List of prime numbers
This is a list of articles about prime numbers. A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. By Euclid's theorem, there are an infinite number of prime numbers. Subsets of the prime numbers may be generated with various formulas for primes.
The first 1,000 primes are listed below, followed by lists of notable types of prime numbers in alphabetical order, giving their respective first terms. The number 1 is neither prime nor composite.
The first 1,000 prime numbers
The following table lists the first 1,000 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 smaller than 4×10. That means 95,676,260,903,887,607 primes, but they were not stored. There are known formulae to evaluate the prime-counting function faster than computing the primes. This has been used to compute that there are 1,925,320,391,606,803,968,923 primes smaller than 10. A different computation found that there are 18,435,599,767,349,200,867,866 primes smaller than 10, if the Riemann hypothesis is true.
Lists of primes by type
Below are listed the first prime numbers of many named forms and types. More details are in the article for the name. n is a natural number in the definitions.Balanced primes
Balanced primes are primes with equal-sized prime gaps before and after them, making them the arithmetic mean of their next larger and next smaller prime.- 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
Bell primes are primes that are also the number of partitions of some finite set.2, 5, 877, 27644437, 35742549198872617291353508656626642567, 359334085968622831041960188598043661065388726959079837.
The next term has 6,539 digits.
Chen primes
Chen primes are primes p such that 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 is a number that remains prime on any cyclic rotation of its base 10 digits.2, 3, 5, 7, 11, 13, 17, 31, 37, 71, 73, 79, 97, 113, 131, 197, 199, 311, 337, 373, 719, 733, 919, 971, 991, 1193, 1931, 3119, 3779, 7793, 7937, 9311, 9377, 11939, 19391, 19937, 37199, 39119, 71993, 91193, 93719, 93911, 99371, 193939, 199933, 319993, 331999, 391939, 393919, 919393, 933199, 939193, 939391, 993319, 999331
Some sources only include the smallest prime in each cycle. For example, listing 13, but omitting 31.
2, 3, 5, 7, 11, 13, 17, 37, 79, 113, 197, 199, 337, 1193, 3779, 11939, 19937, 193939, 199933, 1111111111111111111, 11111111111111111111111
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 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
Cousin primes are pairs of primes that differ by four.,,,,,,,,,,,,,,,,
Cuban primes
Cuban primes are primes of the form where is a natural number.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
The term is also used to refer to primes of the form where is a natural number.
13, 109, 193, 433, 769, 1201, 1453, 2029, 3469, 3889, 4801, 10093, 12289, 13873, 18253, 20173, 21169, 22189, 28813, 37633, 43201, 47629, 60493, 63949, 65713, 69313, 73009, 76801, 84673, 106033, 108301, 112909, 115249
Cullen primes
Cullen primes are primes p of the form p=''k2 + 1, for some natural number k''.3, 393050634124102232869567034555427371542904833
Delicate primes
Delicate primes are those primes that always become a composite number when any of their base 10 digit is changed.294001, 505447, 584141, 604171, 971767, 1062599, 1282529, 1524181, 2017963, 2474431, 2690201, 3085553, 3326489, 4393139
Dihedral primes
Dihedral primes are primes that satisfy 180° rotational symmetry and mirror symmetry on a seven-segment display.2, 5, 11, 101, 181, 1181, 1811, 18181, 108881, 110881, 118081, 120121,
121021, 121151, 150151, 151051, 151121, 180181, 180811, 181081
Real Eisenstein primes
Real Eisenstein primes are real Eisenstein integers that are irreducible. Equivalently, they are primes of the form 3k − 1, for a positive integer k.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
Emirps are those primes that become a different prime after their base 10 digits have been reversed. The name "emirp" is the reverse of the word "prime".13, 17, 31, 37, 71, 73, 79, 97, 107, 113, 149, 157, 167, 179, 199, 311, 337, 347, 359, 389, 701, 709, 733, 739, 743, 751, 761, 769, 907, 937, 941, 953, 967, 971, 983, 991
Euclid primes
Euclid primes are primes p such that p−1 is a primorial.3, 7, 31, 211, 2311, 200560490131
Euler irregular primes
Euler irregular primes are primes that divide an Euler number for some19, 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
Euler irregular primes are primes p that divide the rd Euler number.149, 241, 2946901
Factorial primes
Factorial primes are primes whose distance to the next factorial number is one.2, 3, 5, 7, 23, 719, 5039, 39916801, 479001599, 87178291199, 10888869450418352160768000001, 265252859812191058636308479999999, 263130836933693530167218012159999999, 8683317618811886495518194401279999999
Fermat primes
Fermat primes are primes p of the form p = 2 + 1, for a non-negative integer k. only five Fermat primes have been discovered.3, 5, 17, 257, 65537
Generalized Fermat primes
Generalized Fermat primes are primes p of the form p = a + 1, for a non-negative integer k and even natural number a.| Generalized Fermat primes with base a | |
