
1880	6, 20	12
1882	22	6, 16, 1393
1884	4, 8	1, 3, 95, 150, 494, 598
1886	1, 15	4, 23, 58, 123, 478, 910
1888	717	3
1890	2, 3, 4, 6, 14, 79, 149	20, 388
1892	3, 22	27, 29
1894	63	1, 12, 36
1896	1	1, 7
1898	1, 99, 100	4
1900	2, 25	11, 23
1902	6, 22, 141	15, 20, 1298
1904	4, 70, 193, 404, 1550	2, 16, 1772
1906	10, 309, 496	2, 3, 15, 23, 71, 158
1908	2, 1409	1, 9, 226, 1702
1910	1, 4, 24	1, 2, 104, 480
1912	1, 2, 3, 4, 7, 10, 36	1, 20, 97
1914	1, 2, 8, 116, 188, 1664	6, 47
1916	4, 6, 120, 779	164
1918	20, 44, 68	2, 5, 549, 1488, 1619
1920	4, 12	2
1922	20, 53, 87, 417	1, 3, 88
1924	1, 4, 6, 117, 431	1, 42, 654, 1537, 1597
1926	1	14
1930	1246	1, 126, 1218
1932	1, 4	171
1934	3, 18	107
1936	15, 45, 644, 12893	741, 4105, 10251, 30106, 47970
1938	4, 14, 154, 536, 1026	2, 29, 302, 309
1940	1881	3, 6, 20, 57, 131
1942	206, 506, 1087	1827
1944	6	24
1946	21, 394, 888	1, 2, 5, 43, 64, 69, 95
1948	1, 7, 739	2
1952	7, 86	1, 3, 993, 1852
1954	1, 3, 180	11, 1776
1956	27	21, 911
1958	10, 15, 69	96, 192
1960	11, 423	50, 133
1964	2, 89, 96, 134, 300, 1089	1, 3, 31, 211
1966	1, 355	15, 753
1970	1, 7	276, 436
1972	11, 90	211, 284, 1179, 1746
1974	2, 25, 326	1, 5, 6, 8, 14
1976	1, 951	50
1978	11, 1986	1
1980	1, 5, 7, 11, 378, 1775	3, 5, 6, 286, 780
1982	2, 4, 1810	52, 74, 126
1984	6	540
1988	87, 667, 1754	1, 11, 12, 243, 370, 1789
1990	1, 4, 61, 127, 669, 828	30, 131, 731
1994	2, 37	3, 5, 27, 114, 854
1996	32, 57, 960	1, 3, 666
1998	1, 2, 15, 55, 150	15
2000	562	1, 2, 27, 284
2002	1, 2, 4, 62, 223, 295, 341, 648, 882, 886	6, 26, 201, 299, 385, 687
2004	3	15, 144
2006	5, 6, 773, 1521	12, 19, 66, 1204, 1717
2008	8, 11, 22, 1003, 1174	3, 18, 30, 41, 78
2012	6, 956	4, 689
2014	5	19
2016	18, 78, 253, 1670	29, 608
2020	6, 9	1
2022	1, 6, 22, 771	351, 697
2024	85, 635, 1470	5, 276, 411, 424
2028	78, 99, 279	27
2030	30	1, 7, 108, 277, 309, 424
2032	1	1160
2036	1, 62	1
2038	1, 78, 120	5, 10, 12, 66, 604
2042	4, 15, 501	1
2044	1, 5, 12, 119	15, 50, 296
2046	282	21, 336, 1011
2048	5, 12, 1299	22, 150
2050	216, 1511	2, 20
2052	265, 1119, 1144	1, 3, 10, 77, 113, 1048, 1608, 1710
2054	1, 9, 12, 24, 674, 1659, 1982	40
2056	5, 813	1, 115
2058	1, 6, 105, 416	1, 3, 231, 989, 1977
2060	1, 16, 21	162
2062	2, 5, 18	1029
2066	2, 12, 192, 203	2, 132
2068	105	1551
2070	310, 472	9, 253
2072	10, 17, 109	1, 17, 25, 138, 531
2074	1, 144	2, 3, 275, 996
2078	1, 5, 8, 31, 38, 114, 360	25, 143, 317, 1891
2080	40, 310	1, 2, 6, 20, 23, 30
2082	1	4, 29, 148
2084	3	2, 63, 75, 110
