Carol and Kynea Prime Search

Created by: Mark Rodenkirch
Updated by: Gary Barnes

Carol and Kynea Primes are a special form of Near Square primesCarol primes are of the form (b^n-1)^2-2 and Kynea primes are of the form (b^n+1)^2-2.

How to participate in this search

First you will need cksieve and pfgw. cksieve is a special program that will remove most candidates (likely over 90%) before you start PRP testing.

cksieve will sieve both Carol and Kynea forms concurrently, since that is faster than doing them separately.  It's fairly easy to use: cksieve -P 1e9 -n 1 -N 1e5 -b 6 will search base 6 from n=1 to n=100,000 for primes up to 1e9. The output is written to ck_6.out. If desired you can restart from a previous sieve as follows: cksieve -P 1e12 -i ck_6.out.

You just need to run it until you have sieved deeply enough. How "deeply" depends upon the base and the range of n. Since cksieve uses a discrete log, the range of n doesn't have a significant impact on the speed.  You will want the removal rate to be about two-thirds the time it takes to PRP test the largest number in the range, so consider running pfgw on the large numbers to see how long a test will take.

If you have multiple cores or machines on a network, then I highly recommend installing PRPNet. It will save time managing the work done by those cores. The most difficult part of setting up PRPNet is installing MySQL or PostgreSQL, but once it is set up, it can be used for all PRPNet projects.

When you are ready to participate, please post a reservation at mersenneforum. It is recommended that you choose a range that can be done within a month. If you know it will take longer, please send monthly updates.

Important Links

cksieve - This includes all source and a 64-bit Windows executable. This will build and run on OS X and Linux.
pfgw - There are Windows, OS X, and Linux executables available.
PRPNet - The Windows executable is available and there is a makefile for OS X and Linux in the repository.
Make a reservation here
Post new primes here

Top 5 Carol Primes

Top 5 Kynea Primes

Prime Digits Found By Date
(2^695631-1)^2-2 418,812 Mark Rodenkirch 2016-07-16
(2^688042-1)^2-2 414,243 Mark Rodenkirch 2016-07-05
(178^87525-1)^2-2 393,937 Serge Batalov 2016-05-21
(2^653490-1)^2-2 393,441 Mark Rodenkirch 2016-06-03
(2^520363-1)^2-2 313,290 Mark Rodenkirch 2016-04-14
Prime Digits Found By Date
(30^157950+1)^2-2 466,623 Serge Batalov 2016-05-22
(2^661478+1)^2-2 398,250 Mark Rodenkirch 2016-06-18
(1968^58533+1)^2-2 385,619 Clint Stillman 2017-11-30
(2^621443+1)^2-2 374,146 Mark Rodenkirch 2016-05-30
(44^95940+1)^2-2 315,346 Karsten Bonath 2018-03-12


Status, Reservations, and Primes

Some bases have the text "power of x". This means that the bases are a subset of another base. For example, searching base 2 already covers bases 4, 8, 16, 32, etc. Reservations will not be accepted for them.

