Daftar bilangan prima
Bilangan prima adalah bilangan asli yang tidak memiliki pembagi positif selain dan dirinya sendiri. Menurut Teorema Euklides, ada bilangan prima yang takhingga. Himpunan bagian dari bilangan prima dapat dibuat dengan berbagai rumus untuk bilangan prima. 1000 bilangan prima pertama tercantum di bawah ini, 1 merupakan bilangan prima atau komposit.
1000 bilangan prima pertama
Tabel berikut mencantumkan 1000 bilangan prima pertama, dengan 20 kolom bilangan prima berurutan di masing-masing dari 50 baris.[1]
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1–20 | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 | 61 | 67 | 71 |
21–40 | 73 | 79 | 83 | 89 | 97 | 101 | 103 | 107 | 109 | 113 | 127 | 131 | 137 | 139 | 149 | 151 | 157 | 163 | 167 | 173 |
41–60 | 179 | 181 | 191 | 193 | 197 | 199 | 211 | 223 | 227 | 229 | 233 | 239 | 241 | 251 | 257 | 263 | 269 | 271 | 277 | 281 |
61–80 | 283 | 293 | 307 | 311 | 313 | 317 | 331 | 337 | 347 | 349 | 353 | 359 | 367 | 373 | 379 | 383 | 389 | 397 | 401 | 409 |
81–100 | 419 | 421 | 431 | 433 | 439 | 443 | 449 | 457 | 461 | 463 | 467 | 479 | 487 | 491 | 499 | 503 | 509 | 521 | 523 | 541 |
101–120 | 547 | 557 | 563 | 569 | 571 | 577 | 587 | 593 | 599 | 601 | 607 | 613 | 617 | 619 | 631 | 641 | 643 | 647 | 653 | 659 |
121–140 | 661 | 673 | 677 | 683 | 691 | 701 | 709 | 719 | 727 | 733 | 739 | 743 | 751 | 757 | 761 | 769 | 773 | 787 | 797 | 809 |
141–160 | 811 | 821 | 823 | 827 | 829 | 839 | 853 | 857 | 859 | 863 | 877 | 881 | 883 | 887 | 907 | 911 | 919 | 929 | 937 | 941 |
161–180 | 947 | 953 | 967 | 971 | 977 | 983 | 991 | 997 | 1009 | 1013 | 1019 | 1021 | 1031 | 1033 | 1039 | 1049 | 1051 | angka | 1063 | 1069 |
181–200 | 1087 | 1091 | 1093 | 1097 | 1103 | 1109 | 1117 | 1123 | 1129 | 1151 | 1153 | 1163 | 1171 | 1181 | 1187 | 1193 | 1201 | 1213 | 1217 | 1223 |
201–220 | 1229 | 1231 | 1237 | 1249 | 1259 | 1277 | 1279 | 1283 | 1289 | 1291 | 1297 | 1301 | 1303 | 1307 | 1319 | 1321 | 1327 | 1361 | 1367 | 1373 |
221–240 | 1381 | 1399 | 1409 | 1423 | 1427 | 1429 | 1433 | 1439 | 1447 | 1451 | 1453 | 1459 | 1471 | 1481 | 1483 | 1487 | 1489 | 1493 | 1499 | 1511 |
241–260 | 1523 | 1531 | 1543 | 1549 | 1553 | 1559 | 1567 | 1571 | 1579 | 1583 | 1597 | 1601 | 1607 | 1609 | 1613 | 1619 | 1621 | 1627 | 1637 | 1657 |
261–280 | 1663 | 1667 | 1669 | 1693 | 1697 | 1699 | 1709 | 1721 | 1723 | 1733 | 1741 | 1747 | 1753 | 1759 | 1777 | 1783 | 1787 | 1789 | 1801 | 1811 |
281–300 | 1823 | 1831 | 1847 | 1861 | 1867 | 1871 | 1873 | 1877 | 1879 | 1889 | 1901 | 1907 | 1913 | 1931 | 1933 | 1949 | 1951 | 1973 | 1979 | 1987 |
301–320 | 1993 | 1997 | 1999 | 2003 | 2011 | 2017 | 2027 | 2029 | 2039 | 2053 | 2063 | 2069 | 2081 | 2083 | 2087 | 2089 | 2099 | 2111 | 2113 | 2129 |
321–340 | 2131 | 2137 | 2141 | 2143 | 2153 | 2161 | 2179 | 2203 | 2207 | 2213 | 2221 | 2237 | 2239 | 2243 | 2251 | 2267 | 2269 | 2273 | 2281 | 2287 |
341–360 | 2293 | 2297 | 2309 | 2311 | 2333 | 2339 | 2341 | 2347 | 2351 | 2357 | 2371 | 2377 | 2381 | 2383 | 2389 | 2393 | 2399 | 2411 | 2417 | 2423 |
361–380 | 2437 | 2441 | 2447 | 2459 | 2467 | 2473 | 2477 | 2503 | 2521 | 2531 | 2539 | 2543 | 2549 | 2551 | 2557 | 2579 | 2591 | 2593 | 2609 | 2617 |
381–400 | 2621 | 2633 | 2647 | 2657 | 2659 | 2663 | 2671 | 2677 | 2683 | 2687 | 2689 | 2693 | 2699 | 2707 | 2711 | 2713 | 2719 | 2729 | 2731 | 2741 |
401–420 | 2749 | 2753 | 2767 | 2777 | 2789 | 2791 | 2797 | 2801 | 2803 | 2819 | 2833 | 2837 | 2843 | 2851 | 2857 | 2861 | 2879 | 2887 | 2897 | 2903 |
421–440 | 2909 | 2917 | 2927 | 2939 | 2953 | 2957 | 2963 | 2969 | 2971 | 2999 | 3001 | 3011 | 3019 | 3023 | 3037 | 3041 | 3049 | 3061 | 3067 | 3079 |
441–460 | 3083 | 3089 | 3109 | 3119 | 3121 | 3137 | 3163 | 3167 | 3169 | 3181 | 3187 | 3191 | 3203 | 3209 | 3217 | 3221 | 3229 | 3251 | 3253 | 3257 |
461–480 | 3259 | 3271 | 3299 | 3301 | 3307 | 3313 | 3319 | 3323 | 3329 | 3331 | 3343 | 3347 | 3359 | 3361 | 3371 | 3373 | 3389 | 3391 | 3407 | 3413 |