2086	141, 572, 1326	1, 2, 8, 288
2088	1	2, 5, 23, 62
2090	45, 93, 258	1, 57, 1588
2092	1, 2	1, 16, 30, 83
2094	1, 14, 16, 110, 277, 314, 1400	1, 16, 31, 58
2096	1, 132	4, 5
2098	3, 16, 34, 1881	6
2100	33, 35, 92, 181, 190	1, 47, 1252
2102	1, 9, 19, 34, 79, 84, 162, 1678	63, 111
2106	1, 119, 148, 700, 1620	1, 109
2108	1, 40, 115	1, 4, 50
2110	1, 2, 6, 162, 302	3, 5, 234, 413
2112	17, 33	1, 8, 9, 15, 44, 1327
2114	1, 2, 280	8
2116	6, 652, 2586	27, 1020
2118	17, 65, 122	25, 31, 220
2120	85	2, 317, 394, 458
2122	55	81
2124	44, 87	660
2130	1, 19	8
2134	1, 1189	3, 10, 1090
2136	26	1, 6, 7, 104
2138	1, 2, 13, 92, 390, 578	28, 39, 1121
2142	24	831
2144	82, 189	3, 77, 156, 482, 1182
2148	1323	17, 724
2150	40, 137, 510	1, 3, 273
2152	1, 2	3, 9, 47, 525
2154	411	2, 24, 33, 120
2156	137, 491	3, 345
2158	193	2, 14, 222, 327
2160	3, 9, 212, 266	10, 58
2162	16, 80, 560, 689	1, 524
2164	1, 172	10, 74, 128, 132, 344
2166	368	27
2170	9, 119	1, 2, 875, 1238
2172	1, 40	12
2174	5	106, 744
2176	15, 125	1, 6, 73
2178	1, 3, 4, 25	1, 2, 3, 543
2180	1, 8, 15, 78, 1014	5, 6, 10
2184	1, 8, 64, 715, 722	45, 60
2186	561, 1035, 1350	2, 1284
2188	6	51
2190	110, 114, 440	49, 1824
2192	201	8, 36, 47
2194	25	15, 1245
2196	57, 100, 612	110, 198
2198	417, 552, 562	3, 10, 91, 425
2200	4, 6, 37	5, 6, 200
2202	33, 200, 330, 627	15, 49, 280, 1024
2206	8	519, 1181, 1701
2208	37, 795	3, 11
2210	4, 347, 437	1, 18, 1110, 1326
2212	1, 2, 6, 15, 62	9, 53, 108
2214	3, 16, 394, 1768	35, 89
2216	17, 50	58
2218	233	5, 114, 647, 828
2220	5, 78, 122, 699	2, 15, 169, 204
2222	2, 13, 242	17, 135, 334
2224	286	1, 97
2226	1	1, 2, 3, 79, 107
2228	1, 6, 10, 13, 378	2, 26
2230	3, 228	13
2232	8, 59, 1874	2, 316
2234	9, 12, 48	7, 97
2236	511	147
2238	273	1
2240	1, 2, 3, 1646	1, 8, 13, 61, 101, 383
2242	1, 3, 312	45, 65, 113, 429
2244	5, 885	46
2246	2, 3, 4, 1134, 1919	1
2248	1, 2, 8, 50, 94	7
2252	1293	50, 122, 181
2254	14	2, 5
2256	555	3
2258	1215	4
2260	4, 454	1, 4, 8, 15, 119
2262	1, 2, 78, 98, 235, 1745	3, 53, 110, 154, 271
2268	2	1, 30, 54, 141, 286, 376
2270	1, 24, 388	795, 1274, 1500
2272	6, 1175, 1736	1, 9, 415
2274	1, 79, 826	5, 13, 69
2276	33, 36, 69, 820	1, 29, 113, 903
2278	1, 108	18, 46, 99, 129, 233, 345
2282	2, 3, 5, 33, 70, 1309, 1327	1, 385
2284	1	18, 275, 1478
2286	20	1, 6, 433, 1416, 1590
2288	1, 193	9
2292	7, 32, 60	6, 258
2294	4	87, 350, 481
2296	15, 25, 85, 535	1, 2, 6, 10, 26, 140, 1485
2302	3, 107	1, 3, 84, 216, 292, 323
2304	1, 2, 3, 6, 1941	6637, 12989
