Daftar bilangan prima 📖 Wikipedia

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.

Tabel berikut mencantumkan 1000 bilangan prima pertama, dengan 20 kolom bilangan prima berurutan di masing-masing dari 50 baris.^[1]

(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]

Di bawah ini terdaftar bilangan prima pertama dari banyak bentuk dan tipe bernama. Lebih jelasnya ada di artikel untuk namanya. ${\displaystyle n}$ adalah bilangan asli (termasuk 0) di definisikan

Bilangan prima yang merupakan bilangan partisi himpunan dengan ${\displaystyle n}$ anggota.

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

Suku selanjutnya memiliki 6539 digit. ( A051131)

Bentuk: ${\displaystyle p-n,\,p,\,p+n}$

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 A006562 dalam OEIS).

Dari bentuk ${\displaystyle (2^{n}-1)^{2}-2}$

7, 47, 223, 3967, 16127, 1046527, 16769023, 1073676287, 68718952447, 274876858367, 4398042316799, 1125899839733759, 18014398241046527, 1298074214633706835075030044377087 ( A091516)

Di mana ${\displaystyle p}$ adalah bilangan prima dan ${\displaystyle p+2}$ 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 ( A109611)

Dari bentuk ${\displaystyle {\frac {x^{3}-y^{3}}{x-y}}}$ di mana ${\displaystyle x=y+1}$.

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 ( A002407)

Dari bentuk ${\displaystyle {\frac {x^{3}-y^{3}}{x-y}}}$ di mana ${\displaystyle x=y+2}$.

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 ( A002648)

Dari bentuk ${\displaystyle n\cdot 2^{n}+1}$.

3, 393050634124102232869567034555427371542904833 ( A050920)

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 bulat Eisenstein yang merupakan bilangan taktereduksi dan bilangan real (bilangan prima dari bentuk ${\displaystyle 3n-1}$).

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 ( A003627)

Dari bentuk ${\displaystyle p_{n}\#+1}$ (sebuah himpunan bagian bilangan prima primorial).

3, 7, 31, 211, 2311, 200560490131 ( A018239^[5])

Dari bentuk n! - 1 atau n! + 1.

2, 3, 5, 7, 23, 719, 5039, 39916801, 479001599, 87178291199, 10888869450418352160768000001, 265252859812191058636308479999999, 263130836933693530167218012159999999, 8683317618811886495518194401279999999 ( A088054)

Dari bentuk ${\displaystyle 2^{2^{n}}+1}$.

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]

Dari bentuk ${\displaystyle a^{2^{n}}+1}$ untuk bilangan bulat tetap ${\displaystyle a}$.

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 ${\displaystyle a\leq 24}$.

Bilangan prima dalam barisan Fibonacci ${\displaystyle F_{0}=0}$, ${\displaystyle F_{1}=1}$, ${\displaystyle F_{n}=F_{n-1}+F_{n-2}}$.

2, 3, 5, 13, 89, 233, 1597, 28657, 514229, 433494437, 2971215073, 99194853094755497, 1066340417491710595814572169, 19134702400093278081449423917 ( A005478)

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)

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.

Di mana ${\displaystyle (p,p+4)}$ 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) ( A023200,  A046132)

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

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 ( A120337)

Bilangan prima ${\displaystyle p}$ sehingga ${\displaystyle (p,p-3)}$ adalah sebuah pasangan takberaturan Euler.

149, 241, 2946901 ( A198245)

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)

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)

Primes p_n for which p_n² > p_n−i p_n+i for all 1 ≤ i ≤ n−1, where p_n 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 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)

Primes p for which there are no solutions to H_k ≡ 0 (mod p) and H_k ≡ −ω_p (mod p) for 1 ≤ k ≤ p−2, where H_k denotes the k-th harmonic number and ω_p 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)

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)

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)

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)

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)

(See Wolstenholme prime)

Primes p such that (p, p−5) is an irregular pair.^[8]

37

Primes p such that (p, p − 9) is an irregular pair.^[8]

67, 877 ( A212557)

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)

Of the form (2ⁿ + 1)² − 2.

2, 7, 23, 79, 1087, 66047, 263167, 16785407, 1073807359, 17180131327, 68720001023, 4398050705407, 70368760954879, 18014398777917439, 18446744082299486207 ( A091514)

Of the form x^y + y^x, with 1 < x < y.

17, 593, 32993, 2097593, 8589935681, 59604644783353249, 523347633027360537213687137, 43143988327398957279342419750374600193 ( A094133)

Primes p for which, in a given base b, ${\displaystyle {\frac {b^{p-1}-1}{p}}}$ 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 ( A001913)

Primes in the Lucas number sequence L₀ = 2, L₁ = 1, L_n = L_n−1 + L_n−2.

2,^[9] 3, 7, 11, 29, 47, 199, 521, 2207, 3571, 9349, 3010349, 54018521, 370248451, 6643838879, 119218851371, 5600748293801, 688846502588399, 32361122672259149 ( A005479)

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)

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: M_82589933, the 51st known Mersenne prime.

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.

Primes p such that 2^p − 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

A subset of Mersenne primes of the form 2^2p−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]}

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

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

Of the form ⌊θ³ⁿ⌋, where θ is Mills' constant. This form is prime for all positive integers n.

2, 11, 1361, 2521008887, 16022236204009818131831320183 ( A051254)

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 numbers that are prime.