481–500 | 3433 | 3449 | 3457 | 3461 | 3463 | 3467 | 3469 | 3491 | 3499 | 3511 | 3517 | 3527 | 3529 | 3533 | 3539 | 3541 | 3547 | 3557 | 3559 | 3571 |
501–520 | 3581 | 3583 | 3593 | 3607 | 3613 | 3617 | 3623 | 3631 | 3637 | 3643 | 3659 | 3671 | 3673 | 3677 | 3691 | 3697 | 3701 | 3709 | 3719 | 3727 |
521–540 | 3733 | 3739 | 3761 | 3767 | 3769 | 3779 | 3793 | 3797 | 3803 | 3821 | 3823 | 3833 | 3847 | 3851 | 3853 | 3863 | 3877 | 3881 | 3889 | 3907 |
541–560 | 3911 | 3917 | 3919 | 3923 | 3929 | 3931 | 3943 | 3947 | 3967 | 3989 | 4001 | 4003 | 4007 | 4013 | 4019 | 4021 | 4027 | 4049 | 4051 | 4057 |
561–580 | 4073 | 4079 | 4091 | 4093 | 4099 | 4111 | 4127 | 4129 | 4133 | 4139 | 4153 | 4157 | 4159 | 4177 | 4201 | 4211 | 4217 | 4219 | 4229 | 4231 |
581–600 | 4241 | 4243 | 4253 | 4259 | 4261 | 4271 | 4273 | 4283 | 4289 | 4297 | 4327 | 4337 | 4339 | 4349 | 4357 | 4363 | 4373 | 4391 | 4397 | 4409 |
601–620 | 4421 | 4423 | 4441 | 4447 | 4451 | 4457 | 4463 | 4481 | 4483 | 4493 | 4507 | 4513 | 4517 | 4519 | 4523 | 4547 | 4549 | 4561 | 4567 | 4583 |
621–640 | 4591 | 4597 | 4603 | 4621 | 4637 | 4639 | 4643 | 4649 | 4651 | 4657 | 4663 | 4673 | 4679 | 4691 | 4703 | 4721 | 4723 | 4729 | 4733 | 4751 |
641–660 | 4759 | 4783 | 4787 | 4789 | 4793 | 4799 | 4801 | 4813 | 4817 | 4831 | 4861 | 4871 | 4877 | 4889 | 4903 | 4909 | 4919 | 4931 | 4933 | 4937 |
661–680 | 4943 | 4951 | 4957 | 4967 | 4969 | 4973 | 4987 | 4993 | 4999 | 5003 | 5009 | 5011 | 5021 | 5023 | 5039 | 5051 | 5059 | 5077 | 5081 | 5087 |
681–700 | 5099 | 5101 | 5107 | 5113 | 5119 | 5147 | 5153 | 5167 | 5171 | 5179 | 5189 | 5197 | 5209 | 5227 | 5231 | 5233 | 5237 | 5261 | 5273 | 5279 |
701–720 | 5281 | 5297 | 5303 | 5309 | 5323 | 5333 | 5347 | 5351 | 5381 | 5387 | 5393 | 5399 | 5407 | 5413 | 5417 | 5419 | 5431 | 5437 | 5441 | 5443 |
721–740 | 5449 | 5471 | 5477 | 5479 | 5483 | 5501 | 5503 | 5507 | 5519 | 5521 | 5527 | 5531 | 5557 | 5563 | 5569 | 5573 | 5581 | 5591 | 5623 | 5639 |
741–760 | 5641 | 5647 | 5651 | 5653 | 5657 | 5659 | 5669 | 5683 | 5689 | 5693 | 5701 | 5711 | 5717 | 5737 | 5741 | 5743 | 5749 | 5779 | 5783 | 5791 |
761–780 | 5801 | 5807 | 5813 | 5821 | 5827 | 5839 | 5843 | 5849 | 5851 | 5857 | 5861 | 5867 | 5869 | 5879 | 5881 | 5897 | 5903 | 5923 | 5927 | 5939 |
781–800 | 5953 | 5981 | 5987 | 6007 | 6011 | 6029 | 6037 | 6043 | 6047 | 6053 | 6067 | 6073 | 6079 | 6089 | 6091 | 6101 | 6113 | 6121 | 6131 | 6133 |
801–820 | 6143 | 6151 | 6163 | 6173 | 6197 | 6199 | 6203 | 6211 | 6217 | 6221 | 6229 | 6247 | 6257 | 6263 | 6269 | 6271 | 6277 | 6287 | 6299 | 6301 |
821–840 | 6311 | 6317 | 6323 | 6329 | 6337 | 6343 | 6353 | 6359 | 6361 | 6367 | 6373 | 6379 | 6389 | 6397 | 6421 | 6427 | 6449 | 6451 | 6469 | 6473 |
841–860 | 6481 | 6491 | 6521 | 6529 | 6547 | 6551 | 6553 | 6563 | 6569 | 6571 | 6577 | 6581 | 6599 | 6607 | 6619 | 6637 | 6653 | 6659 | 6661 | 6673 |
861–880 | 6679 | 6689 | 6691 | 6701 | 6703 | 6709 | 6719 | 6733 | 6737 | 6761 | 6763 | 6779 | 6781 | 6791 | 6793 | 6803 | 6823 | 6827 | 6829 | 6833 |
881–900 | 6841 | 6857 | 6863 | 6869 | 6871 | 6883 | 6899 | 6907 | 6911 | 6917 | 6947 | 6949 | 6959 | 6961 | 6967 | 6971 | 6977 | 6983 | 6991 | 6997 |
901–920 | 7001 | 7013 | 7019 | 7027 | 7039 | 7043 | 7057 | 7069 | 7079 | 7103 | 7109 | 7121 | 7127 | 7129 | 7151 | 7159 | 7177 | 7187 | 7193 | 7207 |
921–940 | 7211 | 7213 | 7219 | 7229 | 7237 | 7243 | 7247 | 7253 | 7283 | 7297 | 7307 | 7309 | 7321 | 7331 | 7333 | 7349 | 7351 | 7369 | 7393 | 7411 |
941–960 | 7417 | 7433 | 7451 | 7457 | 7459 | 7477 | 7481 | 7487 | 7489 | 7499 | 7507 | 7517 | 7523 | 7529 | 7537 | 7541 | 7547 | 7549 | 7559 | 7561 |
961–980 | 7573 | 7577 | 7583 | 7589 | 7591 | 7603 | 7607 | 7621 | 7639 | 7643 | 7649 | 7669 | 7673 | 7681 | 7687 | 7691 | 7699 | 7703 | 7717 | 7723 |
981–1000 | 7727 | 7741 | 7753 | 7757 | 7759 | 7789 | 7793 | 7817 | 7823 | 7829 | 7841 | 7853 | 7867 | 7873 | 7877 | 7879 | 7883 | 7901 | 7907 | 7919 |
(barisan A000040 pada OEIS).
