List of prime numbers


A prime number 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. The first 1000 primes are listed below, followed by lists of notable types of prime numbers in alphabetical order, giving their respective first terms. 1 is neither prime nor composite.

The first 1000 prime numbers

The following table lists the first 1000 primes, with 20 columns of consecutive primes in each of the 50 rows.
1234567891011121314151617181920
1–20235711131719232931374143475359616771
21–407379838997101103107109113127131137139149151157163167173
41–60179181191193197199211223227229233239241251257263269271277281
61–80283293307311313317331337347349353359367373379383389397401409
81–100419421431433439443449457461463467479487491499503509521523541
101–120547557563569571577587593599601607613617619631641643647653659
121–140661673677683691701709719727733739743751757761769773787797809
141–160811821823827829839853857859863877881883887907911919929937941
161–180947953967971977983991997100910131019102110311033103910491051106110631069
181–20010871091109310971103110911171123112911511153116311711181118711931201121312171223
201–22012291231123712491259127712791283128912911297130113031307131913211327136113671373
221–24013811399140914231427142914331439144714511453145914711481148314871489149314991511
241–26015231531154315491553155915671571157915831597160116071609161316191621162716371657
261–28016631667166916931697169917091721172317331741174717531759177717831787178918011811
281–30018231831184718611867187118731877187918891901190719131931193319491951197319791987
301–32019931997199920032011201720272029203920532063206920812083208720892099211121132129
321–34021312137214121432153216121792203220722132221223722392243225122672269227322812287
341–36022932297230923112333233923412347235123572371237723812383238923932399241124172423
361–38024372441244724592467247324772503252125312539254325492551255725792591259326092617
381–40026212633264726572659266326712677268326872689269326992707271127132719272927312741
401–42027492753276727772789279127972801280328192833283728432851285728612879288728972903
421–44029092917292729392953295729632969297129993001301130193023303730413049306130673079
441–46030833089310931193121313731633167316931813187319132033209321732213229325132533257
461–48032593271329933013307331333193323332933313343334733593361337133733389339134073413
481–50034333449345734613463346734693491349935113517352735293533353935413547355735593571
501–52035813583359336073613361736233631363736433659367136733677369136973701370937193727
521–54037333739376137673769377937933797380338213823383338473851385338633877388138893907
541–56039113917391939233929393139433947396739894001400340074013401940214027404940514057
561–58040734079409140934099411141274129413341394153415741594177420142114217421942294231
581–60042414243425342594261427142734283428942974327433743394349435743634373439143974409
601–62044214423444144474451445744634481448344934507451345174519452345474549456145674583
621–64045914597460346214637463946434649465146574663467346794691470347214723472947334751
641–66047594783478747894793479948014813481748314861487148774889490349094919493149334937
661–68049434951495749674969497349874993499950035009501150215023503950515059507750815087
681–70050995101510751135119514751535167517151795189519752095227523152335237526152735279
701–72052815297530353095323533353475351538153875393539954075413541754195431543754415443
721–74054495471547754795483550155035507551955215527553155575563556955735581559156235639
741–76056415647565156535657565956695683568956935701571157175737574157435749577957835791
761–78058015807581358215827583958435849585158575861586758695879588158975903592359275939
781–80059535981598760076011602960376043604760536067607360796089609161016113612161316133
801–82061436151616361736197619962036211621762216229624762576263626962716277628762996301
821–84063116317632363296337634363536359636163676373637963896397642164276449645164696473
841–86064816491652165296547655165536563656965716577658165996607661966376653665966616673
861–88066796689669167016703670967196733673767616763677967816791679368036823682768296833
881–90068416857686368696871688368996907691169176947694969596961696769716977698369916997
901–92070017013701970277039704370577069707971037109712171277129715171597177718771937207
921–94072117213721972297237724372477253728372977307730973217331733373497351736973937411
941–96074177433745174577459747774817487748974997507751775237529753775417547754975597561
961–98075737577758375897591760376077621763976437649766976737681768776917699770377177723
981–100077277741775377577759778977937817782378297841785378677873787778797883790179077919

The Goldbach conjecture verification project reports that it has computed all primes below 4×10. That means 95,676,260,903,887,607 primes, but they were not stored. There are known formulae to evaluate the prime-counting function faster than computing the primes. This has been used to compute that there are 1,925,320,391,606,803,968,923 primes below 10. A different computation found that there are 18,435,599,767,349,200,867,866 primes below 10, if the Riemann hypothesis is true.

