List of prime numbers (original) (raw)

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This is a list of articles about prime numbers. A prime number (or prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself. By Euclid's theorem, there are an infinite number of prime numbers. Subsets of the prime numbers may be generated with various formulas for primes. Below are lists of notable types of prime numbers in alphabetical order, giving their respective first terms.

Note that 1 is neither prime nor composite. A list of the first 1000 prime numbers is also available.

Lists of primes by type

Below are listed the first prime numbers of many named forms and types. More details are in the article for the name. n is a natural number (including 0) in the definitions.

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

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 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.

See also: § Twin primes, § Prime triplets, and § Prime quadruplets

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 x 3 − y 3 x − y {\displaystyle {\tfrac {x^{3}-y^{3}}{x-y}}} ${\tfrac {x^{3}-y^{3}}{x-y}}$ 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 x 3 − y 3 x − y {\displaystyle {\tfrac {x^{3}-y^{3}}{x-y}}} ${\tfrac {x^{3}-y^{3}}{x-y}}$ 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_×2_n + 1.

3, 393050634124102232869567034555427371542904833 (OEIS: A050920)

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 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 primes without imaginary part

Eisenstein integers that are irreducible and real numbers (primes of the form 3_n_ − 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 the reverse of 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 [1])

Euler irregular primes

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

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 p {\displaystyle p} $p$ such that ( p , p − 3 ) {\displaystyle (p,p-3)} $(p,p-3)$ 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 22_n_ + 1.

3, 5, 17, 257, 65537 (OEIS: A019434)

As of June 2024[update] 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.[2]

Generalized Fermat primes

Of the form a_2_n + 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: (none 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

Primes in the Fibonacci sequence _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 (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 4_n_ + 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 _n_2 > p n_−_i p n+i for all 1 ≤ i ≤ _n_−1, where p n 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 (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.[3]

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)

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 (OEIS: A007459)

Highly cototient primes

Primes that are a cototient more often than any integer below it except 1.

2, 23, 47, 59, 83, 89, 113, 167, 269, 389, 419, 509, 659, 839, 1049, 1259, 1889 (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)

(p, p − 3) irregular primes

(p, p − 5) irregular primes

Primes p such that (p, _p_−5) is an irregular pair.[4]

(p, p − 9) irregular primes

Primes p such that (p, p − 9) is an irregular pair.[4]

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, b p − 1 − 1 p {\displaystyle {\frac {b^{p-1}-1}{p}}} ${\frac {b^{p-1}-1}{p}}$ 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 _L_0 = 2, _L_1 = 1,L n = L _n_−1 + L _n_−2.

2,[5] 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 2_n_ − 1.

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

As of 2024[update], there are 52 known Mersenne primes. The 13th, 14th, and 52nd have respectively 157, 183, and 41,024,320 digits. This includes the largest known prime 2136,279,841−1, which is the 52nd Mersenne prime.

Primes p that divide 2_n_ − 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.

Mersenne prime exponents

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, 74207281, 77232917 (OEIS: A000043)

As of September 2025[update], two more are known to be in the sequence, but it is not known whether they are the next:
82589933, 136279841

Double Mersenne primes

A subset of Mersenne primes of the form 22_p_−1 − 1 for prime p.

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

Of the form (a n − 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

Other generalizations and variations

Many generalizations of Mersenne primes have been defined. This include the following:

Primes of the form bn − (b − 1)n,[6][7][8] including the Mersenne primes and the cuban primes as special cases
Williams primes, of the form (b − 1)·bn − 1

Of the form ⌊θ3_n_⌋, 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 primes

Newman–Shanks–Williams numbers that are prime.

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

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.[9]

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)

Palindromic wing primes

Primes of the form a ( 10 m − 1 ) 9 ± b × 10 m − 1 2 {\displaystyle {\frac {a{\big (}10^{m}-1{\big )}}{9}}\pm b\times 10^{\frac {m-1}{2}}} ${\frac {a{\big (}10^{m}-1{\big )}}{9}}\pm b\times 10^{\frac {m-1}{2}}$ with 0 ≤ a ± b < 10 {\displaystyle 0\leq a\pm b<10} $0\leq a\pm b<10$ .[10] 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 P_0 = 0, P 1 = 1,P n = 2_P _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_u_3_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)

Primes of the form _n_4 + 1

Of the form _n_4 + 1.[11][12]

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 [1])

Of the form k_×2_n + 1, with odd k and k < 2_n_.

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 4_n_ + 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)

See also: § Cousin primes, § Twin primes, and § Prime triplets

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 _x_4 + _y_4, 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, and 270343 digits, respectively (OEIS: A004023).