Proyek verifikasi konjektur Goldbach melaporkan bahwa mereka telah menghitung semua bilangan prima di bawah ini 4×1018.[2] That means 95,676,260,903,887,607 primes[3] (nearly 1017), tapi mereka tidak disimpan. Ada rumus yang diketahui untuk mengevaluasi fungsi penghitungan bilangan prima (jumlah bilangan prima di bawah nilai yang diberikan) lebih cepat daripada menghitung bilangan prima. Ini telah digunakan untuk menghitung bahwa ada 1.925.320.391.606.803.968.923 bilangan prima (kira-kira 2×1021) di bawah 1023. Perhitungan yang berbeda menemukan bahwa ada 18.435.599.767.349.200.867.866 bilangan prima (kira-kira 2×1022) di bawah 1024, bila hipotesis Riemann benar.[4]
Daftar bilangan prima menurut tipe
Di bawah ini terdaftar bilangan prima pertama dari banyak bentuk dan tipe bernama. Lebih jelasnya ada di artikel untuk namanya. adalah bilangan asli (termasuk 0) di definisikan
Bilangan prima Bell
Bilangan prima yang merupakan bilangan partisi himpunan dengan anggota.
2, 5, 877, 27644437, 35742549198872617291353508656626642567, 359334085968622831041960188598043661065388726959079837.
Suku selanjutnya memiliki 6539 digit. ( A051131)
Bilangan prima berimbang
Bentuk:
5, 53, 157, 173, 211, 257, 263, 373, 563, 593, 607, 653, 733, 947, 977, 1103, 1123, 1187, 1223, 1367, 1511, 1747, 1753, 1907, 2287, 2417, 2677, 2903, 2963, 3307, 3313, 3637, 3733, 4013, 4409, 4457, 4597, 4657, 4691, 4993, 5107, 5113, 5303, 5387, 5393 (barisan A006562 dalam OEIS).
Bilangan prima Carol
Dari bentuk
7, 47, 223, 3967, 16127, 1046527, 16769023, 1073676287, 68718952447, 274876858367, 4398042316799, 1125899839733759, 18014398241046527, 1298074214633706835075030044377087 ( A091516)
Bilangan prima Chen
Dimana adalah bilangan prima dan adalah baik bilangan prima maupun semiprima.
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 47, 53, 59, 67, 71, 83, 89, 101, 107, 109, 113, 127, 131, 137, 139, 149, 157, 167, 179, 181, 191, 197, 199, 211, 227, 233, 239, 251, 257, 263, 269, 281, 293, 307, 311, 317, 337, 347, 353, 359, 379, 389, 401, 409 ( A109611)
Bilangan prima Cuban
Dari bentuk dimana .
7, 19, 37, 61, 127, 271, 331, 397, 547, 631, 919, 1657, 1801, 1951, 2269, 2437, 2791, 3169, 3571, 4219, 4447, 5167, 5419, 6211, 7057, 7351, 8269, 9241, 10267, 11719, 12097, 13267, 13669, 16651, 19441, 19927, 22447, 23497, 24571, 25117, 26227, 27361, 33391, 35317 ( A002407)
Dari bentuk dimana .
13, 109, 193, 433, 769, 1201, 1453, 2029, 3469, 3889, 4801, 10093, 12289, 13873, 18253, 20173, 21169, 22189, 28813, 37633, 43201, 47629, 60493, 63949, 65713, 69313, 73009, 76801, 84673, 106033, 108301, 112909, 115249 ( A002648)
Bilangan prima Cullen
Dari bentuk .
3, 393050634124102232869567034555427371542904833 ( A050920)
Bilangan prima dihedral
Bilangan prima yang tetap bilangan prima ketika dibaca terbalik atau tercermin dalam sebuah layar tujuh segmen.
2, 5, 11, 101, 181, 1181, 1811, 18181, 108881, 110881, 118081, 120121, 121021, 121151, 150151, 151051, 151121, 180181, 180811, 181081 ( A134996)
Bilangan prima Eisenstein tanpa bagian imajiner/khayal
Bilangan bulat Eisenstein yang merupakan bilangan taktereduksi dan bilangan real (bilangan prima dari bentuk ).
2, 5, 11, 17, 23, 29, 41, 47, 53, 59, 71, 83, 89, 101, 107, 113, 131, 137, 149, 167, 173, 179, 191, 197, 227, 233, 239, 251, 257, 263, 269, 281, 293, 311, 317, 347, 353, 359, 383, 389, 401 ( A003627)
Bilangan prima Euclid
Dari bentuk (sebuah himpunan bagian bilangan prima primorial).
3, 7, 31, 211, 2311, 200560490131 ( A018239[5])
Bilangan prima faktorial
Dari bentuk n! - 1 atau n! + 1.
2, 3, 5, 7, 23, 719, 5039, 39916801, 479001599, 87178291199, 10888869450418352160768000001, 265252859812191058636308479999999, 263130836933693530167218012159999999, 8683317618811886495518194401279999999 ( A088054)
Bilangan prima Fermat
Dari bentuk .
3, 5, 17, 257, 65537 ( A019434)
Hingga Agustus 2019[update], ini hanya dikenal sebagai bilangan prima Fermat, dan secara dugaan hanyalah bilangan prima Fermat. Peluang dari keberadaan bilangan prima Fermat lainnya lebih kecil dari satu miliar.[6]
Bilangan prima Fermat rampat
Dari bentuk untuk bilangan bulat tetap .
a = 2: 3, 5, 17, 257, 65537
a = 4: 5, 17, 257, 65537
a = 6: 7, 37, 1297
a = 8: (tidak ada)
a = 10: 11, 101
a = 12: 13
a = 14: 197
a = 16: 17, 257, 65537
a = 18: 19
a = 20: 401, 160001
a = 22: 23
a = 24: 577, 331777
Hingga April 2017[update], ini haya diketahui bilangan prima Fermat rampat untuk .
Bilangan prima Fibonacci
Bilangan prima dalam barisan Fibonacci , , .
2, 3, 5, 13, 89, 233, 1597, 28657, 514229, 433494437, 2971215073, 99194853094755497, 1066340417491710595814572169, 19134702400093278081449423917 ( A005478)
Bilangan prima fortunate
Bilangan fortunate bahwa semua bilangan prima (ini telah diduga semuanya).
3, 5, 7, 13, 17, 19, 23, 37, 47, 59, 61, 67, 71, 79, 89, 101, 103, 107, 109, 127, 151, 157, 163, 167, 191, 197, 199, 223, 229, 233, 239, 271, 277, 283, 293, 307, 311, 313, 331, 353, 373, 379, 383, 397 ( A046066)
Bilangan prima melingkar
Sebuah bilangan prima melingkar merupakan sebuah bilangan yang tetap bilangan prima pada suatu rotasi siklik mengenai digitnya (dalam basis 10).
2, 3, 5, 7, 11, 13, 17, 31, 37, 71, 73, 79, 97, 113, 131, 197, 199, 311, 337, 373, 719, 733, 919, 971, 991, 1193, 1931, 3119, 3779, 7793, 7937, 9311, 9377, 11939, 19391, 19937, 37199, 39119, 71993, 91193, 93719, 93911, 99371, 193939, 199933, 319993, 331999, 391939, 393919, 919393, 933199, 939193, 939391, 993319, 999331 ( A068652)
Beberapa sumber hanya mencatat bilangan prima terkecil dalam setiap siklus, contohnya, mencatat 13, tetapi menghilangkan 31 (OEIS juga menyebut ini barisan bilangan prima melingkar, tetapi bukan di atas barisan):
2, 3, 5, 7, 11, 13, 17, 37, 79, 113, 197, 199, 337, 1193, 3779, 11939, 19937, 193939, 199933, 1111111111111111111, 11111111111111111111111 ( A016114)
Semua bilangan prima satuan berulang adalah melingkar.