Base Limit n Last Update Reserved by Primes
2 765,830 2018-11-12 Dylan14 (-1) 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, 688042, 695631
(+1) 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
4 power of 2
6 150,000 2018-06-01   (-1) 1, 2, 6, 7, 20, 47, 255, 274, 279, 308, 1162, 2128, 3791, 9028, 9629, 10029, 13202, 38660, 46631, 48257, 117991
(+1) 1, 2, 3, 4, 9, 12, 30, 49, 56, 115, 118, 376, 432, 1045, 1310, 6529, 7768, 8430, 21942, 26930, 33568, 50800
8 power of 2
10 100,000 2017-12-27 (-1) 1, 8, 21, 123, 4299, 6128, 11760, 18884, 40293
(+1) 22, 351, 1061
12 100,000 2018-10-26   (-1) 3, 29, 51, 7824, 15456, 22614, 28312, 47014, 68835
(+1) 1, 2, 8, 60, 513, 1047, 7021, 7506, 78858
14 100,000 2018-04-12   (-1) 1, 6, 13, 45, 74, 240, 553, 12348, 13659, 50603
(+1) 1, 5, 60, 72, 118, 181, 245, 310, 498, 820, 962, 2212, 3928, 5844, 5937
16 power of 2
18 50,000 2018-06-02   (-1) 2, 8, 30, 98, 110, 185, 912, 2514, 4074, 10208, 15123, 19395
(+1) 1, 10, 21, 25, 31, 1083, 40485
20 74,000 2018-01-29   (-1) 1, 2, 53, 183, 1281, 1300, 8041, 29936, 72820
(+1) 1, 15, 44, 77, 141, 208, 304, 1169, 3359, 5050, 22431, 34935
22 30,000 2018-01-30 (-1) 1, 8, 35, 88, 503, 8643, 8743, 14475
(+1) 3, 166, 814, 1851, 2197, 3172, 3865, 19791
24 30,000 2016-08-01 (-1) 2, 27, 92, 4950, 20047
(+1) 24, 321, 971, 984
26 84,239 2018-06-23   (-1) 159, 879, 4744, 5602, 74387
(+1) 1, 2, 8, 78, 79, 111, 5276, 8226, 19545, 75993
28 30,000 2016-05-21 (-1) 1, 22, 127, 165, 2520, 6492, 6577, 22960, 25528
(+1) 1, 2, 11, 15, 586, 993, 5048, 24990
30 65,000 2016-05-05 (-1) 1, 6, 19, 30, 166, 495, 769, 826, 1648, 3993
(+1) 2, 3, 57, 129, 171, 9837, 30359, 157950
32 power of 2
34 30,000 2018-01-24 (-1) 1, 4, 258
(+1) 1, 2, 14, 29, 61, 146, 2901, 6501, 8093
36 power of 6
38 40,000 2016-07-22 (-1) 1, 2, 13, 560, 28933
(+1) 6, 279, 3490
40 60,000 2016-05-05 (-1) 4, 15, 39, 138, 2153, 4084, 5639
(+1) 2, 49, 144, 825, 2856, 2996, 5166, 7824, 9392, 40778
42 30,000 2018-02-13 (-1) 3, 6, 14, 15, 29, 78, 195, 255, 272, 713, 2526, 4852, 10573
(+1) 1, 3, 4, 81, 119, 2046, 2466, 4020, 7907, 8424, 25002
44 100,000 2018-03-21   (-1) 1, 7, 30, 90, 1288, 1947, 12909, 25786
(+1) 3, 195, 1482, 8210, 20502, 60212, 95940
46 30,000 2018-02-18   (-1) 12, 269, 1304, 5172
(+1) 1, 54, 2040, 3063
48 30,000 2018-02-26   (-1) 1, 2, 4, 6, 12, 13, 3882, 6123, 15067, 15085
(+1) 1, 207, 329, 1153, 4687, 13274, 25978
50 100,000 2017-06-27 (-1) 1, 3, 4, 9, 31, 66, 115, 430, 1233, 2546, 2674, 6360, 53351, 69033, 69157
(+1) 4, 38, 93, 120, 4396, 11459, 25887
52 30,000 2018-03-09   (-1) 2, 14, 24, 85, 23234
(+1) 3, 5, 166, 456, 14410
54 30,000 2018-03-17   (-1) 9, 17, 65, 963, 5171, 8068
(+1) 1, 1910, 2575, 7639, 21415
56 30,000 2018-03-29   (-1) 1, 2, 3, 11, 177, 1698, 26698
