List of prime numbers

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. 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.[1]

1234567891011121314151617181920
1–20 235711131719232931374143475359616771
21–40 7379838997101103107109113127131137139149151157163167173
41–60 179181191193197199211223227229233239241251257263269271277281
61–80 283293307311313317331337347349353359367373379383389397401409
81–100 419421431433439443449457461463467479487491499503509521523541
101–120 547557563569571577587593599601607613617619631641643647 653659
121–140 661673677683691701709719727733739743751757761769773787797809
141–160 811821823827829839853857859863877881883887907911919929937941
161–180 947953967971977983991997100910131019102110311033103910491051106110631069
181–200 10871091109310971103110911171123112911511153116311711181118711931201121312171223
201–220 12291231123712491259127712791283128912911297130113031307131913211327136113671373
221–240 13811399140914231427142914331439144714511453145914711481148314871489149314991511
241–260 15231531154315491553155915671571157915831597160116071609161316191621162716371657
261–280 16631667166916931697169917091721172317331741174717531759177717831787178918011811
281–300 18231831184718611867187118731877187918891901190719131931193319491951197319791987
301–320 19931997199920032011201720272029203920532063206920812083208720892099211121132129
321–340 21312137214121432153216121792203220722132221223722392243225122672269227322812287
341–360 22932297230923112333233923412347235123572371237723812383238923932399241124172423
361–380 24372441244724592467247324772503252125312539254325492551255725792591259326092617
381–400 26212633264726572659266326712677268326872689269326992707271127132719272927312741
401–420 27492753276727772789279127972801280328192833283728432851285728612879288728972903
421–440 29092917292729392953295729632969297129993001301130193023303730413049306130673079
441–460 30833089310931193121313731633167316931813187319132033209321732213229325132533257
461–480 32593271329933013307331333193323332933313343334733593361337133733389339134073413
481–500 34333449345734613463346734693491349935113517352735293533353935413547355735593571
501–520 35813583359336073613361736233631363736433659367136733677369136973701370937193727
521–540 37333739376137673769377937933797380338213823383338473851385338633877388138893907
541–560 39113917391939233929393139433947396739894001400340074013401940214027404940514057
561–580 40734079409140934099411141274129413341394153415741594177420142114217421942294231
581–600 42414243425342594261427142734283428942974327433743394349435743634373439143974409
601–620 44214423444144474451445744634481448344934507451345174519452345474549456145674583
621–640 45914597460346214637463946434649465146574663467346794691470347214723472947334751
641–660 47594783478747894793479948014813481748314861487148774889490349094919493149334937
661–680 49434951495749674969497349874993499950035009501150215023503950515059507750815087
681–700 50995101510751135119514751535167517151795189519752095227523152335237526152735279
701–720 52815297530353095323533353475351538153875393539954075413541754195431543754415443
721–740 54495471547754795483550155035507551955215527553155575563556955735581559156235639
741–760 56415647565156535657565956695683568956935701571157175737574157435749577957835791
761–780 58015807581358215827583958435849585158575861586758695879588158975903592359275939
781–800 59535981598760076011602960376043604760536067607360796089609161016113612161316133
801–820 61436151616361736197619962036211621762216229624762576263626962716277628762996301
821–840 63116317632363296337634363536359636163676373637963896397642164276449645164696473
841–860 64816491652165296547655165536563656965716577658165996607661966376653665966616673
861–880 66796689669167016703670967196733673767616763677967816791679368036823682768296833
881–900 68416857686368696871688368996907691169176947694969596961696769716977698369916997
901–920 70017013701970277039704370577069707971037109712171277129715171597177718771937207
921–940 72117213721972297237724372477253728372977307730973217331733373497351736973937411
941–960 74177433745174577459747774817487748974997507751775237529753775417547754975597561
961–980 75737577758375897591760376077621763976437649766976737681768776917699770377177723
981–1000 77277741775377577759778977937817782378297841785378677873787778797883790179077919

(sequence A000040 in the OEIS).