Bilangan prima sepupu
Dimana keduanya bilangan prima.
(3, 7), (7, 11), (13, 17), (19, 23), (37, 41), (43, 47), (67, 71), (79, 83), (97, 101), (103, 107), (109, 113), (127, 131), (163, 167), (193, 197), (223, 227), (229, 233), (277, 281) ( A023200,
A046132)
Sebuah bilangan prima yang membagi bilangan Euler untuk suatu .
19, 31, 43, 47, 61, 67, 71, 79, 101, 137, 139, 149, 193, 223, 241, 251, 263, 277, 307, 311, 349, 353, 359, 373, 379, 419, 433, 461, 463, 491, 509, 541, 563, 571, 577, 587 ( A120337)
Bilangan prima sehingga adalah sebuah pasangan takberaturan Euler.
149, 241, 2946901 ( A198245)
Emirp
Bilangan prima yang menjadi sebuah bilangan prima yang berbeda ketika digit desimalnya terbalik. Nama "emirp" diperoleh dengan membalikkan kata "prime" (yang berarti prima)).
13, 17, 31, 37, 71, 73, 79, 97, 107, 113, 149, 157, 167, 179, 199, 311, 337, 347, 359, 389, 701, 709, 733, 739, 743, 751, 761, 769, 907, 937, 941, 953, 967, 971, 983, 991 ( A006567)
Gaussian primes
Prime elements of the Gaussian integers; equivalently, primes of the form 4n + 3.
3, 7, 11, 19, 23, 31, 43, 47, 59, 67, 71, 79, 83, 103, 107, 127, 131, 139, 151, 163, 167, 179, 191, 199, 211, 223, 227, 239, 251, 263, 271, 283, 307, 311, 331, 347, 359, 367, 379, 383, 419, 431, 439, 443, 463, 467, 479, 487, 491, 499, 503 ( A002145)
Good primes
Primes pn for which pn2 > pn−i pn+i for all 1 ≤ i ≤ n−1, where pn is the nth prime.
5, 11, 17, 29, 37, 41, 53, 59, 67, 71, 97, 101, 127, 149, 179, 191, 223, 227, 251, 257, 269, 307 ( A028388)
Happy primes
Happy numbers that are prime.
7, 13, 19, 23, 31, 79, 97, 103, 109, 139, 167, 193, 239, 263, 293, 313, 331, 367, 379, 383, 397, 409, 487, 563, 617, 653, 673, 683, 709, 739, 761, 863, 881, 907, 937, 1009, 1033, 1039, 1093 ( A035497)
Harmonic primes
Primes p for which there are no solutions to Hk ≡ 0 (mod p) and Hk ≡ −ωp (mod p) for 1 ≤ k ≤ p−2, where Hk denotes the k-th harmonic number and ωp denotes the Wolstenholme quotient.[7]
5, 13, 17, 23, 41, 67, 73, 79, 107, 113, 139, 149, 157, 179, 191, 193, 223, 239, 241, 251, 263, 277, 281, 293, 307, 311, 317, 331, 337, 349 ( A092101)
Higgs primes for squares
Primes p for which p − 1 divides the square of the product of all earlier terms.
2, 3, 5, 7, 11, 13, 19, 23, 29, 31, 37, 43, 47, 53, 59, 61, 67, 71, 79, 101, 107, 127, 131, 139, 149, 151, 157, 173, 181, 191, 197, 199, 211, 223, 229, 263, 269, 277, 283, 311, 317, 331, 347, 349 ( A007459)
Highly cototient primes
Primes that are a cototient more often than any integer below it except 1.
2, 23, 47, 59, 83, 89, 113, 167, 269, 389, 419, 509, 659, 839, 1049, 1259, 1889 ( A105440)
Home primes
For n ≥ 2, write the prime factorization of n in base 10 and concatenate the factors; iterate until a prime is reached.
2, 3, 211, 5, 23, 7, 3331113965338635107, 311, 773, 11, 223, 13, 13367, 1129, 31636373, 17, 233, 19, 3318308475676071413, 37, 211, 23, 331319, 773, 3251, 13367, 227, 29, 547, 31, 241271, 311, 31397, 1129, 71129, 37, 373, 313, 3314192745739, 41, 379, 43, 22815088913, 3411949, 223, 47, 6161791591356884791277 ( A037274)
Irregular primes
Odd primes p that divide the class number of the p-th cyclotomic field.
37, 59, 67, 101, 103, 131, 149, 157, 233, 257, 263, 271, 283, 293, 307, 311, 347, 353, 379, 389, 401, 409, 421, 433, 461, 463, 467, 491, 523, 541, 547, 557, 577, 587, 593, 607, 613 ( A000928)
(p, p − 3) irregular primes
(See Wolstenholme prime)
(p, p − 5) irregular primes
Primes p such that (p, p−5) is an irregular pair.[8]
37
(p, p − 9) irregular primes
Primes p such that (p, p − 9) is an irregular pair.[8]
67, 877 ( A212557)
Isolated primes
Primes p such that neither p − 2 nor p + 2 is prime.
2, 23, 37, 47, 53, 67, 79, 83, 89, 97, 113, 127, 131, 157, 163, 167, 173, 211, 223, 233, 251, 257, 263, 277, 293, 307, 317, 331, 337, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 439, 443, 449, 457, 467, 479, 487, 491, 499, 503, 509, 541, 547, 557, 563, 577, 587, 593, 607, 613, 631, 647, 653, 673, 677, 683, 691, 701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797, 839, 853, 863, 877, 887, 907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997 ( A007510)
Kynea primes
Of the form (2n + 1)2 − 2.
2, 7, 23, 79, 1087, 66047, 263167, 16785407, 1073807359, 17180131327, 68720001023, 4398050705407, 70368760954879, 18014398777917439, 18446744082299486207 ( A091514)
Leyland primes
Of the form xy + yx, with 1 < x < y.
17, 593, 32993, 2097593, 8589935681, 59604644783353249, 523347633027360537213687137, 43143988327398957279342419750374600193 ( A094133)
Long primes
Primes p for which, in a given base b, gives a cyclic number. They are also called full reptend primes. Primes p for base 10:
7, 17, 19, 23, 29, 47, 59, 61, 97, 109, 113, 131, 149, 167, 179, 181, 193, 223, 229, 233, 257, 263, 269, 313, 337, 367, 379, 383, 389, 419, 433, 461, 487, 491, 499, 503, 509, 541, 571, 577, 593 ( A001913)
Lucas primes
Primes in the Lucas number sequence L0 = 2, L1 = 1, Ln = Ln−1 + Ln−2.