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 in the definitions.

[Balanced prime]s

Form: p − n, p, p + n
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.

Carol primes">Carol number">Carol primes

Of the form − 2.
7, 47, 223, 3967, 16127, 1046527, 16769023, 1073676287, 68718952447, 274876858367, 4398042316799, 1125899839733759, 18014398241046527, 1298074214633706835075030044377087

[Chen primes]

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

[Circular prime]s

A circular prime number is a number that remains prime on any cyclic rotation of its digits.
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
Some sources only list the smallest prime in each cycle, for example, listing 13, but omitting 31 :
2, 3, 5, 7, 11, 13, 17, 37, 79, 113, 197, 199, 337, 1193, 3779, 11939, 19937, 193939, 199933, 1111111111111111111, 11111111111111111111111
All [|repunit primes] are circular.

[Cousin primes]

Where are both prime.
,,,,,,,,,,,,,,,,

[Cuban prime]s

Of the form 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
Of the form 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

Cullen primes">Cullen number">Cullen primes

Of the form n×2 + 1.
3, 393050634124102232869567034555427371542904833

[Dihedral prime]s

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

[Eisenstein primes] without imaginary part

s that are irreducible and real numbers.
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

[Emirps]

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

Euclid primes">Euclid number">Euclid primes

Of the form p# + 1.
3, 7, 31, 211, 2311, 200560490131

Euler irregular primes">Regular prime#Euler irregular primes">Euler irregular primes

A prime that divides Euler number for some.
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

Euler (''p'', ''p'' − 3) irregular primes">Regular prime#Irregular pairs">Euler (''p'', ''p'' − 3) irregular primes

Primes such that is an Euler irregular pair.
149, 241, 2946901

[Factorial prime]s

Of the form n! − 1 or n! + 1.
2, 3, 5, 7, 23, 719, 5039, 39916801, 479001599, 87178291199, 10888869450418352160768000001, 265252859812191058636308479999999, 263130836933693530167218012159999999, 8683317618811886495518194401279999999

[Fermat prime]s

Of the form 2 + 1.
3, 5, 17, 257, 65537
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.

Generalized [Fermat primes]

Of the form a + 1 for fixed integer a.
a = 2: 3, 5, 17, 257, 65537
a = 4: 5, 17, 257, 65537
a = 6: 7, 37, 1297
a = 8:
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
these are the only known generalized Fermat primes for a ≤ 24.

[Fibonacci prime]s

Primes in the Fibonacci sequence F = 0, F = 1,
F = F + F.
2, 3, 5, 13, 89, 233, 1597, 28657, 514229, 433494437, 2971215073, 99194853094755497, 1066340417491710595814572169, 19134702400093278081449423917

Fortunate primes">Fortunate number">Fortunate primes

s that are prime.
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

[Gaussian prime]s

s 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

[Good prime]s

Primes p for which p > p p for all 1 ≤ i ≤ n−1, where p 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

[Happy prime]s

Happy numbers that are prime.
7, 13, 19, 23, 31, 79, 97, 103, 109, 139, 167, 193, 239, 263, 293, 313, 331, 367, 379, 383, 397, 409, 487, 563, 617, 653, 673, 683, 709, 739, 761, 863, 881, 907, 937, 1009, 1033, 1039, 1093

Harmonic primes

Primes p for which there are no solutions to H ≡ 0 and H ≡ −ω for 1 ≤ k ≤ p−2, where H denotes the k-th harmonic number and ω denotes the Wolstenholme quotient.
5, 13, 17, 23, 41, 67, 73, 79, 107, 113, 139, 149, 157, 179, 191, 193, 223, 239, 241, 251, 263, 277, 281, 293, 307, 311, 317, 331, 337, 349