7, 41, 239, 9369319, 63018038201, 489133282872437279, 19175002942688032928599 ( A088165)

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)

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)

Primes of the form ${\displaystyle {\frac {a{\big (}10^{m}-1{\big )}}{9}}\pm b\times 10^{\frac {m-1}{2}}}$ with ${\displaystyle 0\leq a\pm b<10}$.^[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 function values that are prime.

2, 3, 5, 7, 11, 101, 17977, 10619863, 6620830889, 80630964769, 228204732751, 1171432692373, 1398341745571, 10963707205259, 15285151248481, 10657331232548839, 790738119649411319, 18987964267331664557 ( A049575)

Primes in the Pell number sequence P₀ = 0, P₁ = 1, P_n = 2P_n−1 + P_n−2.

2, 5, 29, 5741, 33461, 44560482149, 1746860020068409, 68480406462161287469, 13558774610046711780701, 4125636888562548868221559797461449 ( A086383)

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.

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)

Of the form 2^u3^v + 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)

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)

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)

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)

Of the form p_n# ± 1.

3, 5, 7, 29, 31, 211, 2309, 2311, 30029, 200560490131, 304250263527209, 23768741896345550770650537601358309 (union of  A057705 and  A018239^[5])

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)

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)

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)

Of the form x⁴ + y⁴, where x,y > 0.

2, 17, 97, 257, 337, 641, 881 ( A002645)

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

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)

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)

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)

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)

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)

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)

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.

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

3, 5, 7, 11, 13 ( A165255)

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)

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.

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)

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)

There are exactly fifteen supersingular primes:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59, 71 ( A002267)

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)

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)

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)

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)

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)

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)

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)

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)

A prime p > 5, if p² divides the Fibonacci number ${\displaystyle F_{p-\left({\frac {p}{5}}\right)}}$, where the Legendre symbol ${\displaystyle \left({\frac {p}{5}}\right)}$ is defined as

${\displaystyle \left({\frac {p}{5}}\right)={\begin{cases}1&{\textrm {if}}\;p\equiv \pm 1{\pmod {5}}\\-1&{\textrm {if}}\;p\equiv \pm 2{\pmod {5}}.\end{cases}}}$

Hingga 2018^[update], no Wall-Sun-Sun primes are known.

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)

Primes p such that a^{p − 1} ≡ 1 (mod p²) for fixed integer a > 1.

2^{p − 1} ≡ 1 (mod p²): 1093, 3511 ( A001220)
3^{p − 1} ≡ 1 (mod p²): 11, 1006003 ( A014127)^[17][18][19]
4^{p − 1} ≡ 1 (mod p²): 1093, 3511
5^{p − 1} ≡ 1 (mod p²): 2, 20771, 40487, 53471161, 1645333507, 6692367337, 188748146801 ( A123692)
6^{p − 1} ≡ 1 (mod p²): 66161, 534851, 3152573 ( A212583)
7^{p − 1} ≡ 1 (mod p²): 5, 491531 ( A123693)
8^{p − 1} ≡ 1 (mod p²): 3, 1093, 3511
9^{p − 1} ≡ 1 (mod p²): 2, 11, 1006003
10^{p − 1} ≡ 1 (mod p²): 3, 487, 56598313 ( A045616)
11^{p − 1} ≡ 1 (mod p²): 71^[20]
12^{p − 1} ≡ 1 (mod p²): 2693, 123653 ( A111027)
13^{p − 1} ≡ 1 (mod p²): 2, 863, 1747591 ( A128667)^[20]
14^{p − 1} ≡ 1 (mod p²): 29, 353, 7596952219 ( A234810)
15^{p − 1} ≡ 1 (mod p²): 29131, 119327070011 ( A242741)
16^{p − 1} ≡ 1 (mod p²): 1093, 3511
17^{p − 1} ≡ 1 (mod p²): 2, 3, 46021, 48947 ( A128668)^[20]
18^{p − 1} ≡ 1 (mod p²): 5, 7, 37, 331, 33923, 1284043 ( A244260)
19^{p − 1} ≡ 1 (mod p²): 3, 7, 13, 43, 137, 63061489 ( A090968)^[20]
20^{p − 1} ≡ 1 (mod p²): 281, 46457, 9377747, 122959073 ( A242982)
21^{p − 1} ≡ 1 (mod p²): 2
22^{p − 1} ≡ 1 (mod p²): 13, 673, 1595813, 492366587, 9809862296159 ( A298951)
23^{p − 1} ≡ 1 (mod p²): 13, 2481757, 13703077, 15546404183, 2549536629329 ( A128669)
24^{p − 1} ≡ 1 (mod p²): 5, 25633
25^{p − 1} ≡ 1 (mod p²): 2, 20771, 40487, 53471161, 1645333507, 6692367337, 188748146801

Hingga 2018^[update], these are all known Wieferich primes with a ≤ 25.

Primes p for which p² divides (p−1)! + 1.

5, 13, 563 ( A007540)

Hingga 2018^[update], these are the only known Wilson primes.

Primes p for which the binomial coefficient ${\displaystyle {{2p-1} \choose {p-1}}\equiv 1{\pmod {p^{4}}}.}$

16843, 2124679 ( A088164)

Hingga 2018^[update], these are the only known Wolstenholme primes.

Of the form n×2ⁿ − 1.

7, 23, 383, 32212254719, 2833419889721787128217599, 195845982777569926302400511, 4776913109852041418248056622882488319 ( A050918)