2,[9] 3, 7, 11, 29, 47, 199, 521, 2207, 3571, 9349, 3010349, 54018521, 370248451, 6643838879, 119218851371, 5600748293801, 688846502588399, 32361122672259149 ( A005479)
Lucky primes
Lucky numbers that are prime.
3, 7, 13, 31, 37, 43, 67, 73, 79, 127, 151, 163, 193, 211, 223, 241, 283, 307, 331, 349, 367, 409, 421, 433, 463, 487, 541, 577, 601, 613, 619, 631, 643, 673, 727, 739, 769, 787, 823, 883, 937, 991, 997 ( A031157)
Mersenne primes
Of the form 2n − 1.
3, 7, 31, 127, 8191, 131071, 524287, 2147483647, 2305843009213693951, 618970019642690137449562111, 162259276829213363391578010288127, 170141183460469231731687303715884105727 ( A000668)
Hingga 2018[update], there are 51 known Mersenne primes. The 13th, 14th, and 51st have respectively 157, 183, and 24,862,048 digits.
Hingga 2018[update], this class of prime numbers also contains the largest known prime: M82589933, the 51st known Mersenne prime.
Mersenne divisors
Primes p that divide 2n − 1, for some prime number n.
3, 7, 23, 31, 47, 89, 127, 167, 223, 233, 263, 359, 383, 431, 439, 479, 503, 719, 839, 863, 887, 983, 1103, 1319, 1367, 1399, 1433, 1439, 1487, 1823, 1913, 2039, 2063, 2089, 2207, 2351, 2383, 2447, 2687, 2767, 2879, 2903, 2999, 3023, 3119, 3167, 3343 ( A122094)
All Mersenne primes are, by definition, members of this sequence.
Mersenne prime exponents
Primes p such that 2p − 1 is prime.
2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583, 25964951, 30402457, 32582657, 37156667, 42643801, 43112609 ( A000043)
Hingga Desember 2018[update] four more are known to be in the sequence, but it is not known whether they are the next:
57885161, 74207281, 77232917, 82589933
Double Mersenne primes
A subset of Mersenne primes of the form 22p−1 − 1 for prime p.
7, 127, 2147483647, 170141183460469231731687303715884105727 (primes in A077586)
As of June 2017, these are the only known double Mersenne primes, and number theorists think these are probably the only double Mersenne primes.[butuh rujukan]
Generalized repunit primes
Of the form (an − 1) / (a − 1) for fixed integer a.
For a = 2, these are the Mersenne primes, while for a = 10 they are the repunit primes. For other small a, they are given below:
a = 3: 13, 1093, 797161, 3754733257489862401973357979128773, 6957596529882152968992225251835887181478451547013 ( A076481)
a = 4: 5 (the only prime for a = 4)
a = 5: 31, 19531, 12207031, 305175781, 177635683940025046467781066894531, 14693679385278593849609206715278070972733319459651094018859396328480215743184089660644531 ( A086122)
a = 6: 7, 43, 55987, 7369130657357778596659, 3546245297457217493590449191748546458005595187661976371 ( A165210)
a = 7: 2801, 16148168401, 85053461164796801949539541639542805770666392330682673302530819774105141531698707146930307290253537320447270457
a = 8: 73 (the only prime for a = 8)
a = 9: none exist
Other generalizations and variations
Many generalizations of Mersenne primes have been defined. This include the following:
- Primes of the form bn − (b − 1)n,[10][11][12] including the Mersenne primes and the cuban primes as special cases
- Williams primes, of the form (b − 1)·bn − 1
Mills primes
Of the form ⌊θ3n⌋, where θ is Mills' constant. This form is prime for all positive integers n.
2, 11, 1361, 2521008887, 16022236204009818131831320183 ( A051254)
Minimal primes
Primes for which there is no shorter sub-sequence of the decimal digits that form a prime. There are exactly 26 minimal primes:
2, 3, 5, 7, 11, 19, 41, 61, 89, 409, 449, 499, 881, 991, 6469, 6949, 9001, 9049, 9649, 9949, 60649, 666649, 946669, 60000049, 66000049, 66600049 ( A071062)
Newman–Shanks–Williams primes
Newman–Shanks–Williams numbers that are prime.
7, 41, 239, 9369319, 63018038201, 489133282872437279, 19175002942688032928599 ( A088165)
Non-generous primes
Primes p for which the least positive primitive root is not a primitive root of p2. Three such primes are known; it is not known whether there are more.[13]
2, 40487, 6692367337 ( A055578)
Palindromic primes
Primes that remain the same when their decimal digits are read backwards.
2, 3, 5, 7, 11, 101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, 797, 919, 929, 10301, 10501, 10601, 11311, 11411, 12421, 12721, 12821, 13331, 13831, 13931, 14341, 14741 ( A002385)
Palindromic wing primes
Primes of the form with .[14] This means all digits except the middle digit are equal.
101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, 797, 919, 929, 11311, 11411, 33533, 77377, 77477, 77977, 1114111, 1117111, 3331333, 3337333, 7772777, 7774777, 7778777, 111181111, 111191111, 777767777, 77777677777, 99999199999 ( A077798)
Partition primes
Partition function values that are prime.
2, 3, 5, 7, 11, 101, 17977, 10619863, 6620830889, 80630964769, 228204732751, 1171432692373, 1398341745571, 10963707205259, 15285151248481, 10657331232548839, 790738119649411319, 18987964267331664557 ( A049575)
Pell primes
Primes in the Pell number sequence P0 = 0, P1 = 1, Pn = 2Pn−1 + Pn−2.
2, 5, 29, 5741, 33461, 44560482149, 1746860020068409, 68480406462161287469, 13558774610046711780701, 4125636888562548868221559797461449 ( A086383)
Permutable primes
Any permutation of the decimal digits is a prime.
2, 3, 5, 7, 11, 13, 17, 31, 37, 71, 73, 79, 97, 113, 131, 199, 311, 337, 373, 733, 919, 991, 1111111111111111111, 11111111111111111111111 ( A003459)
It seems likely that all further permutable primes are repunits, i.e. contain only the digit 1.
Perrin primes
Primes in the Perrin number sequence P(0) = 3, P(1) = 0, P(2) = 2, P(n) = P(n−2) + P(n−3).
2, 3, 5, 7, 17, 29, 277, 367, 853, 14197, 43721, 1442968193, 792606555396977, 187278659180417234321, 66241160488780141071579864797 ( A074788)
Pierpont primes
Of the form 2u3v + 1 for some integers u,v ≥ 0.
These are also class 1- primes.