(+1) 8, 14, 73, 122, 136, 706, 5411, 7771
58 30,000 2018-03-30   (-1) 88, 1720
(+1) 2, 21, 35, 213, 296, 1734, 2354, 23046, 29505
60 30,000 2016-06-06 (-1) 2, 5, 155, 6665, 11549
(+1) 1, 9, 21, 49, 1717, 3882, 11307
62 30,000 2016-06-06 (-1) 1, 454, 1573, 11080
(+1) 1, 2, 212, 761, 2183, 3818, 8800, 13352, 17625
64 power of 2
66 30,000 2016-05-21 (-1) 12, 475, 1902
(+1) 172, 375, 945, 1751, 9832, 28522
68 30,000 2016-05-21 (-1) 4, 59, 779, 2272, 2634, 16522
(+1) 1, 3, 6, 32, 2960, 3014
70 40,000 2016-05-29 (-1) 1, 5, 9, 18, 24, 277, 337, 431, 516, 10744, 11260, 19370
(+1) 1, 7, 11
72 40,000 2016-05-29 (-1) 1, 9, 187, 37930
(+1) 354, 1920, 6831
74 78,000 2016-06-07 (-1) 183, 2300, 3855, 39227, 56039
(+1) 1, 3, 694
76 40,000 2016-05-30 (-1) 1, 2, 32, 37, 51, 220, 3081, 35617, 36090
(+1) 1, 2, 3, 22, 29, 126, 173, 284, 2306
78 40,000 2016-05-30 (-1) 1, 2, 85, 6429
(+1) 3, 5, 13, 79, 6853
80 30,000 2016-05-21 (-1) 320, 363
(+1) 29, 34, 45, 81, 148, 562, 1445, 22631
82 30,000 2016-05-21 (-1) 24, 1074, 1212, 20148
(+1) 3, 9, 595, 1866
84 30,000 2016-05-21 (-1) 4, 66, 82, 140, 1253, 1922, 2613, 5582
(+1) 423, 4162, 14493
86 30,000 2016-05-21 (-1) 3, 39, 75, 1120, 2053
(+1) 8, 18, 36, 260, 11270
88 30,000 2016-05-21 (-1) 2, 5, 9, 134, 594
(+1) 1, 40, 324, 1072, 5100, 28032
90 30,000 2016-05-21 (-1) 1, 5, 43, 1105, 2186, 6957
(+1) 11, 504, 520, 676, 3120
92 30,000 2016-05-21 (-1) 3, 4, 6, 60, 78, 116, 638, 2831, 4281
(+1) 1, 2, 900, 1795
94 30,000 2016-05-21 (-1) 1, 2, 666, 13278, 16021, 20654
(+1) 5, 3305, 11045
96 30,000 2016-05-21 (-1) 5, 106, 448, 561, 609
(+1) 2, 13, 15, 54
98 30,000 2016-05-21 (-1) 2, 21, 130, 378, 896, 2588, 5876
(+1) 4, 12, 35, 459, 28725
100 power of 10
102 30,000 2018-09-11   (-1) 2, 92, 165, 492, 2628, 2879, 4676, 5603, 13808
(+1) 1, 3, 27, 33, 264, 292, 1630, 1993
104 30,000 2018-09-20   (-1) 1, 53, 59, 3867, 6328
(+1) 6, 85, 87, 638, 832, 2798, 6365
106 10,000 2016-05-21 (-1) 2, 3, 46, 47, 104, 2922
(+1) 1, 6, 25, 27, 219, 539, 1641, 2046
108 10,000 2016-05-21 (-1) 1, 3807
(+1) 3, 5, 24, 1078, 2104, 8292
110 10,000 2016-05-21 (-1) 705, 1870
(+1) 57, 1245, 2852, 6403
112 10,000 2016-05-21 (-1) 2, 6, 542, 1656
(+1) 24, 527, 1813
114 10,000 2016-05-21 (-1) 3, 72, 156, 406, 472, 3270, 6873
(+1) 368
116 10,000 2016-05-21 (-1) 29, 38
(+1) 1, 12, 1272, 1852
118 10,000 2016-05-21 (-1) 1, 4, 55, 150, 223, 886, 3381
(+1) 1, 17, 40
120 80,000 2017-04-17 (-1) 1, 6, 7, 11, 12, 168, 2750, 12418, 42427
(+1) 1, 12, 16, 236, 1501, 2249, 3421, 7112, 8260, 10856, 22918
122 10,000 2016-05-21 (-1) 1, 2, 489, 499, 2552, 2565
(+1) 11, 45, 244, 1474
124 10,000 2016-05-21 (-1) 4836
(+1) 19, 36, 728
126 10,000 2016-05-21 (-1) 20, 93, 5924
(+1) 1, 2, 37, 58, 84, 126, 276, 2318, 2877, 4514, 6212
128 power of 2
130 10,000 2016-05-21 (-1) 135
