List of prime numbers

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A prime number is a number that cannot be divided by a number other than 1 and itself. By Euclid's theorem, there is an infinite number of prime numbers. Subsets of the prime numbers may be generated with various formulas for primes. The first 500 primes are listed below, followed by lists of notable types of prime numbers in alphabetical order, giving their respective first terms.

Contents

The first 500 prime numbers[edit]

The following table lists the first 500 primes; 20 columns of consecutive primes in each of the 25 rows.[1]

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

(sequence A000040 in 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[edit]

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.

Additive primes[edit]

Primes such that the sum of digits is a prime.

2, 3, 5, 7, 11, 23, 29, 41, 43, 47, 61, 67, 83, 89, 101, 113, 131 (OEISA046704)

Annihilating primes[edit]

Let d(p) be the shadow of the sequence f(n) = seq1-1(n) (which gives the number of sequences without repetitions that can be obtained from n distinct objects), i.e. the count of sequence entries f(0), f(1), f(2), ...., f(h-1) divisible by an integer h. If d(p) = 0, then p is an annihilating prime.[5]

3, 7, 11, 17, 47, 53, 61, 67, 73, 79, 89, 101, 139, 151, 157, 191, 199 (OEISA072456)

Bell number primes[edit]

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. (OEISA051131)

Carol primes[edit]

Of the form (2n−1)2 − 2.

7, 47, 223, 3967, 16127, 1046527, 16769023, 1073676287, 68718952447, 274876858367, 4398042316799, 1125899839733759, 18014398241046527, 1298074214633706835075030044377087 (OEISA091516)

Centered decagonal primes[edit]

Of the form 5(n2 − n) + 1.

11, 31, 61, 101, 151, 211, 281, 661, 911, 1051, 1201, 1361, 1531, 1901, 2311, 2531, 3001, 3251, 3511, 4651, 5281, 6301, 6661, 7411, 9461, 9901, 12251, 13781, 14851, 15401, 18301, 18911, 19531, 20161, 22111, 24151, 24851, 25561, 27011, 27751 (OEISA090562)

Centered heptagonal primes[edit]

Of the form (7n2 − 7n + 2) / 2.

43, 71, 197, 463, 547, 953, 1471, 1933, 2647, 2843, 3697, 4663, 5741, 8233, 9283, 10781, 11173, 12391, 14561, 18397, 20483, 29303, 29947, 34651, 37493, 41203, 46691, 50821, 54251, 56897, 57793, 65213, 68111, 72073, 76147, 84631, 89041, 93563 (primes in OEISA069099)

Centered square primes[edit]

Of the form n2 + (n+1)2.

5, 13, 41, 61, 113, 181, 313, 421, 613, 761, 1013, 1201, 1301, 1741, 1861, 2113, 2381, 2521, 3121, 3613, 4513, 5101, 7321, 8581, 9661, 9941, 10513, 12641, 13613, 14281, 14621, 15313, 16381, 19013, 19801, 20201, 21013, 21841, 23981, 24421, 26681 (OEISA027862)

Centered triangular primes[edit]

Of the form (3n2 + 3n + 2) / 2.

19, 31, 109, 199, 409, 571, 631, 829, 1489, 1999, 2341, 2971, 3529, 4621, 4789, 7039, 7669, 8779, 9721, 10459, 10711, 13681, 14851, 16069, 16381, 17659, 20011, 20359, 23251, 25939, 27541, 29191, 29611, 31321, 34429, 36739, 40099, 40591, 42589 (OEISA125602)

Chen primes[edit]

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 (OEISA109611)

Circular primes[edit]

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 (OEISA068652)

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 (OEISA016114)

All repunit primes are circular.

Cousin primes[edit]

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) (OEISA023200, OEISA046132)

Cuban primes[edit]

Of the form \tfrac{x^3-y^3}{x-y}, 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 (OEISA002407)

Of the form \tfrac{x^3-y^3}{x-y}, 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 (OEISA002648)

Cullen primes[edit]

Of the form n×2n + 1.

3, 393050634124102232869567034555427371542904833 (OEISA050920)

Dihedral primes[edit]

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 (OEISA134996)

Double factorial primes[edit]

Of the form n!! + 1. Values of n:

0, 1, 2, 518, 33416, 37310, 52608 (OEISA080778)

Note that n = 0 and n = 1 produce the same prime, namely 2.

