List of prime numbers#Wieferich primes

The following table lists the first 1000 primes, with 20 columns of consecutive primes in each of the 50 rows.{{Cite book | last = Lehmer | first = D. N. | author-link = Derrick Norman Lehmer | title = List of prime numbers from 1 to 10,006,721 | publisher = Carnegie Institution of Washington | volume = 165 | year = 1982 | location = Washington D.C. | id = OL16553580M | ol = 16553580M }}

{{OEIS|A000040}}.

The Goldbach conjecture verification project reports that it has computed all primes smaller than 4×10{{sup|18}}.Tomás Oliveira e Silva, [http://www.ieeta.pt/~tos/goldbach.html Goldbach conjecture verification] {{webarchive|url=https://web.archive.org/web/20110524084452/http://www.ieeta.pt/~tos/goldbach.html |date=24 May 2011 }}. Retrieved 16 July 2013 That means 95,676,260,903,887,607 primes{{OEIS|id=A080127}} (nearly 10{{sup|17}}), but they were not stored. There are known formulae to evaluate the prime-counting function (the number of primes smaller than a given value) faster than computing the primes. This has been used to compute that there are 1,925,320,391,606,803,968,923 primes (roughly 2{{e|21}}) smaller than 10{{sup|23}}. A different computation found that there are 18,435,599,767,349,200,867,866 primes (roughly 2{{e|22}}) smaller than 10{{sup|24}}, if the Riemann hypothesis is true.{{cite web|title=Conditional Calculation of pi(10{{sup|24}})|url=http://primes.utm.edu/notes/pi(10%5E24).html|author=Jens Franke|date=29 July 2010|access-date=17 May 2011|url-status=live|archive-url=http://archive.wikiwix.com/cache/20140824032441/http://primes.utm.edu/notes/pi(10%5E24).html|archive-date=24 August 2014}}

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 ({{OEIS2C|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. ({{OEIS2C|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 ({{OEIS2C|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 ({{OEIS2C|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 ({{OEIS2C|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, ... ({{OEIS2C|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|#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) ({{OEIS2C|A023200}}, {{OEIS2C|A046132}})

Of the form $\backslash tfrac\{x^3-y^3\}\{x-y\}$ 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 ({{OEIS2C|A002407}})

Of the form $\backslash tfrac\{x^3-y^3\}\{x-y\}$ 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 ({{OEIS2C|A002648}})

Of the form n×2{{sup|n}} + 1.

3, 393050634124102232869567034555427371542904833 ({{OEIS2C|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 ({{OEIS2C|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 ({{OEIS2C|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 ({{OEIS2C|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 ({{OEIS2C|A006567}})

Of the form p{{sub|n}}# + 1 (a subset of primorial primes).

3, 7, 31, 211, 2311, 200560490131 ({{OEIS2C|A018239}}{{OEIS2C|A018239}} includes 2 = empty product of first 0 primes plus 1, but 2 is excluded in this list.)

A prime $p$ that divides Euler number $E\_\{2n\}$ for some $0\backslash leq\; 2n\backslash leq\; p-3$.

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 ({{OEIS2C|A120337}})

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

149, 241, 2946901 ({{OEIS2C|A198245}})

Of the form n! − 1 or n! + 1.

2, 3, 5, 7, 23, 719, 5039, 39916801, 479001599, 87178291199, 10888869450418352160768000001, 265252859812191058636308479999999, 263130836933693530167218012159999999, 8683317618811886495518194401279999999 ({{OEIS2C|A088054}})

Of the form 2{{sup|2{{sup|n}}}} + 1.

3, 5, 17, 257, 65537 ({{OEIS2C|A019434}})

{{As of|2024|6}} 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.{{Cite arXiv|last1=Boklan|first1=Kent D.|last2=Conway|first2=John H.|date=2016|title=Expect at most one billionth of a new Fermat Prime!|eprint=1605.01371|class=math.NT}}

Of the form a{{sup|2{{sup|n}}}} + 1 for fixed integer a.

a = 2: 3, 5, 17, 257, 65537 ({{OEIS2C|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{{sub|0}} = 0, F{{sub|1}} = 1,

F{{sub|n}} = F{{sub|n−1}} + F{{sub|n−2}}.