(+1) 1, 178, 672
132 10,000 2016-05-21 (-1) 1, 2, 10, 57, 236, 665, 1876
(+1) 13
134 10,000 2016-05-21 (-1) 4, 16, 65, 81, 93, 161, 1932, 2677, 4681, 4695
(+1) 1, 11, 42, 204, 1171
136 10,000 2016-05-21 (-1) 1, 6
(+1) 12, 47
138 10,000 2016-05-21 (-1) 6, 10, 105, 4534, 9489
(+1) 1, 2, 69, 1716, 5407, 6300
140 10,000 2016-05-21 (-1) 1, 4, 5, 29, 155, 382, 485, 1375, 1528, 5757
(+1) 41, 395, 4267, 4456, 5120, 5396
142 10,000 2016-05-21 (-1) 15
(+1) 3, 48, 4869
144 power of 12
146 10,000 2016-05-21 (-1) 1, 35, 37, 6965
(+1) 3, 20, 403, 2475
148 10,000 2016-05-21 (-1) 2, 60, 255, 422, 528, 1300
(+1) 4, 20, 30, 43, 112
150 10,000 2016-05-21 (-1) 8, 20, 30, 260
(+1) 4, 34
152 10,000 2016-05-21 (-1) 3, 5, 1263, 1317
(+1) 2, 51, 156, 444
154 10,000 2016-05-21 (-1) 24, 68, 1495
(+1) 1, 49, 75, 82, 371, 612
156 10,000 2016-05-21 (-1) 1, 136, 1663
(+1) 152, 461
158 10,000 2016-05-21 (-1) 14
(+1) 1893
160 10,000 2016-05-21 (-1) 4, 11, 85, 127, 1104
(+1) 1, 5, 24, 30, 104, 135, 148
162 10,000 2016-05-21 (-1) 1, 82, 386, 3198
(+1) 12, 447, 8342
164 10,000 2016-05-21 (-1) 2, 1358
(+1) 6, 15, 4967
166 10,000 2016-05-21 (-1) 321, 9492
(+1) 2, 3, 78
168 10,000 2016-05-21 (-1) 11, 24, 380, 1140
(+1) 1, 16, 44, 230, 6988
170 10,000 2016-05-21 (-1) 1, 18, 19, 135, 460, 1059, 1528, 1653, 9373
(+1) 11, 53, 692, 2921, 5141
172 10,000 2016-05-21 (-1) 174, 246, 302, 1191, 1302, 3680
(+1) 1, 3, 24, 126, 492, 745, 778
174 10,000 2016-05-21 (-1) 1, 15, 20, 96, 354
(+1) 2, 10, 396, 1100, 3894
176 10,000 2016-05-21 (-1) 6, 29, 2001, 7360
(+1) 1, 7, 3385, 7566, 8866
178 10,000 2016-05-21 (-1) 1, 9, 174, 313, 609, 87525
(+1) 3, 4, 6, 839
180 10,000 2016-05-21 (-1) 42, 251
(+1) 14, 189, 218, 360
182 10,000 2016-05-21 (-1) 310, 1258
(+1) 1, 5, 6, 227, 1348
184 10,000 2016-05-21 (-1) 1, 160, 2907, 3417
(+1) 2, 12, 51, 671
186 10,000 2016-05-21 (-1) 2, 12, 206, 293
(+1) 6, 459, 811, 1968
188 10,000 2016-05-21 (-1) 27, 39, 55, 77, 1406, 2544, 6052, 6292
(+1) 2, 5, 926, 2225, 3863, 5495
190 10,000 2016-05-21 (-1) 2, 6, 18, 40, 431
(+1) 1, 4, 71, 262, 2841
192 10,000 2016-05-21 (-1) 1, 2, 7, 26, 39
(+1) 1017, 7989
194 10,000 2016-05-21 (-1) 29, 180, 5007
(+1) 3
196 power of 14
198 10,000 2016-05-21 (-1) 103, 118
(+1) 6, 12
200 10,000 2016-05-21 (-1) 20, 36
(+1) 3, 33, 37
202 10,000 2018-01-14 (-1) 63
(+1) 3, 5, 17, 68, 580, 858, 1100, 2323, 3893, 4942
204 30,000 2016-05-21 (-1) 2, 5, 40, 11867, 14458, 17522, 18929
(+1) 1, 7, 40, 3645, 23750
206 10,000 2018-02-27   (-1) 1, 2, 103, 2607
(+1) 6, 684, 2529, 2550
208 10,000 2018-02-27   (-1) 3, 10, 24, 660, 1468
(+1) 2, 2244
210 10,000 2018-02-27   (-1) 7, 17, 1197
(+1) 1, 5, 7, 34, 254, 6961
212 10,000 2018-02-27   (-1) 1, 6, 580, 1933, 4050
(+1) 2, 14, 33, 503, 1601, 4230
214 10,000 2017-07-11 (-1) 5, 36, 1571