[Higgs prime]s for squares

Primes p for which p − 1 divides the square of the product of all earlier terms.
2, 3, 5, 7, 11, 13, 19, 23, 29, 31, 37, 43, 47, 53, 59, 61, 67, 71, 79, 101, 107, 127, 131, 139, 149, 151, 157, 173, 181, 191, 197, 199, 211, 223, 229, 263, 269, 277, 283, 311, 317, 331, 347, 349

[Highly [cototient number| Highly cototient primes]]

Primes that are a cototient more often than any integer below it except 1.
2, 23, 47, 59, 83, 89, 113, 167, 269, 389, 419, 509, 659, 839, 1049, 1259, 1889

[Home prime]s

For, write the prime factorization of in base 10 and concatenate the factors; iterate until a prime is reached.
2, 3, 211, 5, 23, 7, 3331113965338635107, 311, 773, 11, 223, 13, 13367, 1129, 31636373, 17, 233, 19, 3318308475676071413, 37, 211, 23, 331319, 773, 3251, 13367, 227, 29, 547, 31, 241271, 311, 31397, 1129, 71129, 37, 373, 313, 3314192745739, 41, 379, 43, 22815088913, 3411949, 223, 47, 6161791591356884791277

[Irregular prime]s

Odd primes p that divide the class number of the p-th cyclotomic field.
37, 59, 67, 101, 103, 131, 149, 157, 233, 257, 263, 271, 283, 293, 307, 311, 347, 353, 379, 389, 401, 409, 421, 433, 461, 463, 467, 491, 523, 541, 547, 557, 577, 587, 593, 607, 613

(''p'', ''p'' − 3) irregular primes">Regular prime#Irregular pairs">(''p'', ''p'' − 3) irregular primes

(''p'', ''p'' − 5) irregular primes">Regular prime#Irregular pairs">(''p'', ''p'' − 5) irregular primes

Primes p such that is an irregular pair.
37

(''p'', ''p'' − 9) irregular primes">Regular prime#Irregular pairs">(''p'', ''p'' − 9) irregular primes

Primes p such that is an irregular pair.
67, 877

[Isolated prime]s

Primes p such that neither p − 2 nor p + 2 is prime.
2, 23, 37, 47, 53, 67, 79, 83, 89, 97, 113, 127, 131, 157, 163, 167, 173, 211, 223, 233, 251, 257, 263, 277, 293, 307, 317, 331, 337, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 439, 443, 449, 457, 467, 479, 487, 491, 499, 503, 509, 541, 547, 557, 563, 577, 587, 593, 607, 613, 631, 647, 653, 673, 677, 683, 691, 701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797, 839, 853, 863, 877, 887, 907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997

Kynea primes">Kynea number">Kynea primes

Of the form − 2.
2, 7, 23, 79, 1087, 66047, 263167, 16785407, 1073807359, 17180131327, 68720001023, 4398050705407, 70368760954879, 18014398777917439, 18446744082299486207

Leyland primes">Leyland number">Leyland primes

Of the form x + y, with 1 < x < y.
17, 593, 32993, 2097593, 8589935681, 59604644783353249, 523347633027360537213687137, 43143988327398957279342419750374600193

Long primes">Full reptend prime">Long primes

Primes p for which, in a given base b, gives a cyclic number. They are also called full reptend primes. Primes p for base 10:
7, 17, 19, 23, 29, 47, 59, 61, 97, 109, 113, 131, 149, 167, 179, 181, 193, 223, 229, 233, 257, 263, 269, 313, 337, 367, 379, 383, 389, 419, 433, 461, 487, 491, 499, 503, 509, 541, 571, 577, 593

[Lucas prime]s

Primes in the Lucas number sequence L = 2, L = 1,
L = L + L.
2, 3, 7, 11, 29, 47, 199, 521, 2207, 3571, 9349, 3010349, 54018521, 370248451, 6643838879, 119218851371, 5600748293801, 688846502588399, 32361122672259149

