List Of Special Classes 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.

.

The Goldbach conjecture verification project reports that it has computed all primes below 4×10. That means 95,676,260,903,887,607 primes (nearly 10), 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) below 10. A different computation found that there are 18,435,599,767,349,200,867,866 primes (roughly 2) 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 (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 ().

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. ()

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 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 ()

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 ()

All repunit primes are circular.

Cluster primes

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

3, 5, 7, 11, 13, 17, 19, 23, ...

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) (, )

Cuban primes

Of the form $\tfrac$ 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 $\tfrac$ 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

Of the form ''n''×2 + 1.

3, 393050634124102232869567034555427371542904833 ()

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 ()

Eisenstein primes without imaginary part

Eisenstein integers that are irreducible and real numbers (primes of the form 3''n'' âˆ’ 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 ()

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

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

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

Euler irregular primes

A prime $p$ that divides Euler number $E_$ for some $0\leq 2n\leq p-3$.

19, 31, 43, 47, 61, 67, 71, 79, 101, 137, 139, 149, 193, 223, 241, 251, 263, 277, 307, 311, 349, 353, 359, 373, 379, 419, 433, 461, 463, 491, 509, 541, 563, 571, 577, 587 ()

Euler (''p'', ''p'' − 3) irregular primes

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

149, 241, 2946901 ()

Factorial primes

Of the form ''n'' ! − 1 or ''n'' ! + 1.

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

Fermat primes

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: (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

these are the only known generalized Fermat primes for ''a'' ≤ 24.

Fibonacci primes

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 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 ()

Gaussian primes

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

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

Good primes

Primes ''p'' for which ''p'' > ''p'' ''p'' for all 1 â‰¤ ''i'' â‰¤ ''n''−1, where ''p'' is the ''n''th prime.

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

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 ()

Harmonic primes

Primes ''p'' for which there are no solutions to ''H'' â‰¡ 0 (mod ''p'') and ''H'' â‰¡ âˆ’''ω'' (mod ''p'') for 1 â‰¤ ''k'' â‰¤ ''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 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 ()

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 primes

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 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 ()

(''p'', ''p'' − 3) irregular primes

(See Wolstenholme prime)

(''p'', ''p'' − 5) irregular primes

Primes ''p'' such that (''p'', ''p''−5) is an irregular pair.

37

(''p'', ''p'' − 9) irregular primes

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

67, 877 ()

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 ()

Leyland primes

Of the form ''x'' + ''y'', with 1 < ''x'' < ''y''.

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

Long primes

Primes ''p'' for which, in a given base ''b'', $\frac$ 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 primes

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 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 primes

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: M_82589933, 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, 57885161 ()

, 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 2 âˆ’ 1 for prime ''p''.

7, 127, 2147483647, 170141183460469231731687303715884105727 (primes in )

Generalized repunit primes

Of the form (''a'' âˆ’ 1) / (''a'' âˆ’ 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 ()

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

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

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

''a'' = 7: 2801, 16148168401, 85053461164796801949539541639542805770666392330682673302530819774105141531698707146930307290253537320447270457

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

''a'' = 9: none exist

Other generalizations and variations

Many generalizations of Mersenne primes have been defined. This include the following:
* Primes of the form , including the Mersenne primes and the cuban primes as special cases
* Williams primes, of the form

Mills primes

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

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 primes

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 ''p''². 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 $\frac \pm b \times 10^$ with $0 \le a \pm b < 10$. This means all digits except the middle digit are equal.

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

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

Primes in the Pell number sequence ''P'' = 0, ''P'' = 1,
''P'' = 2''P'' + ''P''.

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

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 ()

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 ()

Pierpont primes

Of the form 23 + 1 for some integers ''u'',''v'' â‰¥ 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 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 ()

Primes of the form ''n''⁴ + 1

Of the form ''n''⁴ + 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 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 ()

Primorial primes

Of the form ''p''# Â± 1.

3, 5, 7, 29, 31, 211, 2309, 2311, 30029, 200560490131, 304250263527209, 23768741896345550770650537601358309 (union of and )

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 primes

Of the form 4''n'' + 1.

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

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) (, , , )

Quartan primes

Of the form ''x'' + ''y'', where ''x'',''y'' > 0.

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

Ramanujan primes

Integers ''R'' that are the smallest to give at least ''n'' primes from ''x''/2 to ''x'' for all ''x'' â‰¥ ''R'' (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 ()

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 ()

Repunit primes

Primes containing only the decimal digit 1.

11, 1111111111111111111 (19 digits), 11111111111111111111111 (23 digits) ()

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

Residue classes of primes

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

The primes of the form 2''n''+1 are the odd primes, including all primes other than 2. Some sequences have alternate names: 4''n''+1 are Pythagorean primes, 4''n''+3 are the integer Gaussian primes, and 6''n''+5 are the Eisenstein primes (with 2 omitted). The classes 10''n''+''d'' (''d'' = 1, 3, 7, 9) are primes ending in the decimal digit ''d''.