2306	69, 165	29
2310	1, 2, 1308	3, 4, 1252
2314	1257	1, 52
2316	1, 13, 73	43, 148, 387
2318	38, 39, 299, 306	12, 88, 94, 325, 375
2320	2, 63, 200, 252	15, 78
2322	3, 5, 6, 22, 90, 154, 1762	2, 67
2324	2, 5, 37, 813	28, 39, 237
2326	120	833
2328	51	3, 6
2332	3	109, 156, 1089
2336	15, 333	1, 7, 432
2338	1, 1286	204, 759
2340	4, 7, 10, 24, 106	14, 1665
2344	4	1, 10, 239, 635
2346	1, 14, 141	9, 12, 1930
2348	13	4, 149
2350	3, 106, 267, 453	3
2352	2, 29, 57, 944, 1270	4, 8, 12
2354	154	20
2356	140	45, 1948
2358	5, 207, 605	1, 28, 98, 1735
2360	1, 2, 11, 108	7, 182, 286, 1113
2362	939	53, 1422
2364	3, 6, 405	151, 1646
2366	24	7, 109, 370, 625
2368	156, 864	2, 29
2370	2, 3, 80, 959, 1475	12
2372	7, 13	4, 171
2374	4	1, 200, 214
2376	1, 9, 84, 378	54
2378	3, 18	2
2380	346	1, 435, 521, 1723
2386	20, 36, 40, 180, 196	24, 112, 1034
2388	110	1, 4, 6
2390	1, 14, 18	36, 69, 402
2392	21	1, 8, 384
2394	1, 205, 905	3, 4, 9, 54, 140
2396	18, 1233	11, 161
2402	1, 10, 23, 24, 661	156, 366
2406	9, 1212	2, 3, 6, 26, 153
2408	21	7, 40
2410	156, 316, 493, 1582	2, 3, 113
2412	56	1, 3, 517
2414	1, 3	3, 21, 41, 87, 647
2416	3	625
2418	2, 247, 1279, 1975	51, 57
2420	4, 900	1, 2, 21, 66, 1916
2422	1, 3, 10, 332	1, 10
2424	1, 420	827
2426	6, 51, 495	7, 193
2428	2	68
2430	3, 5, 12, 27, 122, 147, 155, 404	1, 11, 104
2432	1, 7, 510	3, 6, 30
2434	4	8, 66, 73, 434
2436	4, 19, 30, 133	590
2440	3, 15, 65, 101	3, 7, 375
2442	64, 115	33, 455
2444	26, 70, 205	34, 104, 287
2446	6, 61, 63	4, 190
2448	1240	1, 3, 25, 98
2450	1, 34, 96, 126	2, 21, 225, 423, 1344
2452	127	3, 770
2454	15	1
2456	1, 5, 17	21, 425, 561
2458	4, 17, 19, 86, 318	77
2460	13, 78	3, 6
2462	70, 1360	74
2464	3, 482, 1882	1
2466	1, 7	35, 908
2468	2, 1250	1, 9, 76, 223, 1272
2470	1, 6, 11, 14, 49, 195	5
2472	76, 1853	3, 332
2476	3, 435	42, 169, 1524
2478	3, 22	2, 6, 275
2484	713, 953	1, 207, 344, 866
2486	1, 40, 79, 217, 290	41, 298, 562
2488	9, 128, 1249	310
2490	83	1
2492	1, 2, 81, 978, 1429	2, 3, 6, 1090
2494	670, 1212	182
2496	38, 126, 495	3
2498	2, 9	33, 39, 251, 535
2500	2, 33, 215, 1273, 1337, 3180	2, 19, 60, 2198
2502	2, 3	1329
2504	16, 59	2, 20, 198, 717
2506	3, 10, 13, 1755	1, 43
2508	1, 106, 331, 721	38, 107
2510	9, 12, 1271, 1496	1, 76, 489
2512	1, 2, 10, 276, 583	1, 5, 97, 362, 759, 1193, 1289
2514	1, 14, 270	1, 7, 10, 13, 933
2516	1, 54, 144, 339	4, 159
2520	3, 10, 82, 819	3, 8
2522	90	2, 6, 101, 710, 819
2526	8, 38, 1219	2, 37
2528	18, 29, 188, 204, 434, 554, 581	1, 142