Lucky primes">Lucky number">Lucky primes

Lucky numbers that are prime.
3, 7, 13, 31, 37, 43, 67, 73, 79, 127, 151, 163, 193, 211, 223, 241, 283, 307, 331, 349, 367, 409, 421, 433, 463, 487, 541, 577, 601, 613, 619, 631, 643, 673, 727, 739, 769, 787, 823, 883, 937, 991, 997

[Mersenne prime]s

Of the form 2 − 1.
3, 7, 31, 127, 8191, 131071, 524287, 2147483647, 2305843009213693951, 618970019642690137449562111, 162259276829213363391578010288127, 170141183460469231731687303715884105727
, there are 51 known Mersenne primes. The 13th, 14th, and 51st have respectively 157, 183, and 24,862,048 digits.
, this class of prime numbers also contains the largest known prime: M82589933, the 51st known Mersenne prime.

Mersenne divisors

Primes p that divide 2 − 1, for some prime number n.
3, 7, 23, 31, 47, 89, 127, 167, 223, 233, 263, 359, 383, 431, 439, 479, 503, 719, 839, 863, 887, 983, 1103, 1319, 1367, 1399, 1433, 1439, 1487, 1823, 1913, 2039, 2063, 2089, 2207, 2351, 2383, 2447, 2687, 2767, 2879, 2903, 2999, 3023, 3119, 3167, 3343
All Mersenne primes are, by definition, members of this sequence.

Mersenne prime exponents

Primes p such that 2 − 1 is prime.
2, 3, 5, 7, 13, 17, 19, 31, 61, 89,
107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423,
9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049,
216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011,
24036583, 25964951, 30402457, 32582657, 37156667, 42643801, 43112609
four more are known to be in the sequence, but it is not known whether they are the next:

57885161, 74207281, 77232917, 82589933

[Double Mersenne prime]s

A subset of Mersenne primes of the form 2 − 1 for prime p.
7, 127, 2147483647, 170141183460469231731687303715884105727
As of June 2017, these are the only known double Mersenne primes, and number theorists think these are probably the only double Mersenne primes.

Generalized repunit primes">Repunit#Repunit primes">repunit primes

Of the form / for fixed integer a.
For a = 2, these are the Mersenne primes, while for a = 10 they are the repunit primes. For other small a, they are given below:
a = 3: 13, 1093, 797161, 3754733257489862401973357979128773, 6957596529882152968992225251835887181478451547013
a = 4: 5
a = 5: 31, 19531, 12207031, 305175781, 177635683940025046467781066894531, 14693679385278593849609206715278070972733319459651094018859396328480215743184089660644531
a = 6: 7, 43, 55987, 7369130657357778596659, 3546245297457217493590449191748546458005595187661976371
a = 7: 2801, 16148168401, 85053461164796801949539541639542805770666392330682673302530819774105141531698707146930307290253537320447270457
a = 8: 73
a = 9: none exist

Other generalizations and variations

Many generalizations of Mersenne primes have been defined. This include the following:
Of the form ⌊θ⌋, where θ is Mills' constant. This form is prime for all positive integers n.
2, 11, 1361, 2521008887, 16022236204009818131831320183

Minimal primes">Minimal prime (number theory)">Minimal primes

Primes for which there is no shorter sub-sequence of the decimal digits that form a prime. There are exactly 26 minimal primes:
2, 3, 5, 7, 11, 19, 41, 61, 89, 409, 449, 499, 881, 991, 6469, 6949, 9001, 9049, 9649, 9949, 60649, 666649, 946669, 60000049, 66000049, 66600049

[Newman–Shanks–Williams prime]s

Newman–Shanks–Williams numbers that are prime.
7, 41, 239, 9369319, 63018038201, 489133282872437279, 19175002942688032928599

Non-generous primes

Primes p for which the least positive primitive root is not a primitive root of p2. Three such primes are known; it is not known whether there are more.
2, 40487, 6692367337

[Palindromic primes]