2, 3, 5, 13, 89, 233, 1597, 28657, 514229, 433494437, 2971215073, 99194853094755497, 1066340417491710595814572169, 19134702400093278081449423917 ({{OEIS2C|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 ({{OEIS2C|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 ({{OEIS2C|A002145}})

Primes p{{sub|n}} for which p{{sub|n}}{{sup|2}} > p{{sub|n−i}} p{{sub|n+i}} for all 1 ≤ i ≤ n−1, where p{{sub|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 ({{OEIS2C|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 ({{OEIS2C|A035497}})

Primes p for which there are no solutions to H{{sub|k}} ≡ 0 (mod p) and H{{sub|k}} ≡ −ω{{sub|p}} (mod p) for 1 ≤ k ≤ p−2, where H{{sub|k}} denotes the k-th harmonic number and ω{{sub|p}} denotes the Wolstenholme quotient.{{Cite journal | last1 = Boyd | first1 = D. W. | title = A p-adic Study of the Partial Sums of the Harmonic Series | url = http://projecteuclid.org/euclid.em/1048515811 | doi = 10.1080/10586458.1994.10504298 | journal = Experimental Mathematics | volume = 3 | issue = 4 | pages = 287–302 | year = 1994 | id = CiteSeerX: {{URL|1=citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.56.7026|2=10.1.1.56.7026}} | zbl = 0838.11015 | url-status = live | archive-url = https://web.archive.org/web/20160127080246/http://projecteuclid.org/euclid.em/1048515811 | archive-date = 27 January 2016}}

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 ({{OEIS2C|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 ({{OEIS2C|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 ({{OEIS2C|A105440}})

For {{math|n ≥ 2}}, write the prime factorization of {{mvar|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 ({{OEIS2C|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 ({{OEIS2C|A000928}})

(See Wolstenholme prime)

Primes p such that (p, p−5) is an irregular pair.{{cite journal

|last=Johnson

|first=W.

|title=Irregular Primes and Cyclotomic Invariants

|journal=Mathematics of Computation

|volume=29

|issue=129

|pages=113–120

|publisher=AMS

|year=1975

|doi=10.2307/2005468

|jstor=2005468

|doi-access=free

}}

37

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

67, 877 ({{OEIS2C|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 ({{OEIS2C|A007510}})

Of the form x{{sup|y}} + y{{sup|x}}, with 1 < x < y.

17, 593, 32993, 2097593, 8589935681, 59604644783353249, 523347633027360537213687137, 43143988327398957279342419750374600193 ({{OEIS2C|A094133}})

Primes p for which, in a given base b, $\backslash frac\{b^\{p-1\}-1\}\{p\}$ 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 ({{OEIS2C|A001913}})

Primes in the Lucas number sequence L{{sub|0}} = 2, L{{sub|1}} = 1,

L{{sub|n}} = L{{sub|n−1}} + L{{sub|n−2}}.

2,It varies whether L{{sub|0}} = 2 is included in the Lucas numbers. 3, 7, 11, 29, 47, 199, 521, 2207, 3571, 9349, 3010349, 54018521, 370248451, 6643838879, 119218851371, 5600748293801, 688846502588399, 32361122672259149 ({{OEIS2C|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 ({{OEIS2C|A031157}})

Of the form 2{{sup|n}} − 1.

3, 7, 31, 127, 8191, 131071, 524287, 2147483647, 2305843009213693951, 618970019642690137449562111, 162259276829213363391578010288127, 170141183460469231731687303715884105727 ({{OEIS2C|A000668}})

{{As of|2024}}, 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 2^136,279,841−1, which is the 52nd Mersenne prime.

Primes p that divide 2{{sup|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 ({{OEIS2C|A122094}})

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

Primes p such that 2{{sup|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 ({{OEIS2C|A000043}})

{{As of|2024|10}}, four more are known to be in the sequence, but it is not known whether they are the next:

74207281, 77232917, 82589933, 136279841

A subset of Mersenne primes of the form 2{{sup|2{{sup|p}}−1}} − 1 for prime p.