2532	4	1, 271, 381
2536	9, 646	56, 795, 1610
2538	2	10, 1081
2540	122, 306	1
2542	1, 7, 11, 538, 670, 1012	1, 19, 132, 1782
2544	1, 2, 8, 21, 368	34, 42
2546	52	18
2548	2	1, 3, 7, 44, 199
2550	1	38, 1346, 1644
2552	9	4, 61, 1455
2554	3, 4, 15, 121, 205, 231	1, 2, 182
2556	1, 6, 34, 90	57
2558	332	22, 285
2562	61	1, 2, 5, 12, 200, 995
2564	1, 93, 100, 268	5, 17, 62, 1310
2566	2, 11	114, 175, 535
2568	105, 130, 859	1, 2, 290, 1676
2570	1, 8	6, 486
2572	3	3, 47
2576	108, 119, 305, 648, 834, 1669	1, 2, 35, 196, 717, 1144, 1514
2578	1	168
2580	101, 497, 1085	55
2582	3, 27, 92, 305, 1278	3, 4, 6, 342
2584	29, 46, 67, 512	1, 405
2590	1, 12, 25, 32, 335, 347	1, 2, 33, 89, 383
2592	1, 373, 481, 756	2, 9, 287, 312
2594	200, 476	16, 28, 271, 310, 1380
2596	13, 18, 75, 82	1, 6
2598	1, 24, 104, 367, 719, 798	1, 4
2600	1, 3, 9, 39, 57	3, 130, 420
2602	77, 84, 317	1, 2, 7, 20, 87, 145
2604	1, 5, 9, 17, 79, 1220	135, 191
2608	2, 27, 32	12
2612	1, 102, 1042	4, 8
2618	81, 854	1, 29, 207, 976
2620	1, 217	209, 261
2628	115	558, 748
2630	10, 837	2
2632	7, 196, 342	3, 10, 15, 351, 1065
2636	2, 18, 81, 357	1
2638	1, 18	2, 14, 23, 46, 462, 1053, 1786
2640	2	79, 1960
2642	2	132
2644	41, 267, 1155	8, 282
2648	1, 3, 12, 34, 751	20, 1157
2652	25, 154, 173, 299	1, 2, 7, 10, 86
2654	1, 2, 354, 1139	2
2656	12	16, 293
2662	3	2, 5, 56
2664	3	1, 10, 118, 124
2666	1, 6, 12, 24, 264	1, 124
2668	1, 2, 4, 6	20, 1772
2670	12, 63	42, 113
2672	51, 575	2, 20
2674	305, 435	8
2676	280	11, 435
2680	1	1, 2, 10, 15, 50, 107
2682	1, 34, 84, 106, 118	78, 1708
2684	2, 44	5, 366
2686	257, 888	12, 42, 1112
2690	165	2
2692	143, 893	90, 1870
2694	36, 578, 1739	2, 242, 470, 944, 1118
2696	1134	2, 35, 59, 330, 372
2698	2, 73, 93	51
2700	159	1, 2, 50
2702	1	1
2704	1, 7, 12, 11617	83, 228, 7205
2706	5	1, 7
2708	1, 13, 61, 722	2, 3, 52, 77, 160
2710	21, 284, 424, 542	50
2712	103, 111	15, 102
2714	21, 149, 179, 216	1, 12, 25
2716	1, 13, 79, 99	1, 10, 50, 790
2718	1, 3, 4, 9, 57, 312, 828	9, 14, 1235
2720	24, 39, 305, 1560	1, 177, 1168
2722	1, 2, 13, 22, 65, 1032	3, 4, 33, 343, 1102
2724	1300	1, 23, 25
2726	1, 30, 43, 752	35, 58, 324
2728	10, 11	1, 6, 14, 73
2730	1, 120, 340	2, 42, 124, 200
2732	3, 4, 6	6, 27
2734	17	1, 3, 343
2736	1, 2, 16, 256	1, 18, 245, 610, 796
2738	1, 3	1, 6, 15, 1267
2740	1, 374	6, 39, 210, 522
2744	2, 15, 80, 4116, 4553	20, 24, 166, 1948, 1979
2746	28, 252	2, 45, 452, 537
2748	8, 26, 42	1, 33, 109
2750	1, 2, 10, 44, 218	2, 53, 64, 414, 526