Primes that remain the same when their decimal digits are read backwards.
2, 3, 5, 7, 11, 101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, 797, 919, 929, 10301, 10501, 10601, 11311, 11411, 12421, 12721, 12821, 13331, 13831, 13931, 14341, 14741

Palindromic wing primes

Primes of the form with. This means all digits except the middle digit are equal.
101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, 797, 919, 929, 11311, 11411, 33533, 77377, 77477, 77977, 1114111, 1117111, 3331333, 3337333, 7772777, 7774777, 7778777, 111181111, 111191111, 777767777, 77777677777, 99999199999

Partition primes">Partition (number theory)#Partition function">Partition primes

Partition function values that are prime.
2, 3, 5, 7, 11, 101, 17977, 10619863, 6620830889, 80630964769, 228204732751, 1171432692373, 1398341745571, 10963707205259, 15285151248481, 10657331232548839, 790738119649411319, 18987964267331664557

Pell primes">Pell number">Pell primes

Primes in the Pell number sequence P = 0, P = 1,
P = 2P + P.
2, 5, 29, 5741, 33461, 44560482149, 1746860020068409, 68480406462161287469, 13558774610046711780701, 4125636888562548868221559797461449

[Permutable prime]s

Any permutation of the decimal digits is a prime.
2, 3, 5, 7, 11, 13, 17, 31, 37, 71, 73, 79, 97, 113, 131, 199, 311, 337, 373, 733, 919, 991, 1111111111111111111, 11111111111111111111111
It seems likely that all further permutable primes are repunits, i.e. contain only the digit 1.

Perrin primes">Perrin number">Perrin primes

Primes in the Perrin number sequence P = 3, P = 0, P = 2,
P = P + P.
2, 3, 5, 7, 17, 29, 277, 367, 853, 14197, 43721, 1442968193, 792606555396977, 187278659180417234321, 66241160488780141071579864797

[Pierpont prime]s

Of the form 23 + 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

[Pillai prime]s

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

Primes of the form ''n''4 + 1

Of the form n4 + 1.
2, 17, 257, 1297, 65537, 160001, 331777, 614657, 1336337, 4477457, 5308417, 8503057, 9834497, 29986577, 40960001, 45212177, 59969537, 65610001, 126247697, 193877777, 303595777, 384160001, 406586897, 562448657, 655360001

[Primeval prime]s

Primes for which there are more prime permutations of some or all the decimal digits than for any smaller number.
2, 13, 37, 107, 113, 137, 1013, 1237, 1367, 10079

[Primorial prime]s

Of the form p# ± 1.
3, 5, 7, 29, 31, 211, 2309, 2311, 30029, 200560490131, 304250263527209, 23768741896345550770650537601358309

Proth primes">Proth number">Proth primes

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

[Pythagorean prime]s

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

Prime quadruplets">Prime quadruplet">Prime quadruplets

Where are all prime.
,,,,,,,,,,,

[Quartan prime]s

Of the form x + y, where x,y > 0.
2, 17, 97, 257, 337, 641, 881

[Ramanujan prime]s

Integers R that are the smallest to give at least n primes from x/2 to x for all x ≥ R.
2, 11, 17, 29, 41, 47, 59, 67, 71, 97, 101, 107, 127, 149, 151, 167, 179, 181, 227, 229, 233, 239, 241, 263, 269, 281, 307, 311, 347, 349, 367, 373, 401, 409, 419, 431, 433, 439, 461, 487, 491

[Regular prime]s

Primes p that do not divide the class number of the p-th cyclotomic field.
3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 43, 47, 53, 61, 71, 73, 79, 83, 89, 97, 107, 109, 113, 127, 137, 139, 151, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 239, 241, 251, 269, 277, 281

Repunit primes">Repunit#Decimal repunit primes">Repunit primes

Primes containing only the decimal digit 1.
11, 1111111111111111111, 11111111111111111111111
The next have 317, 1031, 49081, 86453, 109297, 270343 digits

Residue classes of primes">Dirichlet's theorem on arithmetic progressions">Residue classes of primes

