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  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 1317  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

.

Proyek verifikasi konjektur Goldbach melaporkan bahwa mereka telah menghitung semua bilangan prima di bawah ini 4×10.Tomás Oliveira e Silva, Goldbach conjecture verification . Retrieved 16 July 2013 That means 95,676,260,903,887,607 primes (nearly 10), 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) di bawah 10. Perhitungan yang berbeda menemukan bahwa ada 18.435.599.767.349.200.867.866 bilangan prima (kira-kira 2) di bawah 10, bila hipotesis Riemann benar.

 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. ()

 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 ()

 Bilangan prima Chen 
Di mana  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 ()

 Bilangan prima Cuban 
Dari bentuk  di mana .

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 ()

Dari bentuk  di mana .

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 ()

 Bilangan prima Cullen 
Dari bentuk .

3, 393050634124102232869567034555427371542904833 ()

 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 ()

 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 ()

 Bilangan prima Euclid 
Dari bentuk  (sebuah himpunan bagian bilangan prima primorial).

3, 7, 31, 211, 2311, 200560490131 ( termasuk includes 2 = darab kosong mengenai 0 prima pertama ditambah of 1, tetapi 2 dikecualikan dalam daftar ini.)

 Bilangan prima faktorial 
Dari bentuk  atau .

2, 3, 5, 7, 23, 719, 5039, 39916801, 479001599, 87178291199, 10888869450418352160768000001, 265252859812191058636308479999999, 263130836933693530167218012159999999, 8683317618811886495518194401279999999 ()

 Bilangan prima Fermat 
Dari bentuk .

3, 5, 17, 257, 65537 ()

, 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.

 Bilangan prima Fermat rampat 
Dari bentuk  untuk bilangan bulat tetap .

 (tidak ada)

, 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 ()

 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 ()

 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 ()

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 ()

Semua bilangan prima satuan berulang adalah melingkar.

 Bilangan prima sepupu 

Di mana  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) (, )

 Bilangan prima takberaturan Euler 

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 ()

 Bilangan prima takberaturan  Euler 
Bilangan prima  sehingga  adalah sebuah pasangan takberaturan Euler.

149, 241, 2946901 ()

 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 ()

 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 ()

 Good primes 
Primes p for which p > p p for all 1 ≤ i ≤ n−1, where p 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 ()

 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 ()

 Harmonic primes 
Primes p for which there are no solutions to H ≡ 0 (mod p) and H ≡ −ω (mod p) for 1 ≤ k ≤ p−2, where H denotes the k-th harmonic number and ω denotes the Wolstenholme quotient.

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 ()

 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 ()

 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 ()

 Home primes 
For , write the prime factorization of  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 ()

 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 ()

 (p, p − 3) irregular primes 

(See Wolstenholme prime)

 (p, p − 5) irregular primes 
Primes p such that (p, p−5) is an irregular pair.

37

 (p, p − 9) irregular primes 
Primes p such that (p, p − 9) is an irregular pair.

67, 877 ()

 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 ()

 Kynea primes 
Of the form (2 + 1) − 2.

2, 7, 23, 79, 1087, 66047, 263167, 16785407, 1073807359, 17180131327, 68720001023, 4398050705407, 70368760954879, 18014398777917439, 18446744082299486207 ()

 Leyland primes 
Of the form x + y, with 1 < x < y.

17, 593, 32993, 2097593, 8589935681, 59604644783353249, 523347633027360537213687137, 43143988327398957279342419750374600193 ()

 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 ()

 Lucas primes 
Primes in the Lucas number sequence L = 2, L = 1,
L = L + L.

2,It varies whether L = 2 is included in the Lucas numbers. 3, 7, 11, 29, 47, 199, 521, 2207, 3571, 9349, 3010349, 54018521, 370248451, 6643838879, 119218851371, 5600748293801, 688846502588399, 32361122672259149 ()

 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 ()

 Mersenne primes 
Of the form 2 − 1.

3, 7, 31, 127, 8191, 131071, 524287, 2147483647, 2305843009213693951, 618970019642690137449562111, 162259276829213363391578010288127, 170141183460469231731687303715884105727 ()

, there are 51 known Mersenne primes. The 13th, 14th, and 51st have respectively 157, 183, and 24,862,048 digits.

, this class of prime numbers also contains the largest known prime: M82589933, the 51st known Mersenne prime.

 Mersenne divisors 
Primes p that divide 2 − 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 ()

All Mersenne primes are, by definition, members of this sequence.