2752	6, 57, 157, 167, 247	17, 27
2754	2, 42, 73	11, 12, 82
2756	3, 27	2, 9, 27
2758	24, 798	1, 5, 35
2760	1	98
2762	3, 234	22, 94, 1450
2764	5, 34, 1261	27, 91, 627, 644
2766	3, 5, 278	3
2772	1130	1, 63
2776	141	27, 397
2778	747	1, 2, 1349
2780	1, 4	69
2784	1386	1, 3, 51, 174, 339, 1071, 1574
2786	1	125, 137, 702, 965
2788	3, 24, 402, 435, 718, 1416	2, 3, 36, 935
2790	5	127
2792	42, 1821	37, 89, 494
2796	1, 2, 147	9, 29, 370, 1895
2800	22, 1240	20
2806	33, 133	1
2808	1, 28	1, 12, 584, 1547
2810	1, 691, 866	190, 1390
2812	9	26
2814	6, 11, 16	12, 1626
2816	453	2
2818	74	4, 15, 420
2820	21, 194, 1718	1, 5, 11, 63, 303, 710, 1180
2822	1, 8	3, 1682
2824	3, 26	117
2826	9, 400, 753	7, 50
2828	9, 113	3, 490, 542
2832	18, 72, 227, 1518	30
2834	11, 157, 223, 386, 463	6, 40, 42, 1814
2836	4	1, 3, 185
2838	1, 2, 12, 25	186, 316
2840	4, 5, 34, 658	2, 3, 111
2842	22, 196, 609	8, 29, 79, 121, 311
2844	42	24, 384
2848	8, 30, 589, 650	1, 9, 59, 636, 829
2850	1	4, 372
2852	2, 13, 63, 1040	108
2854	5	1, 97, 98
2856	1, 2	20
2858	7, 49, 393	2, 291, 576
2860	50	12
2862	7, 16, 97, 145	9, 121, 180, 265
2864	34, 58, 85, 1315	2
2866	192	21
2868	46, 108	1, 3
2870	1, 765	1, 22, 118, 1221
2872	1, 7, 24, 37, 58	5, 8
2874	11, 332	1776
2876	51, 135	12, 93
2878	4, 33, 256	44, 59, 99, 325, 765
2880	14, 243	1594
2882	681	1, 24, 57, 170
2884	1, 680, 852, 1048	4, 14, 44
2888	5, 20, 749	1
2890	1, 16, 1931	7, 501
2892	6, 802, 1788	154
2894	57, 584	30, 53, 198, 1071, 1906
2896	17, 45, 850	1, 3, 60, 206
2898	1, 11, 13, 21, 633	13
2900	40	5, 6, 9
2902	15, 29, 62, 1142	10, 103
2906	7, 243, 820	13, 29, 33, 76
2908	39, 475	3, 9
2910	4, 226, 1974	1, 2
2912	1, 34, 333	6, 8, 13
2914	4, 34	2, 113, 285, 411
2916	4034	955
2920	17, 34, 59, 136	2, 6, 688
2926	1, 2, 6, 362, 1232	5, 11, 20
2928	7, 217	17, 20, 299, 884
2930	134	1, 49, 1152, 1881
2932	1, 79, 468	1, 2, 41
2934	1, 2, 35	2
2936	12	48, 65
2938	454	31, 36
2940	3, 16, 90	1, 2, 48
2942	1, 45	3, 551
2946	2, 22, 138	20, 118
2948	31, 69, 426, 1577, 1586	66, 128
2950	21	57, 130
2952	4, 12, 420, 455	2, 207, 1920
2954	20	1, 129, 1487
2956	1, 25	5, 24, 41, 80, 839
2958	3, 18, 798	4
2960	25, 444, 1793	25, 367
2964	2, 27, 49, 129	15, 196
2966	3, 52	54
2968	1475	3, 12
2972	2, 32, 153	1, 169
2976	61, 91, 157	1, 1673
2980	1066, 1420	2
2982	2	4, 78, 82, 156, 606
2984	16, 64, 421	18
2986	3, 6	237
2988	3, 121	120, 990
2990	23, 28, 212	2, 53
2992	3, 8, 1800	6, 84, 102
2996	237	1, 4, 30, 70
2998	1, 12, 111, 378, 733	84
3000	312, 366	180