Of the form an + d for fixed integers a and d. Also called primes congruent to d modulo a.
The primes of the form 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. The classes 10n+d 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
4n+1: 5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101, 109, 113, 137
4n+3: 3, 7, 11, 19, 23, 31, 43, 47, 59, 67, 71, 79, 83, 103, 107
6n+1: 7, 13, 19, 31, 37, 43, 61, 67, 73, 79, 97, 103, 109, 127, 139
6n+5: 5, 11, 17, 23, 29, 41, 47, 53, 59, 71, 83, 89, 101, 107, 113
8n+1: 17, 41, 73, 89, 97, 113, 137, 193, 233, 241, 257, 281, 313, 337, 353
8n+3: 3, 11, 19, 43, 59, 67, 83, 107, 131, 139, 163, 179, 211, 227, 251
8n+5: 5, 13, 29, 37, 53, 61, 101, 109, 149, 157, 173, 181, 197, 229, 269
8n+7: 7, 23, 31, 47, 71, 79, 103, 127, 151, 167, 191, 199, 223, 239, 263
10n+1: 11, 31, 41, 61, 71, 101, 131, 151, 181, 191, 211, 241, 251, 271, 281
10n+3: 3, 13, 23, 43, 53, 73, 83, 103, 113, 163, 173, 193, 223, 233, 263
10n+7: 7, 17, 37, 47, 67, 97, 107, 127, 137, 157, 167, 197, 227, 257, 277
10n+9: 19, 29, 59, 79, 89, 109, 139, 149, 179, 199, 229, 239, 269, 349, 359
12n+1: 13, 37, 61, 73, 97, 109, 157, 181, 193, 229, 241, 277, 313, 337, 349
12n+5: 5, 17, 29, 41, 53, 89, 101, 113, 137, 149, 173, 197, 233, 257, 269
12n+7: 7, 19, 31, 43, 67, 79, 103, 127, 139, 151, 163, 199, 211, 223, 271
12n+11: 11, 23, 47, 59, 71, 83, 107, 131, 167, 179, 191, 227, 239, 251, 263

[Safe prime]s

Where p and / 2 are both prime.
5, 7, 11, 23, 47, 59, 83, 107, 167, 179, 227, 263, 347, 359, 383, 467, 479, 503, 563, 587, 719, 839, 863, 887, 983, 1019, 1187, 1283, 1307, 1319, 1367, 1439, 1487, 1523, 1619, 1823, 1907

Self primes">Self number">Self primes in base 10

Primes that cannot be generated by any integer added to the sum of its decimal digits.
3, 5, 7, 31, 53, 97, 211, 233, 277, 367, 389, 457, 479, 547, 569, 613, 659, 727, 839, 883, 929, 1021, 1087, 1109, 1223, 1289, 1447, 1559, 1627, 1693, 1783, 1873

[Sexy prime]s

Where are both prime.
,,,,,,,,,,,,,,,,,,,,,,,,,

Smarandache–Wellin primes">Smarandache–Wellin number">Smarandache–Wellin primes

Primes that are the concatenation of the first n primes written in decimal.
2, 23, 2357
The fourth Smarandache-Wellin prime is the 355-digit concatenation of the first 128 primes that end with 719.

[Solinas prime]s

Of the form 2 ± 2 ± 1, where 0 < b < a.
3, 5, 7, 11, 13

[Sophie Germain prime]s

Where p and 2p + 1 are both prime.
2, 3, 5, 11, 23, 29, 41, 53, 83, 89, 113, 131, 173, 179, 191, 233, 239, 251, 281, 293, 359, 419, 431, 443, 491, 509, 593, 641, 653, 659, 683, 719, 743, 761, 809, 911, 953

[Stern prime]s

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

Strobogrammatic primes

Primes that are also a prime number when rotated upside down.
Using 0, 1, 8 and 6/9:
11, 101, 181, 619, 16091, 18181, 19861, 61819, 116911, 119611, 160091, 169691, 191161, 196961, 686989, 688889

[Super-prime]s

Primes with a prime index in the sequence of prime numbers.
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

Supersingular primes">Supersingular prime (moonshine theory)">Supersingular primes