2, 3, 5, 7, 13, 17, 19, 37, 73, 97, 109, 163, 193, 257, 433, 487, 577, 769, 1153, 1297, 1459, 2593, 2917, 3457, 3889, 10369, 12289, 17497, 18433, 39367, 52489, 65537, 139969, 147457 ( A005109)
Pillai primes
Primes p for which there exist n > 0 such that p divides n! + 1 and n does not divide p − 1.
23, 29, 59, 61, 67, 71, 79, 83, 109, 137, 139, 149, 193, 227, 233, 239, 251, 257, 269, 271, 277, 293, 307, 311, 317, 359, 379, 383, 389, 397, 401, 419, 431, 449, 461, 463, 467, 479, 499 ( A063980)
Primes of the form n4 + 1
Of the form n4 + 1.[15][16]
2, 17, 257, 1297, 65537, 160001, 331777, 614657, 1336337, 4477457, 5308417, 8503057, 9834497, 29986577, 40960001, 45212177, 59969537, 65610001, 126247697, 193877777, 303595777, 384160001, 406586897, 562448657, 655360001 ( A037896)
Primeval primes
Primes for which there are more prime permutations of some or all the decimal digits than for any smaller number.
2, 13, 37, 107, 113, 137, 1013, 1237, 1367, 10079 ( A119535)
Primorial primes
Of the form pn# ± 1.
3, 5, 7, 29, 31, 211, 2309, 2311, 30029, 200560490131, 304250263527209, 23768741896345550770650537601358309 (union of A057705 and
A018239[5])
Proth primes
Of the form k×2n + 1, with odd k and k < 2n.
3, 5, 13, 17, 41, 97, 113, 193, 241, 257, 353, 449, 577, 641, 673, 769, 929, 1153, 1217, 1409, 1601, 2113, 2689, 2753, 3137, 3329, 3457, 4481, 4993, 6529, 7297, 7681, 7937, 9473, 9601, 9857 ( A080076)
Pythagorean primes
Of the form 4n + 1.
5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101, 109, 113, 137, 149, 157, 173, 181, 193, 197, 229, 233, 241, 257, 269, 277, 281, 293, 313, 317, 337, 349, 353, 373, 389, 397, 401, 409, 421, 433, 449 ( A002144)
Prime quadruplets
Where (p, p+2, p+6, p+8) are all prime.
(5, 7, 11, 13), (11, 13, 17, 19), (101, 103, 107, 109), (191, 193, 197, 199), (821, 823, 827, 829), (1481, 1483, 1487, 1489), (1871, 1873, 1877, 1879), (2081, 2083, 2087, 2089), (3251, 3253, 3257, 3259), (3461, 3463, 3467, 3469), (5651, 5653, 5657, 5659), (9431, 9433, 9437, 9439) ( A007530,
A136720,
A136721,
A090258)
Quartan primes
Of the form x4 + y4, where x,y > 0.
2, 17, 97, 257, 337, 641, 881 ( A002645)
Ramanujan primes
Integers Rn that are the smallest to give at least n primes from x/2 to x for all x ≥ Rn (all such integers are primes).
2, 11, 17, 29, 41, 47, 59, 67, 71, 97, 101, 107, 127, 149, 151, 167, 179, 181, 227, 229, 233, 239, 241, 263, 269, 281, 307, 311, 347, 349, 367, 373, 401, 409, 419, 431, 433, 439, 461, 487, 491 ( A104272)
Regular primes
Primes p that do not divide the class number of the p-th cyclotomic field.
3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 43, 47, 53, 61, 71, 73, 79, 83, 89, 97, 107, 109, 113, 127, 137, 139, 151, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 239, 241, 251, 269, 277, 281 ( A007703)
Repunit primes
Primes containing only the decimal digit 1.
11, 1111111111111111111 (19 digits), 11111111111111111111111 (23 digits) ( A004022)
The next have 317, 1031, 49081, 86453, 109297, 270343 digits ( A004023)
Residue classes of primes
Of the form an + d for fixed integers a and d. Also called primes congruent to d modulo a.
The primes of the form 2n+1 are the odd primes, including all primes other than 2. Some sequences have alternate names: 4n+1 are Pythagorean primes, 4n+3 are the integer Gaussian primes, and 6n+5 are the Eisenstein primes (with 2 omitted). The classes 10n+d (d = 1, 3, 7, 9) are primes ending in the decimal digit d.
2n+1: 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53 ( A065091)
4n+1: 5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101, 109, 113, 137 ( A002144)
4n+3: 3, 7, 11, 19, 23, 31, 43, 47, 59, 67, 71, 79, 83, 103, 107 ( A002145)
6n+1: 7, 13, 19, 31, 37, 43, 61, 67, 73, 79, 97, 103, 109, 127, 139 ( A002476)
6n+5: 5, 11, 17, 23, 29, 41, 47, 53, 59, 71, 83, 89, 101, 107, 113 ( A007528)
8n+1: 17, 41, 73, 89, 97, 113, 137, 193, 233, 241, 257, 281, 313, 337, 353 ( A007519)
8n+3: 3, 11, 19, 43, 59, 67, 83, 107, 131, 139, 163, 179, 211, 227, 251 ( A007520)
8n+5: 5, 13, 29, 37, 53, 61, 101, 109, 149, 157, 173, 181, 197, 229, 269 ( A007521)
8n+7: 7, 23, 31, 47, 71, 79, 103, 127, 151, 167, 191, 199, 223, 239, 263 ( A007522)
10n+1: 11, 31, 41, 61, 71, 101, 131, 151, 181, 191, 211, 241, 251, 271, 281 ( A030430)
10n+3: 3, 13, 23, 43, 53, 73, 83, 103, 113, 163, 173, 193, 223, 233, 263 ( A030431)
10n+7: 7, 17, 37, 47, 67, 97, 107, 127, 137, 157, 167, 197, 227, 257, 277 ( A030432)
10n+9: 19, 29, 59, 79, 89, 109, 139, 149, 179, 199, 229, 239, 269, 349, 359 ( A030433)
12n+1: 13, 37, 61, 73, 97, 109, 157, 181, 193, 229, 241, 277, 313, 337, 349 ( A068228)
12n+5: 5, 17, 29, 41, 53, 89, 101, 113, 137, 149, 173, 197, 233, 257, 269 ( A040117)
12n+7: 7, 19, 31, 43, 67, 79, 103, 127, 139, 151, 163, 199, 211, 223, 271 ( A068229)
12n+11: 11, 23, 47, 59, 71, 83, 107, 131, 167, 179, 191, 227, 239, 251, 263 ( A068231)
Safe primes
Where p and (p−1) / 2 are both prime.
5, 7, 11, 23, 47, 59, 83, 107, 167, 179, 227, 263, 347, 359, 383, 467, 479, 503, 563, 587, 719, 839, 863, 887, 983, 1019, 1187, 1283, 1307, 1319, 1367, 1439, 1487, 1523, 1619, 1823, 1907 ( A005385)
Self primes in base 10
Primes that cannot be generated by any integer added to the sum of its decimal digits.