How to participate?

Reserving

Reserve your base(s)/range(s) in this thread.

Sieving

Use cksieve (from Mtsieve) and
- run a new sieve by calling cksieve -b 12 -n 1 -N 10000 -P 1000000000 (for base=12, n-range=1-10000, max prime factor 10⁹). The sieve file will be written to ck_12.pfgw.
- rerun an old sieve by calling cksieve -P 1e12 -i ck_12.pfgw -o ck_12.pfgw -f factors.txt (for base=12, max prime factor 10¹², input/output files given, storing factors to "factors.txt").

PRP testing

Use PFGW calling pfgw64.exe -f0 ck_12.pfgw (running candidates file for base 12, no further factoring).

Prime testing

After testing with PFGW higher probable primes will be written in "pfgw.log". These have to be checked prime by calling like pfgw64 -tp -q"(12^68835-1)^2-2".

Reporting

Once you have completed your range, report any primes found in this thread. Then report the completed range in the reservation thread and specify whether you will continue with the base or release it.

References

External links

Current

Results maintained by Gary Barnes
"Carol / Kynea Coordinated Search - Reservations" at MersenneForum up to post #284 (2021-05-19)
"Carol / Kynea Primes" at MersenneForum up to post #250 (2021-05-19 )
2021 project goals

Others

Number classes

General numbers

Special numbers

Mersenne 2ⁿ-1
Fermat 2^2ⁿ+1
Proth k•2ⁿ+1
Riesel k•2ⁿ-1
Woodall n•2ⁿ-1
Cullen n•2ⁿ+1
Double Mersenne 2^2p-1-1
Perfect 2^n-1(2ⁿ-1)
Leyland x^y ± y^x
Gen.Repunit (b^n-1)/(b-1)

Prime numbers

Mersenne
Proth
Riesel
Williams (b±1)bⁿ±1
Carol/Kynea (bⁿ±1)²-2
Multifactorial n!_j±1

[1] Yahoo Group "Primenumbers", 2002-02-04

[2] Yahoo Group "Primenumbers", 2004-02-23

[3] Thread "Carol / Kynea search (Near-power primes)"

[1]

[2]

[3]

Carol-Kynea prime

Contents

Definitions

History

Top 5 Carol primes

Top 5 Kynea primes

OEIS sequences

Data

All bases

Bases which are a power of

Bases without a Carol prime

Bases without a Kynea prime

Bases without a Carol and Kynea prime

Remaining data

How to participate?

Reserving

Sieving

PRP testing

Prime testing

Reporting

References

External links

Current

Others

Navigation menu

Personal tools

Namespaces

Variants

Views

More

Search

Navigation

Prime (direct)

Prime Wiki

Tools