7, 127, 2147483647, 170141183460469231731687303715884105727 (primes in {{OEIS2C|A077586}})

Of the form (a{{sup|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 ({{OEIS2C|A076481}})

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

a = 5: 31, 19531, 12207031, 305175781, 177635683940025046467781066894531, 14693679385278593849609206715278070972733319459651094018859396328480215743184089660644531 ({{OEIS2C|A086122}})

a = 6: 7, 43, 55987, 7369130657357778596659, 3546245297457217493590449191748546458005595187661976371 ({{OEIS2C|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 {{math|bⁿ − (b − 1)ⁿ}},{{Cite OEIS |A121091|Smallest nexus prime of the form n^p - (n-1)^p, where p is an odd prime}}{{Cite OEIS |A121616|Primes of form (n+1)^5 - n^5}}{{Cite OEIS |A121618|Nexus primes of order 7 or primes of form n^7 - (n-1)^7}} including the Mersenne primes and the cuban primes as special cases
• Williams primes, of the form {{math|(b − 1)·bⁿ − 1}}

Of the form ⌊θ{{sup|3{{sup|n}}}}⌋, where θ is Mills' constant. This form is prime for all positive integers n.

2, 11, 1361, 2521008887, 16022236204009818131831320183 ({{OEIS2C|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 ({{OEIS2C|A071062}})

Newman–Shanks–Williams numbers that are prime.

7, 41, 239, 9369319, 63018038201, 489133282872437279, 19175002942688032928599 ({{OEIS2C|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.{{cite journal

|last = Paszkiewicz

| first = Andrzej

| title = A new prime $p$ for which the least primitive root $(\backslash textrm\{mod\; \}\; p)$ and the least primitive root $(\backslash textrm\{mod\; \}\; p^2)$ are not equal

| journal = Math. Comp.

| volume = 78

| year = 2009

| issue = 266

| pages = 1193–1195

| url = https://www.ams.org/journals/mcom/2009-78-266/S0025-5718-08-02090-5/S0025-5718-08-02090-5.pdf

| doi = 10.1090/S0025-5718-08-02090-5

| publisher = American Mathematical Society

| bibcode = 2009MaCom..78.1193P

| doi-access = free

}}

2, 40487, 6692367337 ({{OEIS2C|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 ({{OEIS2C|A002385}})

Primes of the form $\backslash frac\{a\; \backslash big(\; 10^m-1\; \backslash big)\}\{9\}\; \backslash pm\; b\; \backslash times\; 10^\{\backslash frac\{\; m-1\; \}\{2\}\}$ with .{{Cite journal | last1 = Caldwell | first1 = C. | author1-link = Chris Caldwell (mathematician)| last2 = Dubner | first2 = H. | author2-link = Harvey Dubner | title = The near repdigit primes $A\_\{n-k-1\}B\_1A\_k$, especially $9\_\{n-k-1\}8\_19\_k$ | journal = Journal of Recreational Mathematics | volume = 28 | issue = 1 | pages = 1–9 | year = 1996–97}} 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 ({{OEIS2C|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 ({{OEIS2C|A049575}})

Primes in the Pell number sequence P{{sub|0}} = 0, P{{sub|1}} = 1,

P{{sub|n}} = 2P{{sub|n−1}} + P{{sub|n−2}}.

2, 5, 29, 5741, 33461, 44560482149, 1746860020068409, 68480406462161287469, 13558774610046711780701, 4125636888562548868221559797461449 ({{OEIS2C|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 ({{OEIS2C|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 ({{OEIS2C|A074788}})

Of the form 2{{sup|u}}3{{sup|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 ({{OEIS2C|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 ({{OEIS2C|A063980}})

Of the form n⁴ + 1.{{cite journal | last = Lal | first = M. | title = Primes of the Form n⁴ + 1 | journal = Mathematics of Computation | volume = 21 | pages = 245–247 | publisher = AMS | date = 1967 | url = https://www.ams.org/journals/mcom/1967-21-098/S0025-5718-1967-0222007-9/S0025-5718-1967-0222007-9.pdf | issn = 1088-6842 | doi = 10.1090/S0025-5718-1967-0222007-9 | url-status = live | archive-url = https://web.archive.org/web/20150113214845/http://www.ams.org/journals/mcom/1967-21-098/S0025-5718-1967-0222007-9/S0025-5718-1967-0222007-9.pdf | archive-date = 13 January 2015| doi-access = free }}{{cite journal | last = Bohman | first = J. | title = New primes of the form n⁴ + 1 | journal = BIT Numerical Mathematics | volume = 13 | issue = 3 | pages = 370–372 | publisher = Springer | date = 1973 | issn = 1572-9125 | doi = 10.1007/BF01951947| s2cid = 123070671 }}

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 ({{OEIS2C|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 ({{OEIS2C|A119535}})

Of the form p{{sub|n}}# ± 1.