3, 5, 7, 31, 53, 97, 211, 233, 277, 367, 389, 457, 479, 547, 569, 613, 659, 727, 839, 883, 929, 1021, 1087, 1109, 1223, 1289, 1447, 1559, 1627, 1693, 1783, 1873 ( A006378)
Sexy primes
Where (p, p + 6) are both prime.
(5, 11), (7, 13), (11, 17), (13, 19), (17, 23), (23, 29), (31, 37), (37, 43), (41, 47), (47, 53), (53, 59), (61, 67), (67, 73), (73, 79), (83, 89), (97, 103), (101, 107), (103, 109), (107, 113), (131, 137), (151, 157), (157, 163), (167, 173), (173, 179), (191, 197), (193, 199) ( A023201,
A046117)
Smarandache–Wellin primes
Primes that are the concatenation of the first n primes written in decimal.
2, 23, 2357 ( A069151)
The fourth Smarandache-Wellin prime is the 355-digit concatenation of the first 128 primes that end with 719.
Solinas primes
Of the form 2a ± 2b ± 1, where 0 < b < a.
Sophie Germain primes
Where p and 2p + 1 are both prime.
2, 3, 5, 11, 23, 29, 41, 53, 83, 89, 113, 131, 173, 179, 191, 233, 239, 251, 281, 293, 359, 419, 431, 443, 491, 509, 593, 641, 653, 659, 683, 719, 743, 761, 809, 911, 953 ( A005384)
Stern primes
Primes that are not the sum of a smaller prime and twice the square of a nonzero integer.
2, 3, 17, 137, 227, 977, 1187, 1493 ( A042978)
Hingga 2011[update], these are the only known Stern primes, and possibly the only existing.
Strobogrammatic primes
Primes that are also a prime number when rotated upside down. (This, as with its alphabetic counterpart the ambigram, is dependent upon the typeface.)
Using 0, 1, 8 and 6/9:
11, 101, 181, 619, 16091, 18181, 19861, 61819, 116911, 119611, 160091, 169691, 191161, 196961, 686989, 688889 (barisan A007597 pada OEIS)
Super-primes
Primes with a prime index in the sequence of prime numbers (the 2nd, 3rd, 5th, ... prime).
3, 5, 11, 17, 31, 41, 59, 67, 83, 109, 127, 157, 179, 191, 211, 241, 277, 283, 331, 353, 367, 401, 431, 461, 509, 547, 563, 587, 599, 617, 709, 739, 773, 797, 859, 877, 919, 967, 991 ( A006450)
Supersingular primes
There are exactly fifteen supersingular primes:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59, 71 ( A002267)
Thabit primes
Of the form 3×2n − 1.
2, 5, 11, 23, 47, 191, 383, 6143, 786431, 51539607551, 824633720831, 26388279066623, 108086391056891903, 55340232221128654847, 226673591177742970257407 ( A007505)
The primes of the form 3×2n + 1 are related.
7, 13, 97, 193, 769, 12289, 786433, 3221225473, 206158430209, 6597069766657 ( A039687)
Prime triplets
Where (p, p+2, p+6) or (p, p+4, p+6) are all prime.
(5, 7, 11), (7, 11, 13), (11, 13, 17), (13, 17, 19), (17, 19, 23), (37, 41, 43), (41, 43, 47), (67, 71, 73), (97, 101, 103), (101, 103, 107), (103, 107, 109), (107, 109, 113), (191, 193, 197), (193, 197, 199), (223, 227, 229), (227, 229, 233), (277, 281, 283), (307, 311, 313), (311, 313, 317), (347, 349, 353) ( A007529,
A098414,
A098415)
Truncatable prime
Left-truncatable
Primes that remain prime when the leading decimal digit is successively removed.
2, 3, 5, 7, 13, 17, 23, 37, 43, 47, 53, 67, 73, 83, 97, 113, 137, 167, 173, 197, 223, 283, 313, 317, 337, 347, 353, 367, 373, 383, 397, 443, 467, 523, 547, 613, 617, 643, 647, 653, 673, 683 ( A024785)
Right-truncatable
Primes that remain prime when the least significant decimal digit is successively removed.
2, 3, 5, 7, 23, 29, 31, 37, 53, 59, 71, 73, 79, 233, 239, 293, 311, 313, 317, 373, 379, 593, 599, 719, 733, 739, 797, 2333, 2339, 2393, 2399, 2939, 3119, 3137, 3733, 3739, 3793, 3797 ( A024770)
Two-sided
Primes that are both left-truncatable and right-truncatable. There are exactly fifteen two-sided primes:
2, 3, 5, 7, 23, 37, 53, 73, 313, 317, 373, 797, 3137, 3797, 739397 ( A020994)
Twin primes
Where (p, p+2) are both prime.
(3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), (59, 61), (71, 73), (101, 103), (107, 109), (137, 139), (149, 151), (179, 181), (191, 193), (197, 199), (227, 229), (239, 241), (269, 271), (281, 283), (311, 313), (347, 349), (419, 421), (431, 433), (461, 463) ( A001359,
A006512)
Unique primes
The list of primes p for which the period length of the decimal expansion of 1/p is unique (no other prime gives the same period).
3, 11, 37, 101, 9091, 9901, 333667, 909091, 99990001, 999999000001, 9999999900000001, 909090909090909091, 1111111111111111111, 11111111111111111111111, 900900900900990990990991 ( A040017)
Wagstaff primes
Of the form (2n + 1) / 3.
3, 11, 43, 683, 2731, 43691, 174763, 2796203, 715827883, 2932031007403, 768614336404564651, 201487636602438195784363, 845100400152152934331135470251, 56713727820156410577229101238628035243 ( A000979)
Values of n:
3, 5, 7, 11, 13, 17, 19, 23, 31, 43, 61, 79, 101, 127, 167, 191, 199, 313, 347, 701, 1709, 2617, 3539, 5807, 10501, 10691, 11279, 12391, 14479, 42737, 83339, 95369, 117239, 127031, 138937, 141079, 267017, 269987, 374321 ( A000978)
Wall–Sun–Sun primes
A prime p > 5, if p2 divides the Fibonacci number , where the Legendre symbol is defined as
Hingga 2018[update], no Wall-Sun-Sun primes are known.
Weakly prime numbers
Primes that having any one of their (base 10) digits changed to any other value will always result in a composite number.
294001, 505447, 584141, 604171, 971767, 1062599, 1282529, 1524181, 2017963, 2474431, 2690201, 3085553, 3326489, 4393139 ( A050249)
Wieferich primes
Primes p such that ap − 1 ≡ 1 (mod p2) for fixed integer a > 1.
