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.

.

Proyek verifikasi konjektur Goldbach melaporkan bahwa mereka telah menghitung semua bilangan prima di bawah ini 4×10. 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. n adalah bilangan asli (termasuk 0) di definisikan

### [[Bilangan prima Bell]]
Bilangan prima yang merupakan bilangan partisi himpunan dengan n anggota.

2, 5, 877, 27644437, 35742549198872617291353508656626642567, 359334085968622831041960188598043661065388726959079837.

Suku selanjutnya memiliki 6539 digit. ()

### [[Bilangan prima berimbang]]
Bentuk: p - n, \, p , \, p + n

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  dalam OEIS).

### [[Bilangan prima Carol]]
Dari bentuk (2^n - 1)^2 - 2

7, 47, 223, 3967, 16127, 1046527, 16769023, 1073676287, 68718952447, 274876858367, 4398042316799, 1125899839733759, 18014398241046527, 1298074214633706835075030044377087 ()

### [[Bilangan prima Chen]]
Di mana p adalah bilangan prima dan p+2 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 \frac{x^3-y^3}{x-y} di mana  x = y + 1 .

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 \frac{x^3-y^3}{x-y} di mana  x = y + 2 .

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  n \cdot 2^n + 1 .

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  3n - 1 ).

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 Euclid|Bilangan prima Euclid]]
Dari bentuk  p_n \# + 1  (sebuah himpunan bagian bilangan prima primorial).

3, 7, 31, 211, 2311, 200560490131 ()

### [[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 2^{2^n} + 1.

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 a^{2^n} + 1 untuk bilangan bulat tetap a.

 (tidak ada)

, ini haya diketahui bilangan prima Fermat rampat untuk a \le 24.

### [[Bilangan prima Fibonacci]]
Bilangan prima dalam barisan Fibonacci F_0 = 0*,* F_1 = 1*,* F_n = F_{n - 1} + F_{n - 2}.

2, 3, 5, 13, 89, 233, 1597, 28657, 514229, 433494437, 2971215073, 99194853094755497, 1066340417491710595814572169, 19134702400093278081449423917 ()

### [[Bilangan prima Fortunate|Bilangan prima ''fortunate'']]
Bilangan f''ortunate'' 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 (p,p+4) 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 beraturan#Bilangan prima takberaturan Euler|Bilangan prima takberaturan Euler]]

Sebuah bilangan prima p yang membagi bilangan Euler E_{2n} untuk suatu 0\leq 2n\leq p-3.

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 beraturan#Bilangan prima takberaturan|Bilangan prima takberaturan]] <math>(p,\,p-3)</math> [[Bilangan prima beraturan#Bilangan prima takberaturan|Euler]]
Bilangan prima p sehingga (p, p-3) 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 prime]]s
Prime elements of the Gaussian integers; equivalently, primes of the form 4*n*&nbsp;+&nbsp;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 prime]]s
Primes *p* for which *p*&nbsp;>&nbsp;*p*&nbsp;*p* for all 1&nbsp;≤&nbsp;*i*&nbsp;≤&nbsp;*n*−1, where *p* is the *n*th prime.

5, 11, 17, 29, 37, 41, 53, 59, 67, 71, 97, 101, 127, 149, 179, 191, 223, 227, 251, 257, 269, 307 ()

### [[Happy prime]]s
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*&nbsp;≡&nbsp;0&nbsp;(mod&nbsp;*p*) and *H*&nbsp;≡&nbsp;−*ω*&nbsp;(mod&nbsp;*p*) for 1&nbsp;≤&nbsp;*k*&nbsp;≤&nbsp;*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 prime]]s 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 number|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 prime]]s
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 prime]]s
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 ()

#### [[Regular prime#Irregular pairs|(''p'', ''p'' − 3) irregular primes]]

(See Wolstenholme prime)

#### [[Regular prime#Irregular pairs|(''p'', ''p'' − 5) irregular primes]]
Primes *p* such that (*p*, *p*−5) is an irregular pair.

37

#### [[Regular prime#Irregular pairs|(''p'', ''p'' − 9) irregular primes]]
Primes *p* such that (*p*, *p* − 9) is an irregular pair.

67, 877 ()

### [[Isolated prime]]s
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 number|Kynea primes]]
Of the form (2&nbsp;+&nbsp;1)&nbsp;−&nbsp;2.