Of the form n!! − 1. Values of n:

3, 4, 6, 8, 16, 26, 64, 82, 90, 118, 194, 214, 728, 842, 888, 2328, 3326, 6404, 8670, 9682, 27056, 44318 (OEISA007749)

Double Mersenne primes[edit]

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

7, 127, 2147483647, 170141183460469231731687303715884105727 (primes in OEISA077586)

As of 2011, these are the only known double Mersenne primes, and number theorists think these are probably the only double Mersenne primes.

Eisenstein primes without imaginary part[edit]

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 (OEISA003627)

Emirps[edit]

Primes which become a different prime when their decimal digits are reversed.

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 (OEISA006567)

Euclid primes[edit]

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

3, 7, 31, 211, 2311, 200560490131 (OEISA018239[6])

Even prime[edit]

Of the form 2n.

2

The only even prime is 2. It is therefore sometimes called "the oddest prime" as a pun on the non-mathematical meaning of "odd".[7]

Factorial primes[edit]

Of the form n! − 1 or n! + 1.

2, 3, 5, 7, 23, 719, 5039, 39916801, 479001599, 87178291199, 10888869450418352160768000001, 265252859812191058636308479999999, 263130836933693530167218012159999999, 8683317618811886495518194401279999999 (OEISA088054)

Fermat primes[edit]

Of the form 22n + 1.

3, 5, 17, 257, 65537 (OEISA019434)

As of 2013 these are the only known Fermat primes, and conjecturally the only Fermat primes.

Fibonacci primes[edit]

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 (OEISA005478)

Fortunate primes[edit]

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 (OEISA046066)

Gaussian primes[edit]

Prime elements of the Gaussian integers (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 (OEISA002145)

Generalized Fermat primes base 10[edit]

Of the form 10n + 1, where n > 0.

11, 101

As of April 2011, these are the only known generalized Fermat primes in base 10.[8]

Genocchi number primes[edit]

17

The only positive prime Genocchi number is 17.[9]

Gilda's primes[edit]

Gilda's numbers that are prime. A number n is a Gilda's number, if when a Fibonacci sequence is formed with the first term equal to the absolute value of the successive differences between consecutive digits of n and the second term equal to the sum of the decimal digits of n, n itself appears as a term in this Fibonacci sequence.[10]

29, 683, 997, 2207, 30571351 (OEISA046850; another entry OEISA135995 is erroneous)

Good primes[edit]

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 (OEISA028388)

Happy primes[edit]

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 (OEISA035497)

Harmonic primes[edit]

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.[11]

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 (OEISA092101)

Higgs primes for squares[edit]

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 (OEISA007459)

Highly cototient number primes[edit]

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 (OEISA105440)

Irregular primes[edit]

Odd primes p which 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, 617, 619 (OEISA000928)

(p, p−5) irregular primes[edit]

Primes p such that (p, p−5) is an irregular pair.[12]

37

(p, p−9) irregular primes[edit]

Primes p such that (p, p−9) is an irregular pair.[12]

67, 877 (OEISA212557)

Isolated primes[edit]

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 (OEISA007510)

Kynea primes[edit]

Of the form (2n + 1)2 − 2.

2, 7, 23, 79, 1087, 66047, 263167, 16785407, 1073807359, 17180131327, 68720001023, 4398050705407, 70368760954879, 18014398777917439, 18446744082299486207 (OEISA091514)

Left-truncatable primes[edit]

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 (OEISA024785)

Leyland primes[edit]

Of the form xy + yx, with 1 < x ≤ y.

17, 593, 32993, 2097593, 8589935681, 59604644783353249, 523347633027360537213687137, 43143988327398957279342419750374600193 (OEISA094133)

Long primes[edit]

Primes p for which, in a given base b, \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 (OEISA001913)

Lucas primes[edit]

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

2,[13] 3, 7, 11, 29, 47, 199, 521, 2207, 3571, 9349, 3010349, 54018521, 370248451, 6643838879, 119218851371, 5600748293801, 688846502588399, 32361122672259149 (OEISA005479)

Lucky primes[edit]

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 (OEISA031157)

Markov primes[edit]

Primes p for which there exist integers x and y such that x2 + y2 + p2 = 3xyp.

2, 5, 13, 29, 89, 233, 433, 1597, 2897, 5741, 7561, 28657, 33461, 43261, 96557, 426389, 514229, 1686049, 2922509, 3276509, 94418953, 321534781, 433494437, 780291637, 1405695061, 2971215073, 19577194573, 25209506681 (primes in OEISA002559)

Mersenne primes[edit]

Of the form 2n − 1.