The Goldbach conjecture verification project reports that it has computed all primes below 4×1018.[2] That means 95,676,260,903,887,607 primes[3] (nearly 1017), but they were not stored. There are known formulae to evaluate the prime-counting function (the number of primes below 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 2×1021) below 1023. A different computation found that there are 18,435,599,767,349,200,867,866 primes (roughly 2×1022) below 1024, if the Riemann hypothesis is true.[4]

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.

Balanced primes

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

  • 5, 53, 157, 173, 211, 257, 263, 373, 563, 593, 607, 653, 733, 947, 977, 1103, 1123, 1187, 1223, 1367, 1511, 1747, 1753, 1907, 2287, 2417, 2677, 2903, 2963, 3307, 3313, 3637, 3733, 4013, 4409, 4457, 4597, 4657, 4691, 4993, 5107, 5113, 5303, 5387, 5393 (OEIS: A006562).

Bell primes

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)

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 (OEIS: A109611)

Circular primes

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.

Cluster primes

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

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

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

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

Cousin primes

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)

Cuban primes

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 (OEIS: A002407)

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 (OEIS: A002648)

Cullen primes

Of the form n×2n + 1.

3, 393050634124102232869567034555427371542904833 (OEIS: A050920)

Dihedral primes

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 3n  1).

2, 5, 11, 17, 23, 29, 41, 47, 53, 59, 71, 83, 89, 101, 107, 113, 131, 137, 149, 167, 173, 179, 191, 197, 227, 233, 239, 251, 257, 263, 269, 281, 293, 311, 317, 347, 353, 359, 383, 389, 401 (OEIS: A003627)

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 (OEIS: A006567)

Euclid primes

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

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

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 (OEIS: A120337)

Euler (p, p − 3) irregular primes

Primes such that is an Euler irregular pair.

149, 241, 2946901 (OEIS: A198245)

Factorial primes

Of the form n! − 1 or n! + 1.

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

Fermat primes

Of the form 22n + 1.

3, 5, 17, 257, 65537 (OEIS: A019434)

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

Generalized Fermat primes

Of the form a2n + 1 for fixed integer a.

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

a = 4: 5, 17, 257, 65537

a = 6: 7, 37, 1297

a = 8: (does not exist)

a = 10: 11, 101

a = 12: 13

a = 14: 197

a = 16: 17, 257, 65537

a = 18: 19

a = 20: 401, 160001

a = 22: 23

a = 24: 577, 331777

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

Fibonacci primes

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

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

Fortunate primes

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)

Gaussian primes

Prime elements of the Gaussian integers; equivalently, primes of the form 4n + 3.

3, 7, 11, 19, 23, 31, 43, 47, 59, 67, 71, 79, 83, 103, 107, 127, 131, 139, 151, 163, 167, 179, 191, 199, 211, 223, 227, 239, 251, 263, 271, 283, 307, 311, 331, 347, 359, 367, 379, 383, 419, 431, 439, 443, 463, 467, 479, 487, 491, 499, 503 (OEIS: A002145)

Good primes

Primes pn for which pn2 > pni pn+i for all 1  i  n−1, where pn is the nth prime.

5, 11, 17, 29, 37, 41, 53, 59, 67, 71, 97, 101, 127, 149, 179, 191, 223, 227, 251, 257, 269, 307 (OEIS: A028388)

Happy primes

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)

Harmonic primes

Primes p for which there are no solutions to Hk  0 (mod p) and Hk  ωp (mod p) for 1  k  p−2, where Hk denotes the k-th harmonic number and ωp denotes the Wolstenholme quotient.[7]

5, 13, 17, 23, 41, 67, 73, 79, 107, 113, 139, 149, 157, 179, 191, 193, 223, 239, 241, 251, 263, 277, 281, 293, 307, 311, 317, 331, 337, 349 (OEIS: A092101)

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)

Home primes

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)

Irregular primes

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

(See Wolstenholme prime)

(p, p − 5) irregular primes

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

37

(p, p − 9) irregular primes

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

67, 877 (OEIS: A212557)

Isolated primes

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)

Leyland primes

Of the form xy + yx, with 1 < x < y.