3, 5, 7, 29, 31, 211, 2309, 2311, 30029, 200560490131, 304250263527209, 23768741896345550770650537601358309 (union of {{OEIS2C|A057705}} and {{OEIS2C|A018239}})

Of the form k×2{{sup|n}} + 1, with odd k and k < 2{{sup|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 ({{OEIS2C|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 ({{OEIS2C|A002144}})

{{see also|#Cousin primes|#Twin primes|#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) ({{OEIS2C|A007530}}, {{OEIS2C|A136720}}, {{OEIS2C|A136721}}, {{OEIS2C|A090258}})

Of the form x{{sup|4}} + y{{sup|4}}, where x,y > 0.

2, 17, 97, 257, 337, 641, 881 ({{OEIS2C|A002645}})

Integers R{{sub|n}} that are the smallest to give at least n primes from x/2 to x for all x ≥ R{{sub|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 ({{OEIS2C|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 ({{OEIS2C|A007703}})

Primes containing only the decimal digit 1.

11, 1111111111111111111 (19 digits), 11111111111111111111111 (23 digits) ({{OEIS2C|A004022}})

The next have 317, 1031, 49081, 86453, 109297, and 270343 digits, respectively ({{OEIS2C|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 ({{OEIS2C|A065091}})

4n+1: 5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101, 109, 113, 137 ({{OEIS2C|A002144}})

4n+3: 3, 7, 11, 19, 23, 31, 43, 47, 59, 67, 71, 79, 83, 103, 107 ({{OEIS2C|A002145}})

6n+1: 7, 13, 19, 31, 37, 43, 61, 67, 73, 79, 97, 103, 109, 127, 139 ({{OEIS2C|A002476}})

6n+5: 5, 11, 17, 23, 29, 41, 47, 53, 59, 71, 83, 89, 101, 107, 113 ({{OEIS2C|A007528}})

8n+1: 17, 41, 73, 89, 97, 113, 137, 193, 233, 241, 257, 281, 313, 337, 353 ({{OEIS2C|A007519}})

8n+3: 3, 11, 19, 43, 59, 67, 83, 107, 131, 139, 163, 179, 211, 227, 251 ({{OEIS2C|A007520}})

8n+5: 5, 13, 29, 37, 53, 61, 101, 109, 149, 157, 173, 181, 197, 229, 269 ({{OEIS2C|A007521}})

8n+7: 7, 23, 31, 47, 71, 79, 103, 127, 151, 167, 191, 199, 223, 239, 263 ({{OEIS2C|A007522}})

10n+1: 11, 31, 41, 61, 71, 101, 131, 151, 181, 191, 211, 241, 251, 271, 281 ({{OEIS2C|A030430}})

10n+3: 3, 13, 23, 43, 53, 73, 83, 103, 113, 163, 173, 193, 223, 233, 263 ({{OEIS2C|A030431}})

10n+7: 7, 17, 37, 47, 67, 97, 107, 127, 137, 157, 167, 197, 227, 257, 277 ({{OEIS2C|A030432}})

10n+9: 19, 29, 59, 79, 89, 109, 139, 149, 179, 199, 229, 239, 269, 349, 359 ({{OEIS2C|A030433}})

12n+1: 13, 37, 61, 73, 97, 109, 157, 181, 193, 229, 241, 277, 313, 337, 349 ({{OEIS2C|A068228}})

12n+5: 5, 17, 29, 41, 53, 89, 101, 113, 137, 149, 173, 197, 233, 257, 269 ({{OEIS2C|A040117}})

12n+7: 7, 19, 31, 43, 67, 79, 103, 127, 139, 151, 163, 199, 211, 223, 271 ({{OEIS2C|A068229}})

12n+11: 11, 23, 47, 59, 71, 83, 107, 131, 167, 179, 191, 227, 239, 251, 263 ({{OEIS2C|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 ({{OEIS2C|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 ({{OEIS2C|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) ({{OEIS2C|A023201}}, {{OEIS2C|A046117}})

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

2, 23, 2357 ({{OEIS2C|A069151}})

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

Of the form 2{{sup|k}} − c{{sub|1}}·2{{sup|k−1}} − c{{sub|2}}·2{{sup|k−2}} − ... − c{{sub|k}}.