Residue classes of primes

Of the form an + d for fixed integers a and d. Also called primes congruent to d modulo a.

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

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 (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)

Smarandache–Wellin primes

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_k_ − c _1_·2_k_−1 − c _2_·2_k_−2 − ... − c k.

3, 5, 7, 11, 13 (OEIS: A165255)
232 − 5, the largest prime that fits into 32 bits of memory.[13]
264 − 59, the largest prime that fits into 64 bits of memory.

Sophie Germain primes

Where p and 2_p_ + 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[update], these are the only known Stern primes, and possibly the only existing.

Primes with prime-numbered indexes 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)

Supersingular primes

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 3×2_n_ − 1.

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

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

7, 13, 97, 193, 769, 12289, 786433, 3221225473, 206158430209, 6597069766657 (OEIS: A039687)

See also: § Cousin primes, § Twin primes, and § Prime quadruplets

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)

See also: § Cousin primes, § Prime triplets, and § Prime quadruplets

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 (2_n_ + 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)

Wall–Sun–Sun primes

A prime p > 5, if _p_2 divides the Fibonacci number F p − ( p 5 ) {\displaystyle F_{p-\left({\frac {p}{5}}\right)}} $F_{p-\left({\frac {p}{5}}\right)}$ , where the Legendre symbol ( p 5 ) {\displaystyle \left({\frac {p}{5}}\right)} $\left({\frac {p}{5}}\right)$ is defined as

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

As of 2022[update], no Wall-Sun-Sun primes have been found below 2 64 {\displaystyle 2^{64}} $2^{64}$ (about 18 ⋅ 10 18 {\displaystyle 18\cdot 10^{18}} $18\cdot 10^{18}$ ).[14]

	This section appears to contradict the article Wieferich prime on The definition of a Weiferich prime. Please see the talk page for more information. (October 2025)

Primes p such that a p − 1 ≡ 1 (mod _p_2) for fixed integer a > 1.

2_p_ − 1 ≡ 1 (mod p_2): 1093, 3511 (OEIS: A001220)
3_p − 1 ≡ 1 (mod p_2): 11, 1006003 (OEIS: A014127)[15][16][17]
4_p − 1 ≡ 1 (mod p_2): 1093, 3511
5_p − 1 ≡ 1 (mod p_2): 2, 20771, 40487, 53471161, 1645333507, 6692367337, 188748146801 (OEIS: A123692)
6_p − 1 ≡ 1 (mod p_2): 66161, 534851, 3152573 (OEIS: A212583)
7_p − 1 ≡ 1 (mod p_2): 5, 491531 (OEIS: A123693)
8_p − 1 ≡ 1 (mod p_2): 3, 1093, 3511
9_p − 1 ≡ 1 (mod p_2): 2, 11, 1006003
10_p − 1 ≡ 1 (mod p_2): 3, 487, 56598313 (OEIS: A045616)
11_p − 1 ≡ 1 (mod p_2): 71 [18]
12_p − 1 ≡ 1 (mod p_2): 2693, 123653 (OEIS: A111027)
13_p − 1 ≡ 1 (mod p_2): 2, 863, 1747591 (OEIS: A128667)[18]
14_p − 1 ≡ 1 (mod p_2): 29, 353, 7596952219 (OEIS: A234810)
15_p − 1 ≡ 1 (mod p_2): 29131, 119327070011 (OEIS: A242741)
16_p − 1 ≡ 1 (mod p_2): 1093, 3511
17_p − 1 ≡ 1 (mod p_2): 2, 3, 46021, 48947 (OEIS: A128668)[18]
18_p − 1 ≡ 1 (mod p_2): 5, 7, 37, 331, 33923, 1284043 (OEIS: A244260)
19_p − 1 ≡ 1 (mod p_2): 3, 7, 13, 43, 137, 63061489 (OEIS: A090968)[18]
20_p − 1 ≡ 1 (mod p_2): 281, 46457, 9377747, 122959073 (OEIS: A242982)
21_p − 1 ≡ 1 (mod p_2): 2
22_p − 1 ≡ 1 (mod p_2): 13, 673, 1595813, 492366587, 9809862296159 (OEIS: A298951)
23_p − 1 ≡ 1 (mod p_2): 13, 2481757, 13703077, 15546404183, 2549536629329 (OEIS: A128669)
24_p − 1 ≡ 1 (mod p_2): 5, 25633
25_p − 1 ≡ 1 (mod _p_2): 2, 20771, 40487, 53471161, 1645333507, 6692367337, 188748146801

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

Primes p for which _p_2 divides (_p_−1)! + 1.

5, 13, 563 (OEIS: A007540)

As of 2018[update], these are the only known Wilson primes.