(+1) 52, 132, 2614
216 power of 6
218 10,000 2017-07-11 (-1) 1, 52, 397, 969
(+1) 17, 32, 78, 312, 1216
220 10,000 2017-07-31 (-1) 30, 60, 63, 780
(+1) 3, 1728
222 10,000 2017-07-31 (-1) 3, 118, 186, 208
(+1) 1, 93
224 10,000 2018-02-15 (-1) 1, 5, 6, 167, 460, 488, 1130, 1152
(+1) 129, 1137
226 10,000 2018-02-16 (-1) 22, 588, 2989
(+1) 788, 3775
228 10,000 2017-08-28 (-1) 2123
(+1) 3, 48, 150
230 10,000 2018-02-16 (-1) 3, 5, 59, 243, 256, 1382, 1659
(+1) 1, 2, 27, 6534
232 10,000 2018-02-17 (-1) 1, 5, 6, 50, 177, 390
(+1) 1, 26, 30, 216, 934, 1268, 1384, 2671
234 10,000 2018-02-18 (-1) 1, 2, 6447
(+1) 3, 18, 27, 645, 1854
236 10,000 2018-02-19 (-1) 5
(+1) 1, 80
238 10,000 2018-02-19 (-1) 1, 2, 7, 68, 2345, 4783, 5425
(+1) 1, 5, 114, 137, 213, 270, 5796, 6798
240 10,000 2018-02-23 (-1) 1, 261
(+1) 3, 201, 779, 1211
242 10,000 2018-02-24 (-1) 5, 119
(+1) 4, 13, 34, 93, 396, 514, 2056, 7765
244 10,000 2018-02-24 (-1) 25, 404
(+1) 3, 17, 19, 21, 65, 70, 200, 1128, 6742, 7510
246 10,000 2018-02-24 (-1) 4, 46, 57, 122, 266, 364, 565, 2302, 7589
(+1) 1, 9, 236, 561, 735, 741, 816, 6940
248 10,000 2018-02-24 (-1) 1, 6, 55, 1248
(+1) 45, 100, 3588
250 10,000 2018-02-24 (-1) 4, 34, 132
(+1) 7, 90
252 10,000 2017-04-28 (-1) 88, 337, 717, 1330, 1996, 3095, 6548
(+1) 1, 177, 1468
254 10,000 2018-02-24 (-1) 1, 27, 45, 295
(+1) 3, 56, 144, 2730, 5864
256 power of 2
278 10,000 2017-08-28 (-1) 826
(+1) 1, 3, 7, 174, 2025
290 113,000 2018-12-06 kar_bon (-1) 3, 3386
(+1)
316 50,000 2017-05-29 (-1) 1, 5, 27, 183, 5331, 14854, 17396
(+1) 9, 41, 360, 521, 6421
324 power of 18
326 10,000 2017-08-28 (-1) 288, 290
(+1) 8941
360 10,000 2018-02-24 (-1) 1, 3, 7, 31, 1080
(+1) 4, 15, 16, 63, 108, 2290
362 100,000 2018-12-06 kar_bon (-1)
(+1) 156, 264, 630, 794, 46308
364 10,000 2018-03-01   (-1) 1139
(+1) 4, 14, 16
368 10,000 2018-03-01   (-1) 1856
(+1) 969
394 10,000 2018-02-28   (-1)
(+1) 198
400 power of 20
426 10,000 2018-03-01   (-1) 1, 41, 680, 2042, 2233, 9475
(+1) 9708
432 10,000 2018-02-25 (-1) 8742
(+1) 6, 1227
472 50,000 2018-07-23   (-1) 16077
(+1) 10, 459, 7307, 14542, 17019
484 power of 22
500 10,000 2018-02-19 (-1) 981
(+1) 4, 5, 11, 2195
542 10,000 2018-03-01 (-1) 1, 2, 3, 8, 954, 4669, 7267
(+1)
600 10,000 2018-03-01 (-1) 4606, 6509
(+1) 61
640 44,940 2016-05-26 (-1)
(+1) 44940
688 11,000 2016-05-21 (-1)
(+1) 10160
720 10,000 2017-12-30 (-1) 1, 2, 91, 1703, 7583
(+1) 2, 590
1656 70,000 2016-06-18 (-1)
(+1)
1968 60,000 2017-12-27   (-1) 3, 9, 78, 335, 782, 2737
(+1) 1, 36, 29656, 58533
2010 50,000 2017-03-08 (-1) 3, 35, 1967
(+1) 3, 30505
2018 27,000 2018-02-21 (-1) 1800, 3720
(+1) 2, 3143
2026 10,000 2018-02-13 (-1)
(+1) 5526
2264 10,098 2016-05-26 (-1)
(+1) 10098
2482 18,735 2016-05-31 (-1)
(+1) 18735
2634 66,000 2016-06-18 (-1)
(+1)