3, 7, 31, 127, 8191, 131071, 524287, 2147483647, 2305843009213693951, 618970019642690137449562111, 162259276829213363391578010288127, 170141183460469231731687303715884105727 (OEISA000668)

As of 2013, there are 48 known Mersenne primes. The 13th, 14th, and 48th have respectively 157, 183, and 17,425,170 digits.

Mersenne prime exponents[edit]

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 (OEISA000043)

As of March 2014 five more are known to be in the sequence but it is not known whether they are the next:
32582657, 37156667, 42643801, 43112609, 57885161

Mills primes[edit]

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

2, 11, 1361, 2521008887, 16022236204009818131831320183 (OEISA051254)

Minimal primes[edit]

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 (OEISA071062)

Motzkin primes[edit]

Primes that are the number of different ways of drawing non-intersecting chords on a circle between n points.

2, 127, 15511, 953467954114363 (OEISA092832)

Newman–Shanks–Williams primes[edit]

Newman–Shanks–Williams numbers that are prime.

7, 41, 239, 9369319, 63018038201, 489133282872437279, 19175002942688032928599 (OEISA088165)

Non-generous primes[edit]

Primes p for which the least positive primitive root is not a primitive root of p2.

2, 40487, 6692367337 (OEISA055578)

Odd primes[edit]

Of the form 2n − 1.

3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199... (OEISA065091)

All prime numbers except 2 are odd.

Padovan primes[edit]

Primes in the Padovan sequence P(0) = P(1) = P(2) = 1, P(n) = P(n−2) + P(n−3).

2, 3, 5, 7, 37, 151, 3329, 23833, 13091204281, 3093215881333057, 1363005552434666078217421284621279933627102780881053358473 (OEISA100891)

Palindromic primes[edit]

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 (OEISA002385)

Palindromic wing primes[edit]

Primes of the form \frac{a \big( 10^m-1 \big)}{9} \pm b \times 10^{\frac{m}{2}} with 0 \le a \pm b < 10.[14]

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 (OEISA077798)

Partition primes[edit]

Partition numbers that are prime.

2, 3, 5, 7, 11, 101, 17977, 10619863, 6620830889, 80630964769, 228204732751, 1171432692373, 1398341745571, 10963707205259, 15285151248481, 10657331232548839, 790738119649411319, 18987964267331664557 (OEISA049575)

Pell primes[edit]

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 (OEISA086383)

Permutable primes[edit]

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 (OEISA003459)

It seems likely that all further permutable primes are repunits, i.e. contain only the digit 1.

Perrin primes[edit]

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 (OEISA074788)

Pierpont primes[edit]

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 (OEISA005109)

Pillai primes[edit]

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 (OEISA063980)

Primes of the form n4 + 1[edit]

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 (OEISA037896)

Primeval primes[edit]

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 (OEISA119535)

Primorial primes[edit]

Of the form pn# ± 1.

3, 5, 7, 29, 31, 211, 2309, 2311, 30029, 200560490131, 304250263527209, 23768741896345550770650537601358309 (union of OEISA057705 and OEISA018239[6])

Proth primes[edit]

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 (OEISA080076)

Pythagorean primes[edit]

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 (OEISA002144)

Prime quadruplets[edit]

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) (OEISA007530, OEISA136720, OEISA136721, OEISA090258)

Primes of binary quadratic form[edit]

Of the form x2 + xy + 2y2, with non-negative integers x and y.

2, 11, 23, 37, 43, 53, 71, 79, 107, 109, 127, 137, 149, 151, 163, 193, 197, 211, 233, 239, 263, 281, 317, 331, 337, 373, 389, 401, 421, 431, 443, 463, 487, 491, 499, 541, 547, 557, 569, 599, 613, 617, 641, 653, 659, 673, 683, 739, 743, 751, 757, 809, 821 (OEISA106856)

Quartan primes[edit]

Of the form x4 + y4, where x,y > 0.

2, 17, 97, 257, 337, 641, 881 (OEISA002645)

Ramanujan primes[edit]

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 (OEISA104272)

Regular primes[edit]

Primes p which 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 (OEISA007703)

Repunit primes[edit]

Primes containing only the decimal digit 1.

11, 1111111111111111111, 11111111111111111111111 (OEISA004022)

The next have 317 and 1,031 digits.

Primes in residue classes[edit]

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

Three cases have their own entry: 2n+1 are the odd primes, 4n+1 are Pythagorean primes, 4n+3 are the integer Gaussian primes.