 Mersenne prime exponents 
Primes p such that 2 − 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 ()

 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 2 − 1 for prime p.

7, 127, 2147483647, 170141183460469231731687303715884105727 (primes in )

As of June 2017, these are the only known double Mersenne primes, and number theorists think these are probably the only double Mersenne primes.

 Generalized repunit primes 
Of the form (a − 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 ()

a = 4: 5 (the only prime for a = 4)

a = 5: 31, 19531, 12207031, 305175781, 177635683940025046467781066894531, 14693679385278593849609206715278070972733319459651094018859396328480215743184089660644531 ()

a = 6: 7, 43, 55987, 7369130657357778596659, 3546245297457217493590449191748546458005595187661976371 ()

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 , including the Mersenne primes and the cuban primes as special cases
 Williams primes, of the form 

 Mills primes 
Of the form ⌊θ⌋, where θ is Mills' constant. This form is prime for all positive integers n.

2, 11, 1361, 2521008887, 16022236204009818131831320183 ()

 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 ()

 Newman–Shanks–Williams primes 
Newman–Shanks–Williams numbers that are prime.

7, 41, 239, 9369319, 63018038201, 489133282872437279, 19175002942688032928599 ()

 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.

2, 40487, 6692367337 ()

 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 ()

 Palindromic wing primes 
Primes of the form  with . 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 ()

 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 ()

 Pell primes 
Primes in the Pell number sequence P = 0, P = 1,
P = 2P + P.

2, 5, 29, 5741, 33461, 44560482149, 1746860020068409, 68480406462161287469, 13558774610046711780701, 4125636888562548868221559797461449 ()

 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 ()

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 ()

 Pierpont primes 
Of the form 23 + 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 ()

 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 ()

 Primes of the form n4 + 1 
Of the form n4 + 1.

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 ()

 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 ()

 Primorial primes 
Of the form p# ± 1.

3, 5, 7, 29, 31, 211, 2309, 2311, 30029, 200560490131, 304250263527209, 23768741896345550770650537601358309 (union of  and )

 Proth primes 
Of the form k×2 + 1, with odd k and k < 2.

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 ()

 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 ()

 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) (, , , )

 Quartan primes 
Of the form x + y, where x,y > 0.

2, 17, 97, 257, 337, 641, 881 ()

 Ramanujan primes 
Integers R that are the smallest to give at least n primes from x/2 to x for all x ≥ R (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 ()

 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 ()

 Repunit primes 
Primes containing only the decimal digit 1.

11, 1111111111111111111 (19 digits), 11111111111111111111111  (23 digits) ()

The next have 317, 1031, 49081, 86453, 109297, 270343 digits ()

 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 ()
4n+1: 5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101, 109, 113, 137 ()
4n+3: 3, 7, 11, 19, 23, 31, 43, 47, 59, 67, 71, 79, 83, 103, 107 ()
6n+1: 7, 13, 19, 31, 37, 43, 61, 67, 73, 79, 97, 103, 109, 127, 139 ()
6n+5: 5, 11, 17, 23, 29, 41, 47, 53, 59, 71, 83, 89, 101, 107, 113 ()
8n+1: 17, 41, 73, 89, 97, 113, 137, 193, 233, 241, 257, 281, 313, 337, 353 ()
8n+3: 3, 11, 19, 43, 59, 67, 83, 107, 131, 139, 163, 179, 211, 227, 251 ()
8n+5: 5, 13, 29, 37, 53, 61, 101, 109, 149, 157, 173, 181, 197, 229, 269 ()
8n+7: 7, 23, 31, 47, 71, 79, 103, 127, 151, 167, 191, 199, 223, 239, 263 ()
10n+1: 11, 31, 41, 61, 71, 101, 131, 151, 181, 191, 211, 241, 251, 271, 281 ()
10n+3: 3, 13, 23, 43, 53, 73, 83, 103, 113, 163, 173, 193, 223, 233, 263 ()
10n+7: 7, 17, 37, 47, 67, 97, 107, 127, 137, 157, 167, 197, 227, 257, 277 ()
10n+9: 19, 29, 59, 79, 89, 109, 139, 149, 179, 199, 229, 239, 269, 349, 359 ()
12n+1: 13, 37, 61, 73, 97, 109, 157, 181, 193, 229, 241, 277, 313, 337, 349 ()
12n+5: 5, 17, 29, 41, 53, 89, 101, 113, 137, 149, 173, 197, 233, 257, 269 ()
12n+7: 7, 19, 31, 43, 67, 79, 103, 127, 139, 151, 163, 199, 211, 223, 271 ()
12n+11: 11, 23, 47, 59, 71, 83, 107, 131, 167, 179, 191, 227, 239, 251, 263 ()

 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 ()

 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 ()

 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) (, )

 Smarandache–Wellin primes 
Primes that are the concatenation of the first n primes written in decimal.