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

### [[Leyland number|Leyland primes]]
Of the form *x*&nbsp;+&nbsp;*y*, with 1&nbsp;\frac{b^{p-1}-1}{p} 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 prime]]s
Primes in the Lucas number sequence *L*&nbsp;=&nbsp;2, *L*&nbsp;=&nbsp;1,
*L*&nbsp;=&nbsp;*L*&nbsp;+&nbsp;*L*.

2, 3, 7, 11, 29, 47, 199, 521, 2207, 3571, 9349, 3010349, 54018521, 370248451, 6643838879, 119218851371, 5600748293801, 688846502588399, 32361122672259149 ()

### [[Lucky number|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 prime]]s
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 prime]]s
A subset of Mersenne primes of the form 2&nbsp;−&nbsp;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#Repunit primes|repunit primes]]
Of the form (*a*&nbsp;−&nbsp;1)&nbsp;/&nbsp;(*a*&nbsp;−&nbsp;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' constant|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 prime (number theory)|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 prime]]s
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 *p*2.  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 \frac{a \big( 10^m-1 \big)}{9} \pm b \times 10^{\frac{ m-1 }{2}} with 0 \le a \pm b . 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 (number theory)#Partition function|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 number|Pell primes]]
Primes in the Pell number sequence *P*&nbsp;=&nbsp;0, *P*&nbsp;=&nbsp;1,
*P*&nbsp;=&nbsp;2*P*&nbsp;+&nbsp;*P*.

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

### [[Permutable prime]]s
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 number|Perrin primes]]
Primes in the Perrin number sequence *P*(0)&nbsp;=&nbsp;3, *P*(1)&nbsp;=&nbsp;0, *P*(2)&nbsp;=&nbsp;2,
*P*(*n*)&nbsp;=&nbsp;*P*(*n*−2)&nbsp;+&nbsp;*P*(*n*−3).

2, 3, 5, 7, 17, 29, 277, 367, 853, 14197, 43721, 1442968193, 792606555396977, 187278659180417234321, 66241160488780141071579864797 ()

### [[Pierpont prime]]s
Of the form 23&nbsp;+&nbsp;1 for some integers *u*,*v*&nbsp;≥&nbsp;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 prime]]s
Primes *p* for which there exist *n*&nbsp;>&nbsp;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 ''n''<sup>4</sup> + 1
Of the form *n*4 + 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 prime]]s
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 prime]]s
Of the form *p*#&nbsp;± 1.

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

### [[Proth number|Proth primes]]
Of the form *k*×2&nbsp;+&nbsp;1, with odd *k* and *k*&nbsp;&nbsp;0.

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

### [[Ramanujan prime]]s
Integers *R* that are the smallest to give at least *n* primes from *x*/2 to *x* for all *x*&nbsp;≥&nbsp;*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 prime]]s
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#Decimal repunit primes|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 ()

### [[Dirichlet's theorem on arithmetic progressions|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 2*n*+1 are the odd primes, including all primes other than 2.  Some sequences have alternate names: 4*n*+1 are Pythagorean primes, 4*n*+3 are the integer Gaussian primes, and 6*n*+5 are the Eisenstein primes (with 2 omitted).  The classes 10*n*+*d* (*d* = 1, 3, 7, 9) are primes ending in the decimal digit *d*.

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

### [[Safe prime]]s
Where *p* and (*p*−1)&nbsp;/&nbsp;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 number|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 prime]]s
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 number|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 prime]]s
Of the form 2&nbsp;±&nbsp;2&nbsp;±&nbsp;1, where 0&nbsp;&nbsp;5, if *p* divides the Fibonacci number F_{p - \left(\frac\right)}, where the Legendre symbol \left(\frac\right) is defined as
:\left(\frac{p}{5}\right) = \begin{cases} 1 &\textrm{if}\;p \equiv \pm1 \pmod 5\\ -1 &\textrm{if}\;p \equiv \pm2 \pmod 5. \end{cases}

, no Wall-Sun-Sun primes are known.

### [[Weakly prime number]]s
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 prime]]s
Primes *p* such that  for fixed integer *a* > 1.

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

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

### [[Wilson prime]]s
Primes *p* for which *p* divides (*p*−1)! + 1.

5, 13, 563 ()

, these are the only known Wilson primes.

### [[Wolstenholme prime]]s
Primes *p* for which the binomial coefficient  \equiv 1 \pmod{p^4}.

16843, 2124679 ()

, these are the only known Wolstenholme primes.

### [[Woodall number|Woodall primes]]
Of the form *n*×2 − 1.

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

## Referensi