2n+1: 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53 (OEISA065091)
4n+1: 5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101, 109, 113, 137 (OEISA002144)
4n+3: 3, 7, 11, 19, 23, 31, 43, 47, 59, 67, 71, 79, 83, 103, 107 (OEISA002145)
6n+1: 7, 13, 19, 31, 37, 43, 61, 67, 73, 79, 97, 103, 109, 127, 139 (OEISA002476)
6n+5: 5, 11, 17, 23, 29, 41, 47, 53, 59, 71, 83, 89, 101, 107, 113 (OEISA007528)
8n+1: 17, 41, 73, 89, 97, 113, 137, 193, 233, 241, 257, 281, 313, 337, 353 (OEISA007519)
8n+3: 3, 11, 19, 43, 59, 67, 83, 107, 131, 139, 163, 179, 211, 227, 251 (OEISA007520)
8n+5: 5, 13, 29, 37, 53, 61, 101, 109, 149, 157, 173, 181, 197, 229, 269 (OEISA007521)
8n+7: 7, 23, 31, 47, 71, 79, 103, 127, 151, 167, 191, 199, 223, 239, 263 (OEISA007522)
10n+1: 11, 31, 41, 61, 71, 101, 131, 151, 181, 191, 211, 241, 251, 271, 281 (OEISA030430)
10n+3: 3, 13, 23, 43, 53, 73, 83, 103, 113, 163, 173, 193, 223, 233, 263 (OEISA030431)
10n+7: 7, 17, 37, 47, 67, 97, 107, 127, 137, 157, 167, 197, 227, 257, 277 (OEISA030432)
10n+9: 19, 29, 59, 79, 89, 109, 139, 149, 179, 199, 229, 239, 269, 349, 359 (OEISA030433)
...

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

Right-truncatable primes[edit]

Primes that remain prime when the last 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 (OEISA024770)

Safe primes[edit]

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 (OEISA005385)

Self primes in base 10[edit]

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 (OEISA006378)

Sexy primes[edit]

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) (OEISA023201, OEISA046117)

Smarandache–Wellin primes[edit]

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

2, 23, 2357 (OEISA069151)

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

Solinas primes[edit]

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

3, 5, 7, 11, 13 (OEISA165255)

Sophie Germain primes[edit]

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 (OEISA005384)

Star primes[edit]

Of the form 6n(n − 1) + 1.

13, 37, 73, 181, 337, 433, 541, 661, 937, 1093, 2053, 2281, 2521, 3037, 3313, 5581, 5953, 6337, 6733, 7561, 7993, 8893, 10333, 10837, 11353, 12421, 12973, 13537, 15913, 18481 (OEISA083577)

Stern primes[edit]

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 (OEISA042978)

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

Super-primes[edit]

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 (OEISA006450)

Supersingular primes[edit]

There are exactly fifteen supersingular primes:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59, 71 (OEISA002267)

Swinging primes[edit]

Primes of the form n \wr \pm 1, where n \wr denotes the swinging factorial, which is defined in terms of the double swinging factorial as[17] n \wr = (n-1) \wr \wr n \wr \wr and n \wr \wr = \begin{cases} 1 \qquad \qquad \qquad \qquad \qquad \qquad \quad \ n \leqslant 0 \\ (n-2) \wr \wr n^{\big[ \text{n odd} \big]} (4/n)^{\big[ \text{n even} \big]} \quad n > 0 \end{cases}

2, 3, 5, 7, 19, 29, 31, 71, 139, 251, 631, 3433, 12011 (OEISA163074)

Thabit number primes[edit]

Of the form 3×2n − 1.

2, 5, 11, 23, 47, 191, 383, 6143, 786431, 51539607551, 824633720831, 26388279066623, 108086391056891903, 55340232221128654847, 226673591177742970257407 (OEISA007505)

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

7, 13, 97, 193, 769, 12289, 786433, 3221225473, 206158430209, 6597069766657 (OEISA039687)

Prime triplets[edit]

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) (OEISA007529, OEISA098414, OEISA098415)

Twin primes[edit]

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) (OEISA001359, OEISA006512)

Two-sided primes[edit]

Primes which 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 (OEISA020994)

Ulam number primes[edit]

Ulam numbers that are prime.