2''n''+1: 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53 ()

4''n''+1: 5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101, 109, 113, 137 ()

4''n''+3: 3, 7, 11, 19, 23, 31, 43, 47, 59, 67, 71, 79, 83, 103, 107 ()

6''n''+1: 7, 13, 19, 31, 37, 43, 61, 67, 73, 79, 97, 103, 109, 127, 139 ()

6''n''+5: 5, 11, 17, 23, 29, 41, 47, 53, 59, 71, 83, 89, 101, 107, 113 ()

8''n''+1: 17, 41, 73, 89, 97, 113, 137, 193, 233, 241, 257, 281, 313, 337, 353 ()

8''n''+3: 3, 11, 19, 43, 59, 67, 83, 107, 131, 139, 163, 179, 211, 227, 251 ()

8''n''+5: 5, 13, 29, 37, 53, 61, 101, 109, 149, 157, 173, 181, 197, 229, 269 ()

8''n''+7: 7, 23, 31, 47, 71, 79, 103, 127, 151, 167, 191, 199, 223, 239, 263 ()

10''n''+1: 11, 31, 41, 61, 71, 101, 131, 151, 181, 191, 211, 241, 251, 271, 281 ()

10''n''+3: 3, 13, 23, 43, 53, 73, 83, 103, 113, 163, 173, 193, 223, 233, 263 ()

10''n''+7: 7, 17, 37, 47, 67, 97, 107, 127, 137, 157, 167, 197, 227, 257, 277 ()

10''n''+9: 19, 29, 59, 79, 89, 109, 139, 149, 179, 199, 229, 239, 269, 349, 359 ()

12''n''+1: 13, 37, 61, 73, 97, 109, 157, 181, 193, 229, 241, 277, 313, 337, 349 ()

12''n''+5: 5, 17, 29, 41, 53, 89, 101, 113, 137, 149, 173, 197, 233, 257, 269 ()

12''n''+7: 7, 19, 31, 43, 67, 79, 103, 127, 139, 151, 163, 199, 211, 223, 271 ()

12''n''+11: 11, 23, 47, 59, 71, 83, 107, 131, 167, 179, 191, 227, 239, 251, 263 ()

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 ()

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 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) (, )

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 primes

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

3, 5, 7, 11, 13 ()

Sophie Germain primes

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

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

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 ()

, 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 ()

Supersingular primes

There are exactly fifteen supersingular primes:

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

Thabit primes

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

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) (, , )

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 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) (, )

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 ()

Wagstaff primes

Of the form (2 + 1) / 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 primes

A prime ''p'' > 5, if ''p'' divides the Fibonacci number $F_$, where the Legendre symbol $\left(\frac\right)$ is defined as
:$\left(\frac\right) = \begin 1 &\textrm\;p \equiv \pm1 \pmod 5\\ -1 &\textrm\;p \equiv \pm2 \pmod 5. \end$

, 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 ()

Wieferich primes

Primes ''p'' such that for fixed integer ''a'' > 1.

2^{''p'' − 1} ≡ 1 (mod ''p''²): 1093, 3511 ()

3^{''p'' − 1} ≡ 1 (mod ''p''²): 11, 1006003 ()

4^{''p'' − 1} ≡ 1 (mod ''p''²): 1093, 3511

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

6^{''p'' − 1} ≡ 1 (mod ''p''²): 66161, 534851, 3152573 ()

7^{''p'' − 1} ≡ 1 (mod ''p''²): 5, 491531 ()

8^{''p'' − 1} ≡ 1 (mod ''p''²): 3, 1093, 3511

9^{''p'' − 1} ≡ 1 (mod ''p''²): 2, 11, 1006003

10^{''p'' − 1} ≡ 1 (mod ''p''²): 3, 487, 56598313 ()

11^{''p'' − 1} ≡ 1 (mod ''p''²): 71

12^{''p'' − 1} ≡ 1 (mod ''p''²): 2693, 123653 ()

13^{''p'' − 1} ≡ 1 (mod ''p''²): 2, 863, 1747591 ()

14^{''p'' − 1} ≡ 1 (mod ''p''²): 29, 353, 7596952219 ()

15^{''p'' − 1} ≡ 1 (mod ''p''²): 29131, 119327070011 ()

16^{''p'' − 1} ≡ 1 (mod ''p''²): 1093, 3511

17^{''p'' − 1} ≡ 1 (mod ''p''²): 2, 3, 46021, 48947 ()

18^{''p'' − 1} ≡ 1 (mod ''p''²): 5, 7, 37, 331, 33923, 1284043 ()

19^{''p'' − 1} ≡ 1 (mod ''p''²): 3, 7, 13, 43, 137, 63061489 ()

20^{''p'' − 1} ≡ 1 (mod ''p''²): 281, 46457, 9377747, 122959073 ()

21^{''p'' − 1} ≡ 1 (mod ''p''²): 2

22^{''p'' − 1} ≡ 1 (mod ''p''²): 13, 673, 1595813, 492366587, 9809862296159 ()

23^{''p'' − 1} ≡ 1 (mod ''p''²): 13, 2481757, 13703077, 15546404183, 2549536629329 ()

24^{''p'' − 1} ≡ 1 (mod ''p''²): 5, 25633

25^{''p'' − 1} ≡ 1 (mod ''p''²): 2, 20771, 40487, 53471161, 1645333507, 6692367337, 188748146801

, these are all known Wieferich primes with ''a'' ≤ 25.

Wilson primes

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

5, 13, 563 ()

, these are the only known Wilson primes.

Wolstenholme primes

Primes ''p'' for which the binomial coefficient $\equiv 1 \pmod.$

16843, 2124679 ()

, these are the only known Wolstenholme primes.

Woodall primes

Of the form ''n''×2 − 1.

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

See also

*
*
*
*
*
*
*
*
*
*

References

External links

Lists of Primes
at the Prime Pages.

The Nth Prime Page
Nth prime through n=10^12, pi(x) through x=3*10^13, Random prime in same range.

Prime Numbers List
Full list for prime numbers below 10,000,000,000, partial list for up to 400 digits.


(primes less than 2,000,000,000)
*

Selected prime related sequences
in OEIS.
* Fischer, R
Thema: Fermatquotient B^(P−1)

1 (mod P^2)
(Lists Wieferich primes in all bases up to 1052)
*

{{Prime number classes

*
*
Prime numbers
Prime