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

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 (OEIS: A001913)

Lucas primes

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

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

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 (OEIS: A031157)

Mersenne primes

Of the form 2n − 1.

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

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

As of 2018, this class of prime numbers also contains the largest known prime: M82589933, the 51st known Mersenne prime.

Mersenne divisors

Primes p that divide 2n − 1, for some prime number n.

3, 7, 23, 31, 47, 89, 127, 167, 223, 233, 263, 359, 383, 431, 439, 479, 503, 719, 839, 863, 887, 983, 1103, 1319, 1367, 1399, 1433, 1439, 1487, 1823, 1913, 2039, 2063, 2089, 2207, 2351, 2383, 2447, 2687, 2767, 2879, 2903, 2999, 3023, 3119, 3167, 3343 (OEIS: A122094)

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

Mersenne prime exponents

Primes p such that 2p − 1 is prime.

2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583, 25964951, 30402457, 32582657, 37156667, 42643801, 43112609, 57885161 (OEIS: A000043)

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

Double Mersenne primes

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

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

Generalized repunit primes

Of the form (an  1) / (a  1) for fixed integer a.

For a = 2, these are the Mersenne primes, while for a = 10 they are the repunit primes. For other small a, they are given below:

a = 3: 13, 1093, 797161, 3754733257489862401973357979128773, 6957596529882152968992225251835887181478451547013 (OEIS: A076481)

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

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

a = 6: 7, 43, 55987, 7369130657357778596659, 3546245297457217493590449191748546458005595187661976371 (OEIS: A165210)

a = 7: 2801, 16148168401, 85053461164796801949539541639542805770666392330682673302530819774105141531698707146930307290253537320447270457

a = 8: 73 (the only prime for a = 8)

a = 9: none exist

Other generalizations and variations

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

Mills primes

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

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

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 (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 p2. Three such primes are known; it is not known whether there are more.[13]

2, 40487, 6692367337 (OEIS: A055578)

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 (OEIS: A002385)

Palindromic wing primes

Primes of the form with .[14] This means all digits except the middle digit are equal.

101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, 797, 919, 929, 11311, 11411, 33533, 77377, 77477, 77977, 1114111, 1117111, 3331333, 3337333, 7772777, 7774777, 7778777, 111181111, 111191111, 777767777, 77777677777, 99999199999 (OEIS: A077798)

Partition 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 (OEIS: A049575)

Pell primes

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

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

Permutable primes

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)

Perrin primes

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)

Pierpont primes

Of the form 2u3v + 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)

Pillai primes

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 n4 + 1

Of the form n4 + 1.[15][16]

2, 17, 257, 1297, 65537, 160001, 331777, 614657, 1336337, 4477457, 5308417, 8503057, 9834497, 29986577, 40960001, 45212177, 59969537, 65610001, 126247697, 193877777, 303595777, 384160001, 406586897, 562448657, 655360001 (OEIS: A037896)

Primeval primes

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)

Primorial primes

Of the form pn# ± 1.

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

Proth primes

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

3, 5, 13, 17, 41, 97, 113, 193, 241, 257, 353, 449, 577, 641, 673, 769, 929, 1153, 1217, 1409, 1601, 2113, 2689, 2753, 3137, 3329, 3457, 4481, 4993, 6529, 7297, 7681, 7937, 9473, 9601, 9857 (OEIS: A080076)

Pythagorean primes

Of the form 4n + 1.

5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101, 109, 113, 137, 149, 157, 173, 181, 193, 197, 229, 233, 241, 257, 269, 277, 281, 293, 313, 317, 337, 349, 353, 373, 389, 397, 401, 409, 421, 433, 449 (OEIS: A002144)

Prime quadruplets

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)

Quartan primes

Of the form x4 + y4, where x,y > 0.

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

Ramanujan primes

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

Regular primes

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)

Repunit primes

Primes containing only the decimal digit 1.

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

The next have 317, 1031, 49081, 86453, 109297, 270343 digits (OEIS: A004023)

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 (with 2 omitted). The classes 10n+d (d = 1, 3, 7, 9) are primes ending in the decimal digit d.