There are exactly fifteen supersingular primes:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59, 71

[Thabit prime]s

Of the form 3×2 − 1.
2, 5, 11, 23, 47, 191, 383, 6143, 786431, 51539607551, 824633720831, 26388279066623, 108086391056891903, 55340232221128654847, 226673591177742970257407
The primes of the form 3×2 + 1 are related.
7, 13, 97, 193, 769, 12289, 786433, 3221225473, 206158430209, 6597069766657

Prime triplets">Prime triplet">Prime triplets

Where or are all prime.
,,,,,,,,,,,,,,,,,,,

[Truncatable prime]

Left-truncatable

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

Right-truncatable

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

Two-sided

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

[Twin prime]s

Where are both prime.
,,,,,,,,,,,,,,,,,,,,,,,

[Unique prime]s

The list of primes p for which the period length of the decimal expansion of 1/p is unique.
3, 11, 37, 101, 9091, 9901, 333667, 909091, 99990001, 999999000001, 9999999900000001, 909090909090909091, 1111111111111111111, 11111111111111111111111, 900900900900990990990991

[Wagstaff prime]s

Of the form / 3.
3, 11, 43, 683, 2731, 43691, 174763, 2796203, 715827883, 2932031007403, 768614336404564651, 201487636602438195784363, 845100400152152934331135470251, 56713727820156410577229101238628035243
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

[Wall–Sun–Sun prime]s

A prime p > 5, if p divides the Fibonacci number, where the Legendre symbol is defined as
, no Wall-Sun-Sun primes are known.

[Weakly prime number]s

Primes that having any one of their digits changed to any other value will always result in a composite number.
294001, 505447, 584141, 604171, 971767, 1062599, 1282529, 1524181, 2017963, 2474431, 2690201, 3085553, 3326489, 4393139

[Wieferich prime]s

Primes p such that for fixed integer a > 1.
2p − 1 ≡ 1 : 1093, 3511
3p − 1 ≡ 1 : 11, 1006003
4p − 1 ≡ 1 : 1093, 3511
5p − 1 ≡ 1 : 2, 20771, 40487, 53471161, 1645333507, 6692367337, 188748146801
6p − 1 ≡ 1 : 66161, 534851, 3152573
7p − 1 ≡ 1 : 5, 491531
8p − 1 ≡ 1 : 3, 1093, 3511
9p − 1 ≡ 1 : 2, 11, 1006003
10p − 1 ≡ 1 : 3, 487, 56598313
11p − 1 ≡ 1 : 71
12p − 1 ≡ 1 : 2693, 123653
13p − 1 ≡ 1 : 2, 863, 1747591
14p − 1 ≡ 1 : 29, 353, 7596952219
15p − 1 ≡ 1 : 29131, 119327070011
16p − 1 ≡ 1 : 1093, 3511
17p − 1 ≡ 1 : 2, 3, 46021, 48947
18p − 1 ≡ 1 : 5, 7, 37, 331, 33923, 1284043
19p − 1 ≡ 1 : 3, 7, 13, 43, 137, 63061489
20p − 1 ≡ 1 : 281, 46457, 9377747, 122959073
21p − 1 ≡ 1 : 2
22p − 1 ≡ 1 : 13, 673, 1595813, 492366587, 9809862296159
23p − 1 ≡ 1 : 13, 2481757, 13703077, 15546404183, 2549536629329
24p − 1 ≡ 1 : 5, 25633
25p − 1 ≡ 1 : 2, 20771, 40487, 53471161, 1645333507, 6692367337, 188748146801
, these are all known Wieferich primes with a ≤ 25.

[Wilson prime]s

Primes p for which p divides ! + 1.
5, 13, 563
, these are the only known Wilson primes.

[Wolstenholme prime]s

Primes p for which the binomial coefficient
16843, 2124679
, these are the only known Wolstenholme primes.

Woodall primes">Woodall number">Woodall primes

Of the form n×2 − 1.
7, 23, 383, 32212254719, 2833419889721787128217599, 195845982777569926302400511, 4776913109852041418248056622882488319