• 3, 5, 7, 11, 13 ({{OEIS2C|A165255}})
• 2{{sup|32}} − 5, the largest prime that fits into 32 bits of memory.https://t5k.org/lists/2small/0bit.html
• 2{{sup|64}} − 59, the largest prime that fits into 64 bits of memory.

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 ({{OEIS2C|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 ({{OEIS2C|A042978}})

{{As of|2011}}, 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 ({{OEIS2C|A006450}})

There are exactly fifteen supersingular primes:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59, 71 ({{OEIS2C|A002267}})

Of the form 3×2{{sup|n}} − 1.

2, 5, 11, 23, 47, 191, 383, 6143, 786431, 51539607551, 824633720831, 26388279066623, 108086391056891903, 55340232221128654847, 226673591177742970257407 ({{OEIS2C|A007505}})

The primes of the form 3×2{{sup|n}} + 1 are related.

7, 13, 97, 193, 769, 12289, 786433, 3221225473, 206158430209, 6597069766657 ({{OEIS2C|A039687}})

{{see also|#Cousin primes|#Twin primes|#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) ({{OEIS2C|A007529}}, {{OEIS2C|A098414}}, {{OEIS2C|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 ({{OEIS2C|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 ({{OEIS2C|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 ({{OEIS2C|A020994}})

{{see also|#Cousin primes|#Prime triplets|#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) ({{OEIS2C|A001359}}, {{OEIS2C|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 ({{OEIS2C|A040017}})

Of the form (2{{sup|n}} + 1) / 3.

3, 11, 43, 683, 2731, 43691, 174763, 2796203, 715827883, 2932031007403, 768614336404564651, 201487636602438195784363, 845100400152152934331135470251, 56713727820156410577229101238628035243 ({{OEIS2C|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 ({{OEIS2C|A000978}})

A prime p > 5, if p{{sup|2}} divides the Fibonacci number $F\_\{p\; -\; \backslash left(\backslash frac\{\{p\}\}\{\{5\}\}\backslash right)\}$, where the Legendre symbol $\backslash left(\backslash frac\{\{p\}\}\{\{5\}\}\backslash right)$ is defined as

:

{{As of|2018}}, no Wall-Sun-Sun primes are known.

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

2^{p − 1} ≡ 1 (mod p²): 1093, 3511 ({{OEIS2C|A001220}})

3^{p − 1} ≡ 1 (mod p²): 11, 1006003 ({{OEIS2C|A014127}}){{cite book | last = Ribenboim | first = P. | author-link = Paulo Ribenboim | title = The new book of prime number records | publisher = Springer-Verlag | location = New York | page = 347 | url = https://books.google.com/books?id=72eg8bFw40kC&q=ribenboim | isbn = 0-387-94457-5| date = 22 February 1996 }}{{cite web | title = Mirimanoff's Congruence: Other Congruences | url = http://www.museumstuff.com/learn/topics/Mirimanoff%27s_congruence::sub::Other_Congruences | access-date = 26 January 2011}}{{cite journal | last1 = Gallot | first1 = Y. | last2 = Moree | first2 = P. | last3 = Zudilin | first3 = W. | title = The Erdös-Moser equation 1{{sup|k}} + 2{{sup|k}} +...+ (m−1){{sup|k}} = m{{sup|k}} revisited using continued fractions | journal = Mathematics of Computation | volume = 80 | pages = 1221–1237 | publisher = American Mathematical Society | year = 2011 | url = http://www.mpim-bonn.mpg.de/preprints/send?bid=4053 | doi = 10.1090/S0025-5718-2010-02439-1 | arxiv=0907.1356| s2cid = 16305654 }}

4^{p − 1} ≡ 1 (mod p²): 1093, 3511

5^{p − 1} ≡ 1 (mod p²): 2, 20771, 40487, 53471161, 1645333507, 6692367337, 188748146801 ({{OEIS2C|A123692}})

6^{p − 1} ≡ 1 (mod p²): 66161, 534851, 3152573 ({{OEIS2C|A212583}})

7^{p − 1} ≡ 1 (mod p²): 5, 491531 ({{OEIS2C|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 ({{OEIS2C|A045616}})