2n+1: 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53 (OEIS: A065091)
4n+1: 5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101, 109, 113, 137 (OEIS: A002144)
4n+3: 3, 7, 11, 19, 23, 31, 43, 47, 59, 67, 71, 79, 83, 103, 107 (OEIS: A002145)
6n+1: 7, 13, 19, 31, 37, 43, 61, 67, 73, 79, 97, 103, 109, 127, 139 (OEIS: A002476)
6n+5: 5, 11, 17, 23, 29, 41, 47, 53, 59, 71, 83, 89, 101, 107, 113 (OEIS: A007528)
8n+1: 17, 41, 73, 89, 97, 113, 137, 193, 233, 241, 257, 281, 313, 337, 353 (OEIS: A007519)
8n+3: 3, 11, 19, 43, 59, 67, 83, 107, 131, 139, 163, 179, 211, 227, 251 (OEIS: A007520)
8n+5: 5, 13, 29, 37, 53, 61, 101, 109, 149, 157, 173, 181, 197, 229, 269 (OEIS: A007521)
8n+7: 7, 23, 31, 47, 71, 79, 103, 127, 151, 167, 191, 199, 223, 239, 263 (OEIS: A007522)
10n+1: 11, 31, 41, 61, 71, 101, 131, 151, 181, 191, 211, 241, 251, 271, 281 (OEIS: A030430)
10n+3: 3, 13, 23, 43, 53, 73, 83, 103, 113, 163, 173, 193, 223, 233, 263 (OEIS: A030431)
10n+7: 7, 17, 37, 47, 67, 97, 107, 127, 137, 157, 167, 197, 227, 257, 277 (OEIS: A030432)
10n+9: 19, 29, 59, 79, 89, 109, 139, 149, 179, 199, 229, 239, 269, 349, 359 (OEIS: A030433)
12n+1: 13, 37, 61, 73, 97, 109, 157, 181, 193, 229, 241, 277, 313, 337, 349 (OEIS: A068228)
12n+5: 5, 17, 29, 41, 53, 89, 101, 113, 137, 149, 173, 197, 233, 257, 269 (OEIS: A040117)
12n+7: 7, 19, 31, 43, 67, 79, 103, 127, 139, 151, 163, 199, 211, 223, 271 (OEIS: A068229)
12n+11: 11, 23, 47, 59, 71, 83, 107, 131, 167, 179, 191, 227, 239, 251, 263 (OEIS: A068231)

Safe primes

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)

Sexy primes

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.

Solinas primes

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

3, 5, 7, 11, 13 (OEIS: A165255)

Sophie Germain primes

Where p and 2p + 1 are both prime. A Sophie Germain prime has a corresponding safe prime.

2, 3, 5, 11, 23, 29, 41, 53, 83, 89, 113, 131, 173, 179, 191, 233, 239, 251, 281, 293, 359, 419, 431, 443, 491, 509, 593, 641, 653, 659, 683, 719, 743, 761, 809, 911, 953 (OEIS: A005384)

Stern primes

Primes that are not the sum of a smaller prime and twice the square of a nonzero integer.

2, 3, 17, 137, 227, 977, 1187, 1493 (OEIS: A042978)

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

Super-primes

Primes with a prime index in the sequence of prime numbers (the 2nd, 3rd, 5th, ... prime).

3, 5, 11, 17, 31, 41, 59, 67, 83, 109, 127, 157, 179, 191, 211, 241, 277, 283, 331, 353, 367, 401, 431, 461, 509, 547, 563, 587, 599, 617, 709, 739, 773, 797, 859, 877, 919, 967, 991 (OEIS: A006450)

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)

Thabit primes

Of the form 3×2n − 1.

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

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

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

Prime triplets

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)

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 (OEIS: A024785)

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 (OEIS: A024770)

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 (OEIS: A020994)

Twin primes

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)

Unique primes

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)

Wagstaff primes

Of the form (2n + 1) / 3.