2p − 1 ≡ 1 (mod p2): 1093, 3511 ( A001220)
3p − 1 ≡ 1 (mod p2): 11, 1006003 ( A014127)[17][18][19]
4p − 1 ≡ 1 (mod p2): 1093, 3511
5p − 1 ≡ 1 (mod p2): 2, 20771, 40487, 53471161, 1645333507, 6692367337, 188748146801 ( A123692)
6p − 1 ≡ 1 (mod p2): 66161, 534851, 3152573 ( A212583)
7p − 1 ≡ 1 (mod p2): 5, 491531 ( A123693)
8p − 1 ≡ 1 (mod p2): 3, 1093, 3511
9p − 1 ≡ 1 (mod p2): 2, 11, 1006003
10p − 1 ≡ 1 (mod p2): 3, 487, 56598313 ( A045616)
11p − 1 ≡ 1 (mod p2): 71[20]
12p − 1 ≡ 1 (mod p2): 2693, 123653 ( A111027)
13p − 1 ≡ 1 (mod p2): 2, 863, 1747591 ( A128667)[20]
14p − 1 ≡ 1 (mod p2): 29, 353, 7596952219 ( A234810)
15p − 1 ≡ 1 (mod p2): 29131, 119327070011 ( A242741)
16p − 1 ≡ 1 (mod p2): 1093, 3511
17p − 1 ≡ 1 (mod p2): 2, 3, 46021, 48947 ( A128668)[20]
18p − 1 ≡ 1 (mod p2): 5, 7, 37, 331, 33923, 1284043 ( A244260)
19p − 1 ≡ 1 (mod p2): 3, 7, 13, 43, 137, 63061489 ( A090968)[20]
20p − 1 ≡ 1 (mod p2): 281, 46457, 9377747, 122959073 ( A242982)
21p − 1 ≡ 1 (mod p2): 2
22p − 1 ≡ 1 (mod p2): 13, 673, 1595813, 492366587, 9809862296159 ( A298951)
23p − 1 ≡ 1 (mod p2): 13, 2481757, 13703077, 15546404183, 2549536629329 ( A128669)
24p − 1 ≡ 1 (mod p2): 5, 25633
25p − 1 ≡ 1 (mod p2): 2, 20771, 40487, 53471161, 1645333507, 6692367337, 188748146801
Hingga 2018[update], these are all known Wieferich primes with a ≤ 25.
Wilson primes
Primes p for which p2 divides (p−1)! + 1.
5, 13, 563 ( A007540)
Hingga 2018[update], these are the only known Wilson primes.
Wolstenholme primes
Primes p for which the binomial coefficient
16843, 2124679 ( A088164)
Hingga 2018[update], these are the only known Wolstenholme primes.
Woodall primes
Of the form n×2n − 1.
7, 23, 383, 32212254719, 2833419889721787128217599, 195845982777569926302400511, 4776913109852041418248056622882488319 ( A050918)
Referensi
- ^ Lehmer, D. N. (1982). List of prime numbers from 1 to 10,006,721. 165. Washington D.C.: Carnegie Institution of Washington. OL 16553580M. OL16553580M.
- ^ Tomás Oliveira e Silva, Goldbach conjecture verification Diarsipkan 24 May 2011 di Wayback Machine.. Retrieved 16 July 2013
- ^ (barisan A080127 pada OEIS)
- ^ Jens Franke (29 Juli 2010). "Conditional Calculation of pi(1024)". Diarsipkan dari versi asli tanggal 24 Agustus 2014. Diakses tanggal 17 Mei 2011. Parameter
|url-status=
yang tidak diketahui akan diabaikan (bantuan) - ^ a b
A018239 termasuk includes 2 = darab kosong mengenai 0 prima pertama ditambah of 1, tetapi 2 dikecualikan dalam daftar ini.
- ^ Boklan, Kent D.; Conway, John H. (2016). "Expect at most one billionth of a new Fermat Prime!". arΧiv:1605.01371 [math.NT].
- ^ Boyd, D. W. (1994). "A p-adic Study of the Partial Sums of the Harmonic Series". Experimental Mathematics. 3 (4): 287–302. doi:10.1080/10586458.1994.10504298. Zbl 0838.11015. CiteSeerX: 10.1.1.56.7026. Diarsipkan dari versi asli tanggal 27 January 2016. Parameter
|url-status=
yang tidak diketahui akan diabaikan (bantuan) - ^ a b Johnson, W. (1975). "Irregular Primes and Cyclotomic Invariants" (PDF). Mathematics of Computation. AMS. 29 (129): 113–120. doi:10.2307/2005468. JSTOR 2005468. Diarsipkan dari versi asli (PDF) tanggal 20 December 2010. Parameter
|url-status=
yang tidak diketahui akan diabaikan (bantuan) - ^ It varies whether L0 = 2 is included in the Lucas numbers.
- ^ Sloane, N.J.A. (ed.). "Sequence A121091 (Smallest nexus prime of the form n^p - (n-1)^p, where p is an odd prime)". On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ Sloane, N.J.A. (ed.). "Sequence A121616 (Primes of form (n+1)^5 - n^5)". On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ Sloane, N.J.A. (ed.). "Sequence A121618 (Nexus primes of order 7 or primes of form n^7 - (n-1)^7)". On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ Paszkiewicz, Andrzej (2009). "A new prime for which the least primitive root and the least primitive root are not equal" (PDF). Math. Comp. American Mathematical Society. 78: 1193–1195. Bibcode:2009MaCom..78.1193P. doi:10.1090/S0025-5718-08-02090-5.
- ^ Caldwell, C.; Dubner, H. (1996–97). "The near repdigit primes , especially ". Journal of Recreational Mathematics. 28 (1): 1–9.
- ^ Lal, M. (1967). "Primes of the Form n4 + 1" (PDF). Mathematics of Computation. AMS. 21: 245–247. doi:10.1090/S0025-5718-1967-0222007-9. ISSN 1088-6842. Diarsipkan dari versi asli (PDF) tanggal 13 January 2015. Parameter
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yang tidak diketahui akan diabaikan (bantuan) - ^ Bohman, J. (1973). "New primes of the form n4 + 1". BIT Numerical Mathematics. Springer. 13 (3): 370–372. doi:10.1007/BF01951947. ISSN 1572-9125.
- ^ Ribenboim, P. (22 February 1996). The new book of prime number records. New York: Springer-Verlag. hlm. 347. ISBN 0-387-94457-5.
- ^ "Mirimanoff's Congruence: Other Congruences". Diakses tanggal 26 January 2011.
- ^ Gallot, Y.; Moree, P.; Zudilin, W. (2011). "The Erdös-Moser equation 1k + 2k +...+ (m−1)k = mk revisited using continued fractions". Mathematics of Computation. American Mathematical Society. 80: 1221–1237. arXiv:0907.1356
. doi:10.1090/S0025-5718-2010-02439-1.
- ^ a b c d Ribenboim, P. (2006). Die Welt der Primzahlen (PDF). Berlin: Springer. hlm. 240. ISBN 3-540-34283-4. Parameter
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yang tidak diketahui akan diabaikan (bantuan)