List of prime numbers

Primes with equal-sized prime gaps above and below them, so that they are equal to the arithmetic mean of the nearest primes above and below.

• 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 (OEIS: A006562).

Primes that are the number of partitions of a set with n members.

2, 5, 877, 27644437, 35742549198872617291353508656626642567, 359334085968622831041960188598043661065388726959079837. The next term has 6,539 digits. (OEIS: A051131)

Where p is prime and p+2 is either a prime or semiprime.

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 (OEIS: A109611)

A circular prime number is a number that remains prime on any cyclic rotation of its digits (in base 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 (OEIS: A068652)

Some sources only list the smallest prime in each cycle, for example, listing 13, but omitting 31 (OEIS really calls this sequence circular primes, but not the above sequence):

2, 3, 5, 7, 11, 13, 17, 37, 79, 113, 197, 199, 337, 1193, 3779, 11939, 19937, 193939, 199933, 1111111111111111111, 11111111111111111111111 (OEIS: A016114)

All repunit primes are circular.

A cluster prime is a prime p such that every even natural number k ≤ p − 3 is the difference of two primes not exceeding p.

3, 5, 7, 11, 13, 17, 19, 23, ... (OEIS: A038134)

All odd primes between 3 and 89, inclusive, are cluster primes. The first 10 primes that are not cluster primes are:

2, 97, 127, 149, 191, 211, 223, 227, 229, 251.

Where (p, p + 4) are both prime.

(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) (OEIS: A023200, OEIS: A046132)

Of the form ${\displaystyle {\tfrac {x^{3}-y^{3}}{x-y}}}$ where 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 (OEIS: A002407)

Of the form ${\displaystyle {\tfrac {x^{3}-y^{3}}{x-y}}}$ where 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 (OEIS: A002648)

Of the form n×2ⁿ + 1.

3, 393050634124102232869567034555427371542904833 (OEIS: A050920)

Primes that remain prime when read upside down or mirrored in a seven-segment display.

2, 5, 11, 101, 181, 1181, 1811, 18181, 108881, 110881, 118081, 120121, 121021, 121151, 150151, 151051, 151121, 180181, 180811, 181081 (OEIS: A134996)

Eisenstein integers that are irreducible and real numbers (primes of the form 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 (OEIS: A003627)

Primes that become a different prime when their decimal digits are reversed. The name "emirp" is obtained by reversing the word "prime".

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 (OEIS: A006567)

Of the form p_n# + 1 (a subset of primorial primes).

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

A prime ${\displaystyle p}$ that divides Euler number ${\displaystyle E_{2n}}$ for some ${\displaystyle 0\leq 2n\leq p-3}$.

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 (OEIS: A120337)

Primes ${\displaystyle p}$ such that ${\displaystyle (p,p-3)}$ is an Euler irregular pair.

149, 241, 2946901 (OEIS: A198245)

Of the form n! − 1 or n! + 1.

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

Of the form 2²ⁿ + 1.

3, 5, 17, 257, 65537 (OEIS: A019434)

As of August 2019 these are the only known Fermat primes, and conjecturally the only Fermat primes. The probability of the existence of another Fermat prime is less than one in a billion.[6]

Of the form a²ⁿ + 1 for fixed integer a.

a = 2: 3, 5, 17, 257, 65537 (OEIS: A019434)

a = 4: 5, 17, 257, 65537

a = 6: 7, 37, 1297

a = 8: (does not exist)

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

As of April 2017 these are the only known generalized Fermat primes for a ≤ 24.

Primes in the Fibonacci sequence F₀ = 0, F₁ = 1, F_n = F_n−1 + F_n−2.

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

Fortunate numbers that are prime (it has been conjectured they all are).

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 (OEIS: A046066)

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 (OEIS: 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 (OEIS: 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 (OEIS: 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 (OEIS: 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 (OEIS: 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 (OEIS: 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 (OEIS: 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 (OEIS: 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 (OEIS: 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 (OEIS: A007510)

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

17, 593, 32993, 2097593, 8589935681, 59604644783353249, 523347633027360537213687137, 43143988327398957279342419750374600193 (OEIS: 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 (OEIS: 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 (OEIS: 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 (OEIS: A031157)

Of the form 2ⁿ − 1.

3, 7, 31, 127, 8191, 131071, 524287, 2147483647, 2305843009213693951, 618970019642690137449562111, 162259276829213363391578010288127, 170141183460469231731687303715884105727 (OEIS: A000668)

As of 2018, there are 51 known Mersenne primes. The 13th, 14th, and 51st have respectively 157, 183, and 24,862,048 digits.

As of 2018, 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 (OEIS: 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, 57885161 (OEIS: A000043)

As of December 2018, three more are known to be in the sequence, but it is not known whether they are the next:
74207281, 77232917, 82589933

A subset of Mersenne primes of the form 2^2p−1 − 1 for prime p.

7, 127, 2147483647, 170141183460469231731687303715884105727 (primes in OEIS: A077586)

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 (OEIS: A076481)

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

a = 5: 31, 19531, 12207031, 305175781, 177635683940025046467781066894531, 14693679385278593849609206715278070972733319459651094018859396328480215743184089660644531 (OEIS: A086122)

a = 6: 7, 43, 55987, 7369130657357778596659, 3546245297457217493590449191748546458005595187661976371 (OEIS: 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 (OEIS: 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 (OEIS: A071062)

Newman–Shanks–Williams numbers that are prime.