11^{p − 1} ≡ 1 (mod p²): 71{{Cite book | last = Ribenboim | first = P. | author-link = Paulo Ribenboim | title = Die Welt der Primzahlen | publisher = Springer | year = 2006 | location = Berlin | page = 240 | url = https://www.gbv.de/dms/bs/toc/495799599.pdf | isbn = 3-540-34283-4 }}

12^{p − 1} ≡ 1 (mod p²): 2693, 123653 ({{OEIS2C|A111027}})

13^{p − 1} ≡ 1 (mod p²): 2, 863, 1747591 ({{OEIS2C|A128667}})

14^{p − 1} ≡ 1 (mod p²): 29, 353, 7596952219 ({{OEIS2C|A234810}})

15^{p − 1} ≡ 1 (mod p²): 29131, 119327070011 ({{OEIS2C|A242741}})

16^{p − 1} ≡ 1 (mod p²): 1093, 3511

17^{p − 1} ≡ 1 (mod p²): 2, 3, 46021, 48947 ({{OEIS2C|A128668}})

18^{p − 1} ≡ 1 (mod p²): 5, 7, 37, 331, 33923, 1284043 ({{OEIS2C|A244260}})

19^{p − 1} ≡ 1 (mod p²): 3, 7, 13, 43, 137, 63061489 ({{OEIS2C|A090968}})

20^{p − 1} ≡ 1 (mod p²): 281, 46457, 9377747, 122959073 ({{OEIS2C|A242982}})

21^{p − 1} ≡ 1 (mod p²): 2

22^{p − 1} ≡ 1 (mod p²): 13, 673, 1595813, 492366587, 9809862296159 ({{OEIS2C|A298951}})

23^{p − 1} ≡ 1 (mod p²): 13, 2481757, 13703077, 15546404183, 2549536629329 ({{OEIS2C|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 p{{sup|2}} divides (p−1)! + 1.

5, 13, 563 ({{OEIS2C|A007540}})

{{As of|2018}}, these are the only known Wilson primes.

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

16843, 2124679 ({{OEIS2C|A088164}})

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

Of the form n×2{{sup|n}} − 1.

7, 23, 383, 32212254719, 2833419889721787128217599, 195845982777569926302400511, 4776913109852041418248056622882488319 ({{OEIS2C|A050918}})

{{Portal|Mathematics}}

{{div col|colwidth=20em|small=yes}}

• {{Annotated link|Illegal prime}}
• {{Annotated link|Largest known prime number}}
• {{Annotated link|List of largest known primes and probable primes}}
• {{Annotated link|List of numbers}}
• {{Annotated link|Prime gap}}
• {{Annotated link|Prime number theorem}}
• {{Annotated link|Probable prime}}
• {{Annotated link|Pseudoprime}}
• {{Annotated link|Strong prime}}
• {{Annotated link|Table of prime factors}}
• {{Annotated link|Wieferich pair}}

{{div col end}}

{{reflist}}

• [https://prime-numbers.de] All prime numbers from 31 to 6,469,693,189 for free download.
• [http://primes.utm.edu/lists/ Lists of Primes] at the Prime Pages.
• [http://primes.utm.edu/nthprime/ The Nth Prime Page] Nth prime through n=10^12, pi(x) through x=3*10^13, Random primes in same range.
• [http://www.rsok.com/~jrm/printprimes.html Interface to a list of the first 98 million primes] (primes less than 2,000,000,000)
• {{MathWorld|title=Prime Number Sequences|urlname=topics/PrimeNumberSequences|ref=none}}
• [http://oeis.org/wiki/Index_to_OEIS:_Section_Pri Selected prime related sequences] in OEIS.
• Fischer, R. [http://www.fermatquotient.com/FermatQuotienten/FermQ_Sort.txt Thema: Fermatquotient B^(P−1) == 1 (mod P^2)] {{in lang|de}} (Lists Wieferich primes in all bases up to 1052)
• {{Cite web |last=Padilla |first=Tony |date=7 February 2013 |title=New Largest Known Prime Number |url=https://www.youtube.com/watch?v=QSEKzFGpCQs |url-status=live |archive-url=https://ghostarchive.org/varchive/youtube/20211102/QSEKzFGpCQs |archive-date=2021-11-02 |website=Numberphile |publisher=Brady Haran |ref=none}}{{cbignore}}

{{Prime number classes}}

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Prime numbers

Prime