2, 3, 11, 13, 47, 53, 97, 131, 197, 241, 409, 431, 607, 673, 739, 751, 983, 991, 1103, 1433, 1489, 1531, 1553, 1709, 1721, 2371, 2393, 2447, 2633, 2789, 2833, 2897 (OEISA068820)

Unique primes[edit]

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 (OEISA040017)

Wagstaff primes[edit]

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

3, 11, 43, 683, 2731, 43691, 174763, 2796203, 715827883, 2932031007403, 768614336404564651, 201487636602438195784363, 845100400152152934331135470251, 56713727820156410577229101238628035243 (OEISA000979)

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 (OEISA000978)

Wall–Sun–Sun primes[edit]

A prime p > 5 if p2 divides the Fibonacci number F_{p - \left(\frac{{p}}{{5}}\right)}, where the Legendre symbol \left(\frac{{p}}{{5}}\right) is defined as

\left(\frac{p}{5}\right) = \begin{cases} 1 &\textrm{if}\;p \equiv \pm1 \pmod 5\\ -1 &\textrm{if}\;p \equiv \pm2 \pmod 5. \end{cases}

As of 2013, no Wall-Sun-Sun primes are known.

Wedderburn-Etherington number primes[edit]

Wedderburn-Etherington numbers that are prime.

2, 3, 11, 23, 983, 2179, 24631, 3626149, 253450711, 596572387 (primes in OEISA001190)

Weakly prime numbers[edit]

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 (OEISA050249)

Wieferich primes[edit]

Primes p such that ap − 1 ≡ 1 (mod p2) where a is not a perfect power.

2p − 1 ≡ 1 (mod p2): 1093, 3511 (OEISA001220)
3p − 1 ≡ 1 (mod p2): 11, 1006003 (OEISA014127)[18][19][20]
5p − 1 ≡ 1 (mod p2): 2, 20771, 40487, 53471161, 1645333507, 6692367337, 188748146801 (OEISA123692)
6p − 1 ≡ 1 (mod p2): 66161, 534851, 3152573 (OEISA212583)
7p − 1 ≡ 1 (mod p2): 5, 491531 (OEISA123693)
10p − 1 ≡ 1 (mod p2): 3, 487, 56598313 (OEISA045616)
11p − 1 ≡ 1 (mod p2): 71[21]
12p − 1 ≡ 1 (mod p2): 2693, 123653 (OEISA111027)
13p − 1 ≡ 1 (mod p2): 863, 1747591 (OEISA128667)[21]
17p − 1 ≡ 1 (mod p2): 2, 3, 46021, 48947 (OEISA128668)[21]
19p − 1 ≡ 1 (mod p2): 3, 7, 13, 43, 137, 63061489 (OEISA090968)[21]

Wilson primes[edit]

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

5, 13, 563 (OEISA007540)

As of 2011, these are the only known Wilson primes.

Wolstenholme primes[edit]

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

16843, 2124679 (OEISA088164)

As of 2011, these are the only known Wolstenholme primes.

Woodall primes[edit]

Of the form n×2n − 1.

7, 23, 383, 32212254719, 2833419889721787128217599, 195845982777569926302400511, 4776913109852041418248056622882488319 (OEISA050918)

Largest primes smaller than 2n[edit]

The following table lists the 5 largest primes smaller than 2n. In computer science hash tables are often roughly doubled in size when they get to full. On the other hand a size that is prime makes the most out of the information contained in a hash resulting in less collisions. Lastly adding some randomness to the size prevents targeted attacks with carefully created values to cause excessive collisions in the hash table.