7, 41, 239, 9369319, 63018038201, 489133282872437279, 19175002942688032928599 (OEIS: 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 (OEIS: 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 (OEIS: 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 (OEIS: 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 (OEIS: 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 (OEIS: 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 (OEIS: A003459)

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 (OEIS: 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 (OEIS: 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 (OEIS: 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 (OEIS: 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 (OEIS: A119535)

Of the form p_n# ± 1.

3, 5, 7, 29, 31, 211, 2309, 2311, 30029, 200560490131, 304250263527209, 23768741896345550770650537601358309 (union of OEIS: A057705 and OEIS: 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 (OEIS: 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 (OEIS: 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) (OEIS: A007530, OEIS: A136720, OEIS: A136721, OEIS: A090258)

Of the form x⁴ + y⁴, where x,y > 0.

2, 17, 97, 257, 337, 641, 881 (OEIS: 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 (OEIS: 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 (OEIS: A007703)

Primes containing only the decimal digit 1.

11, 1111111111111111111 (19 digits), 11111111111111111111111 (23 digits) (OEIS: A004022)

The next have 317, 1031, 49081, 86453, 109297, 270343 digits (OEIS: 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 (OEIS: A065091)
4n+1: 5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101, 109, 113, 137 (OEIS: A002144)
4n+3: 3, 7, 11, 19, 23, 31, 43, 47, 59, 67, 71, 79, 83, 103, 107 (OEIS: A002145)
6n+1: 7, 13, 19, 31, 37, 43, 61, 67, 73, 79, 97, 103, 109, 127, 139 (OEIS: A002476)
6n+5: 5, 11, 17, 23, 29, 41, 47, 53, 59, 71, 83, 89, 101, 107, 113 (OEIS: A007528)
8n+1: 17, 41, 73, 89, 97, 113, 137, 193, 233, 241, 257, 281, 313, 337, 353 (OEIS: A007519)
8n+3: 3, 11, 19, 43, 59, 67, 83, 107, 131, 139, 163, 179, 211, 227, 251 (OEIS: A007520)
8n+5: 5, 13, 29, 37, 53, 61, 101, 109, 149, 157, 173, 181, 197, 229, 269 (OEIS: A007521)
8n+7: 7, 23, 31, 47, 71, 79, 103, 127, 151, 167, 191, 199, 223, 239, 263 (OEIS: A007522)
10n+1: 11, 31, 41, 61, 71, 101, 131, 151, 181, 191, 211, 241, 251, 271, 281 (OEIS: A030430)
10n+3: 3, 13, 23, 43, 53, 73, 83, 103, 113, 163, 173, 193, 223, 233, 263 (OEIS: A030431)
10n+7: 7, 17, 37, 47, 67, 97, 107, 127, 137, 157, 167, 197, 227, 257, 277 (OEIS: A030432)
10n+9: 19, 29, 59, 79, 89, 109, 139, 149, 179, 199, 229, 239, 269, 349, 359 (OEIS: A030433)
12n+1: 13, 37, 61, 73, 97, 109, 157, 181, 193, 229, 241, 277, 313, 337, 349 (OEIS: A068228)
12n+5: 5, 17, 29, 41, 53, 89, 101, 113, 137, 149, 173, 197, 233, 257, 269 (OEIS: A040117)
12n+7: 7, 19, 31, 43, 67, 79, 103, 127, 139, 151, 163, 199, 211, 223, 271 (OEIS: A068229)
12n+11: 11, 23, 47, 59, 71, 83, 107, 131, 167, 179, 191, 227, 239, 251, 263 (OEIS: 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 (OEIS: 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 (OEIS: 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) (OEIS: A023201, OEIS: A046117)

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

2, 23, 2357 (OEIS: 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 (OEIS: A165255)

Where p and 2p + 1 are both prime. A Sophie Germain prime has a corresponding safe 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 (OEIS: 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 (OEIS: A042978)

As of 2011, these are the only known Stern primes, and possibly the only existing.

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 (OEIS: A006450)

There are exactly fifteen supersingular primes:

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

Of the form 3×2ⁿ − 1.

2, 5, 11, 23, 47, 191, 383, 6143, 786431, 51539607551, 824633720831, 26388279066623, 108086391056891903, 55340232221128654847, 226673591177742970257407 (OEIS: A007505)

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

7, 13, 97, 193, 769, 12289, 786433, 3221225473, 206158430209, 6597069766657 (OEIS: 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) (OEIS: A007529, OEIS: A098414, OEIS: 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 (OEIS: 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 (OEIS: 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 (OEIS: 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) (OEIS: A001359, OEIS: 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 (OEIS: A040017)

Of the form (2ⁿ + 1) / 3.

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

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

As of 2018, 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 (OEIS: A050249)

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

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

As of 2018, these are all known Wieferich primes with a ≤ 25.

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

5, 13, 563 (OEIS: A007540)

As of 2018, 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 (OEIS: A088164)

As of 2018, these are the only known Wolstenholme primes.

Of the form n×2ⁿ − 1.

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