3, 11, 43, 683, 2731, 43691, 174763, 2796203, 715827883, 2932031007403, 768614336404564651, 201487636602438195784363, 845100400152152934331135470251, 56713727820156410577229101238628035243 (OEIS: A000979)

Values of n:

3, 5, 7, 11, 13, 17, 19, 23, 31, 43, 61, 79, 101, 127, 167, 191, 199, 313, 347, 701, 1709, 2617, 3539, 5807, 10501, 10691, 11279, 12391, 14479, 42737, 83339, 95369, 117239, 127031, 138937, 141079, 267017, 269987, 374321 (OEIS: A000978)

Wall–Sun–Sun primes

A prime p > 5, if p2 divides the Fibonacci number , where the Legendre symbol is defined as

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

Weakly prime numbers

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)

Wieferich primes

Primes p such that ap − 1 1 (mod p2) for fixed integer a > 1.

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

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

Wilson primes

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

5, 13, 563 (OEIS: A007540)

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

Wolstenholme primes

Primes p for which the binomial coefficient

16843, 2124679 (OEIS: A088164)

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

Woodall primes

Of the form n×2n − 1.

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

See also

References

  1. Lehmer, D. N. (1982). List of prime numbers from 1 to 10,006,721. Vol. 165. Washington D.C.: Carnegie Institution of Washington. OL 16553580M. OL16553580M.
  2. Tomás Oliveira e Silva, Goldbach conjecture verification Archived 24 May 2011 at the Wayback Machine. Retrieved 16 July 2013
  3. (sequence A080127 in the OEIS)
  4. Jens Franke (29 July 2010). "Conditional Calculation of pi(1024)". Archived from the original on 24 August 2014. Retrieved 17 May 2011.
  5. OEIS: A018239 includes 2 = empty product of first 0 primes plus 1, but 2 is excluded in this list.
  6. Boklan, Kent D.; Conway, John H. (2016). "Expect at most one billionth of a new Fermat Prime!". arXiv:1605.01371 [math.NT].
  7. 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.
  8. Johnson, W. (1975). "Irregular Primes and Cyclotomic Invariants". Mathematics of Computation. AMS. 29 (129): 113–120. doi:10.2307/2005468. JSTOR 2005468.
  9. It varies whether L0 = 2 is included in the Lucas numbers.
  10. 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.
  11. Sloane, N. J. A. (ed.). "Sequence A121616 (Primes of form (n+1)^5 - n^5)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  12. 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.
  13. Paszkiewicz, Andrzej (2009). "A new prime for which the least primitive root and the least primitive root are not equal" (PDF). Math. Comp. American Mathematical Society. 78 (266): 1193–1195. Bibcode:2009MaCom..78.1193P. doi:10.1090/S0025-5718-08-02090-5.
  14. Caldwell, C.; Dubner, H. (1996–97). "The near repdigit primes , especially ". Journal of Recreational Mathematics. 28 (1): 1–9.
  15. Lal, M. (1967). "Primes of the Form n4 + 1" (PDF). Mathematics of Computation. AMS. 21: 245–247. doi:10.1090/S0025-5718-1967-0222007-9. ISSN 1088-6842. Archived (PDF) from the original on 13 January 2015.
  16. Bohman, J. (1973). "New primes of the form n4 + 1". BIT Numerical Mathematics. Springer. 13 (3): 370–372. doi:10.1007/BF01951947. ISSN 1572-9125. S2CID 123070671.
  17. Ribenboim, P. (22 February 1996). The new book of prime number records. New York: Springer-Verlag. p. 347. ISBN 0-387-94457-5.
  18. "Mirimanoff's Congruence: Other Congruences". Retrieved 26 January 2011.
  19. Gallot, Y.; Moree, P.; Zudilin, W. (2011). "The Erdös-Moser equation 1k + 2k +...+ (m−1)k = mk revisited using continued fractions". Mathematics of Computation. American Mathematical Society. 80: 1221–1237. arXiv:0907.1356. doi:10.1090/S0025-5718-2010-02439-1. S2CID 16305654.
  20. Ribenboim, P. (2006). Die Welt der Primzahlen (PDF). Berlin: Springer. p. 240. ISBN 3-540-34283-4.
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