2nprime 1prime 2prime 3prime 4prime 5
243
38753
4161311753
5323129231917
6646159534743
7128127113109107103
8256251241239233229
9512509503499491487
1010241021101910131009997
11204820392029202720172011
12409640934091407940734057
13819281918179817181678161
14163841638116369163631636116349
15327683274932719327173271332707
16655366552165519654976547965449
17131072131071131063131059131041131023
18262144262139262133262127262121262111
19524288524287524269524261524257524243
20104857610485731048571104855910485491048517
21209715220971432097133209713120970972097091
22419430441943014194287419427741942714194247
23838860883885938388587838858183885718388547
24167772161677721316777199167771831677715316777141
25335544323355439333554383335543713355434733554341
26671088646710885967108837671088196710877767108763
27134217728134217689134217649134217617134217613134217593
28268435456268435399268435367268435361268435337268435331
29536870912536870909536870879536870869536870849536870839
30107374182410737417891073741783107374174110737417231073741719
31214748364821474836472147483629214748358721474835792147483563
32429496729642949672954294967291429496727942949672314294967197
33858993459285899345918589934589858993458785899345858589934581
34171798691841717986918317179869181171798691791717986917717179869175
35343597383683435973836734359738365343597383633435973836134359738359
36687194767366871947673568719476733687194767296871947672768719476725
37137438953472137438953471137438953469137438953467137438953465137438953463
38274877906944274877906943274877906941274877906939274877906937274877906935
39549755813888549755813887549755813885549755813883549755813879549755813877
40109951162777610995116277751099511627773109951162777110995116277691099511627767
41219902325555221990232555512199023255549219902325554721990232555452199023255543
42439804651110443980465111034398046511101439804651109943980465110974398046511095
43879609302220887960930222078796093022205879609302220387960930222018796093022199
44175921860444161759218604441517592186044413175921860444111759218604440917592186044407
45351843720888323518437208883135184372088829351843720888273518437208882535184372088823
46703687441776647036874417766370368744177661703687441776597036874417765770368744177655
47140737488355328140737488355327140737488355325140737488355323140737488355321140737488355319
48281474976710656281474976710655281474976710653281474976710651281474976710649281474976710647
49562949953421312562949953421311562949953421309562949953421307562949953421305562949953421303
50112589990684262411258999068426231125899906842621112589990684261911258999068426171125899906842615
51225179981368524822517998136852472251799813685245225179981368524322517998136852412251799813685239
52450359962737049645035996273704954503599627370493450359962737049145035996273704894503599627370487
53900719925474099290071992547409919007199254740989900719925474098790071992547409859007199254740983
54180143985094819841801439850948198318014398509481981180143985094819791801439850948197718014398509481975
55360287970189639683602879701896396736028797018963965360287970189639633602879701896396136028797018963959
56720575940379279367205759403792793572057594037927933720575940379279297205759403792792772057594037927925
57144115188075855872144115188075855871144115188075855869144115188075855867144115188075855865144115188075855863
58288230376151711744288230376151711743288230376151711741288230376151711739288230376151711737288230376151711735
59576460752303423488576460752303423487576460752303423485576460752303423483576460752303423481576460752303423479
60115292150460684697611529215046068469751152921504606846973115292150460684697111529215046068469691152921504606846967
61230584300921369395223058430092136939492305843009213693947230584300921369394523058430092136939432305843009213693941
62461168601842738790446116860184273879034611686018427387901461168601842738789946116860184273878974611686018427387895
63922337203685477580892233720368547758079223372036854775805922337203685477580392233720368547758019223372036854775799

See also[edit]

Notes[edit]

  1. ^ Lehmer, D. N. (1982). List of prime numbers from 1 to 10,006,721 165. Washington D.C.: Carnegie Institution of Washington. OL16553580M. 
  2. ^ Tomás Oliveira e Silva, Goldbach conjecture verification. Retrieved 16 July 2013
  3. ^ (sequence A080127 in OEIS)
  4. ^ Jens Franke (29 July 2010). "Conditional Calculation of pi(1024)". Retrieved 2011-05-17. 
  5. ^ L. Halbeisen, N. Hungerbühler, Number theoretic aspects of a combinatorial function
  6. ^ a b OEISA018239 includes 2 = empty product of first 0 primes plus 1, but 2 is excluded in this list.
  7. ^ http://mathworld.wolfram.com/OddPrime.html
  8. ^ Caldwell, C.; Honaker, Jr., G. L. "101". Prime Curios!. Retrieved 1 April 2011. 
  9. ^ Weisstein, Eric W., "Genocchi Number", MathWorld.
  10. ^ Russo, F., A Set of New Samarandache Functions, Sequences and Conjectures in Number Theory, pp. 73–74 
  11. ^ 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.  edit
  12. ^ a b Johnson, W. (1975). "Irregular Primes and Cyclotomic Invariants". Mathematics of Computation (AMS) 29 (129): 113–120. doi:10.2307/2005468. 
  13. ^ It varies whether L0 = 2 is included in the Lucas numbers.
  14. ^ Caldwell, C.; Dubner, H. (1996–97). "The near repdigit primes A_{n-k-1}B_1A_k, especially 9_{n-k-1}8_19_k". Journal of Recreational Mathematics 28 (1): 1–9. 
  15. ^ Lal, M. (1967). "Primes of the Form n4 + 1". Mathematics of Computation (AMS) 21: 245–247. doi:10.1090/S0025-5718-1967-0222007-9. ISSN 1088-6842. 
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  20. ^ 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. 
  21. ^ a b c d Ribenboim, P. (2006). Die Welt der Primzahlen. Berlin: Springer. p. 240. ISBN 3-540-34283-4. 

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