2, 23, 2357 ()

The fourth Smarandache-Wellin prime is the 355-digit concatenation of the first 128 primes that end with 719.

 Solinas primes 
Of the form 2 ± 2 ± 1, where 0 < b < a.

3, 5, 7, 11, 13 ()

 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 ()

 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 ()

, 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 

 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 ()

 Supersingular primes 
There are exactly fifteen supersingular primes:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59, 71 ()

 Thabit primes 
Of the form 3×2 − 1.

2, 5, 11, 23, 47, 191, 383, 6143, 786431, 51539607551, 824633720831, 26388279066623, 108086391056891903, 55340232221128654847, 226673591177742970257407 ()

The primes of the form 3×2 + 1 are related.

7, 13, 97, 193, 769, 12289, 786433, 3221225473, 206158430209, 6597069766657 ()

 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) (, , )

 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 ()

 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 ()

 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 ()

 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) (, )

 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 ()

 Wagstaff primes 
Of the form (2 + 1) / 3.

3, 11, 43, 683, 2731, 43691, 174763, 2796203, 715827883, 2932031007403, 768614336404564651, 201487636602438195784363, 845100400152152934331135470251, 56713727820156410577229101238628035243 ()

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 ()

 Wall–Sun–Sun primes 
A prime p > 5, if p divides the Fibonacci number , where the Legendre symbol  is defined as

, 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 ()

 Wieferich primes 
Primes p such that  for fixed integer a > 1.

2p − 1 ≡ 1 (mod p2): 1093, 3511 ()
3p − 1 ≡ 1 (mod p2): 11, 1006003 ()
4p − 1 ≡ 1 (mod p2): 1093, 3511
5p − 1 ≡ 1 (mod p2): 2, 20771, 40487, 53471161, 1645333507, 6692367337, 188748146801 ()
6p − 1 ≡ 1 (mod p2): 66161, 534851, 3152573 ()
7p − 1 ≡ 1 (mod p2): 5, 491531 ()
8p − 1 ≡ 1 (mod p2): 3, 1093, 3511
9p − 1 ≡ 1 (mod p2): 2, 11, 1006003
10p − 1 ≡ 1 (mod p2): 3, 487, 56598313 ()
11p − 1 ≡ 1 (mod p2): 71
12p − 1 ≡ 1 (mod p2): 2693, 123653 ()
13p − 1 ≡ 1 (mod p2): 2, 863, 1747591 ()
14p − 1 ≡ 1 (mod p2): 29, 353, 7596952219 ()
15p − 1 ≡ 1 (mod p2): 29131, 119327070011 ()
16p − 1 ≡ 1 (mod p2): 1093, 3511
17p − 1 ≡ 1 (mod p2): 2, 3, 46021, 48947 ()
18p − 1 ≡ 1 (mod p2): 5, 7, 37, 331, 33923, 1284043 ()
19p − 1 ≡ 1 (mod p2): 3, 7, 13, 43, 137, 63061489 ()
20p − 1 ≡ 1 (mod p2): 281, 46457, 9377747, 122959073 ()
21p − 1 ≡ 1 (mod p2): 2
22p − 1 ≡ 1 (mod p2): 13, 673, 1595813, 492366587, 9809862296159 ()
23p − 1 ≡ 1 (mod p2): 13, 2481757, 13703077, 15546404183, 2549536629329 ()
24p − 1 ≡ 1 (mod p2): 5, 25633
25p − 1 ≡ 1 (mod p2): 2, 20771, 40487, 53471161, 1645333507, 6692367337, 188748146801

, these are all known Wieferich primes with a ≤ 25.

 Wilson primes 
Primes p for which p divides (p−1)! + 1.

5, 13, 563 ()

, these are the only known Wilson primes.

 Wolstenholme primes 
Primes p for which the binomial coefficient 

16843, 2124679 ()

, these are the only known Wolstenholme primes.

 Woodall primes 
Of the form n×2 − 1.

7, 23, 383, 32212254719, 2833419889721787128217599, 195845982777569926302400511, 4776913109852041418248056622882488319 ()

 Referensi