Wolstenholme primes

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

16843, 2124679 (OEIS: A088164)

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

Of the form n_×2_n − 1.

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

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^ a b OEIS: A018239 includes 2 = empty product of first 0 primes plus 1, but 2 is excluded in this list.
^ Boklan, Kent D.; Conway, John H. (2016). "Expect at most one billionth of a new Fermat Prime!". arXiv:1605.01371 [math.NT].
^ Boyd, D. W. (1994). "A _p_-adic Study of the Partial Sums of the Harmonic Series". Experimental Mathematics. 3 (4): 287–302. doi:10.1080/10586458.1994.10504298. Zbl 0838.11015. CiteSeerX: 10.1.1.56.7026. Archived from the original on 27 January 2016.
^ a b Johnson, W. (1975). "Irregular Primes and Cyclotomic Invariants". Mathematics of Computation. 29 (129). AMS: 113–120. doi:10.2307/2005468. JSTOR 2005468.
^ It varies whether _L_0 = 2 is included in the Lucas numbers.
^ Sloane, N. J. A. (ed.). "Sequence A121091 (Smallest nexus prime of the form n^p - (n-1)^p, where p is an odd prime)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
^ Sloane, N. J. A. (ed.). "Sequence A121616 (Primes of form (n+1)^5 - n^5)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
^ Sloane, N. J. A. (ed.). "Sequence A121618 (Nexus primes of order 7 or primes of form n^7 - (n-1)^7)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
^ Paszkiewicz, Andrzej (2009). "A new prime p {\displaystyle p} $p$ for which the least primitive root ( mod p ) {\displaystyle ({\textrm {mod}}p)} $({\textrm {mod}}p)$ and the least primitive root ( mod p 2 ) {\displaystyle ({\textrm {mod}}p^{2})} $({\textrm {mod}}p^{2})$ are not equal" (PDF). Math. Comp. 78 (266). American Mathematical Society: 1193–1195. Bibcode:2009MaCom..78.1193P. doi:10.1090/S0025-5718-08-02090-5.
^ Caldwell, C.; Dubner, H. (1996–97). "The near repdigit primes A n − k − 1 B 1 A k {\displaystyle A_{n-k-1}B_{1}A_{k}} $A_{n-k-1}B_{1}A_{k}$ , especially 9 n − k − 1 8 1 9 k {\displaystyle 9_{n-k-1}8_{1}9_{k}} $9_{n-k-1}8_{1}9_{k}$ ". Journal of Recreational Mathematics. 28 (1): 1–9.
^ Lal, M. (1967). "Primes of the Form n4 + 1" (PDF). Mathematics of Computation. 21. AMS: 245–247. doi:10.1090/S0025-5718-1967-0222007-9. ISSN 1088-6842. Archived (PDF) from the original on 13 January 2015.
^ Bohman, J. (1973). "New primes of the form _n_4 + 1". BIT Numerical Mathematics. 13 (3). Springer: 370–372. doi:10.1007/BF01951947. ISSN 1572-9125. S2CID 123070671.
^ "Primes just less than a power of two 8 to 100 bits (page 1 of 4)". t5k.org. Retrieved 19 August 2025.
^ Subproject status at PrimeGrid
^ Ribenboim, P. (22 February 1996). The new book of prime number records. New York: Springer-Verlag. p. 347. ISBN 0-387-94457-5.
^ "Mirimanoff's Congruence: Other Congruences". Retrieved 26 January 2011.
^ Gallot, Y.; Moree, P.; Zudilin, W. (2011). "The Erdös-Moser equation 1_k_ + 2_k_ +...+ (m−1)k = m_k_ revisited using continued fractions". Mathematics of Computation. 80. American Mathematical Society: 1221–1237. arXiv:0907.1356. doi:10.1090/S0025-5718-2010-02439-1. S2CID 16305654.
^ a b c d Ribenboim, P. (2006). Die Welt der Primzahlen (PDF). Berlin: Springer. p. 240. ISBN 3-540-34283-4.

Lists of Primes at the Prime Pages.
The Nth Prime Page Nth prime through n=10^12, pi(x) through x=3*10^13, Random primes in same range.
Interface to a list of the first 98 million primes (primes less than 2,000,000,000)
Weisstein, Eric W. "Prime Number Sequences". MathWorld.
Selected prime related sequences in OEIS.
Fischer, R. Thema: Fermatquotient B^(P−1) == 1 (mod P^2) (in German) (Lists Wieferich primes in all bases up to 1052)
Padilla, Tony (7 February 2013). "New Largest Known Prime Number". Numberphile. Brady Haran. Archived from the original on 2 November 2021.