| 2 | 3, 5, 17, 257, 65537,... |
| 4 | 5, 17, 257, 65537,... |
| 6 | 7, 37, 1297,... |
| 8 | |
| 10 | 11, 101,... |
| 12 | 13,... |
| 14 | 197,... |
| 16 | 17, 257, 65537,... |
| 18 | 19,... |
| 20 | 401, 160001,... |
| 22 | 23,... |
| 24 | 577, 331777,... |
Fibonacci primes
Fibonacci primes are primes that appear in the Fibonacci sequence.2, 3, 5, 13, 89, 233, 1597, 28657, 514229, 433494437, 2971215073, 99194853094755497, 1066340417491710595814572169, 19134702400093278081449423917
Fortunate primes
Fortunate primes are primes that are also Fortunate numbers. There are no known composite Fortunate numbers.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
Gaussian primes are primes p of the form p = 4k + 3, for a non-negative integer k.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
Good primes are primes p satisfying ab < p, for all primes a and b such that a,''b < p''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 primes are primes that are also happy numbers.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
Harmonic primes are primes p for which there are no solutions to H ≡ 0 and H ≡ −ω, for 1 ≤ k ≤ p−2, where H denotes the k-th harmonic number and ω denotes the Wolstenholme quotient.5, 13, 17, 23, 41, 67, 73, 79, 107, 113, 139, 149, 157, 179, 191, 193, 223, 239, 241, 251, 263, 277, 281, 293, 307, 311, 317, 331, 337, 349
Higgs primes
Higgs primes are primes p for which p − 1 divides the square of the product of all smaller Higgs primes.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
Highly cototient primes are 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.For a non-negative integer, its home prime is obtained by concatenating its prime factors in increasing order repeatedly, until a prime is achieved.
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
Irregular primes are 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
The irregular primes are primes p such that is an irregular pair.16843, 2124679
(''p'', ''p'' − 5) irregular primes
The irregular primes are primes p such that is an irregular pair.37
(''p'', ''p'' − 9) irregular primes
The irregular primes are primes p such that is an irregular pair.67, 877
Isolated primes
Isolated primes are primes p such that both p − 2 and p + 2 are both composite.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
Leyland primes are primes p of the form p = a + b, where a and b are integers larger than one.17, 593, 32993, 2097593, 8589935681, 59604644783353249, 523347633027360537213687137, 43143988327398957279342419750374600193
Long primes
Long primes, or full reptend primes, are odd primes p for which is a cyclic number. Bases other than 10 are also used.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
Lucas primes are primes that appear in the Lucas sequence.2, 3, 7, 11, 29, 47, 199, 521, 2207, 3571, 9349, 3010349, 54018521, 370248451, 6643838879, 119218851371, 5600748293801, 688846502588399, 32361122672259149
Lucky primes
Lucky primes are primes that are also lucky numbers.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
Mersenne primes are primes p of the form p = 2 − 1, for some non-negative integer k.3, 7, 31, 127, 8191, 131071, 524287, 2147483647, 2305843009213693951, 618970019642690137449562111, 162259276829213363391578010288127, 170141183460469231731687303715884105727
, there are 52 known Mersenne primes. The 13th, 14th, and 52nd have respectively 157, 183, and 41,024,320 digits. The largest known prime 2136,279,841−1 is the 52nd Mersenne prime.
Mersenne divisors
Mersenne divisors are primes that divide 2 − 1, for some prime k. Every Mersenne prime p is also a Mersenne divisor, with k = p.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
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, 74207281, 77232917
, two more are known to be in the sequence, but it is not known whether they are the next:
82589933, 136279841
Double Mersenne primes
A subset of Mersenne primes of the form 2 − 1 for prime p.7, 127, 2147483647, 170141183460469231731687303715884105727
Generalized repunit primes">Repunit">repunit primes
Of the form / for fixed integer a.For a = 2, these are the Mersenne primes, while for a = 10 they are the [|repunit primes]. For other small a, they are given below:
a = 3: 13, 1093, 797161, 3754733257489862401973357979128773, 6957596529882152968992225251835887181478451547013
a = 4: 5
a = 5: 31, 19531, 12207031, 305175781, 177635683940025046467781066894531, 14693679385278593849609206715278070972733319459651094018859396328480215743184089660644531
a = 6: 7, 43, 55987, 7369130657357778596659, 3546245297457217493590449191748546458005595187661976371
a = 7: 2801, 16148168401, 85053461164796801949539541639542805770666392330682673302530819774105141531698707146930307290253537320447270457
a = 8: 73
a = 9: none exist
Other generalizations and variations
Many generalizations of Mersenne primes have been defined. This include the following:- 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 p2. Three such primes are known; it is not known whether there are more.2, 40487, 6692367337
Palindromic primes
Primes that remain the same when their decimal digits are read backwards.2, 3, 5, 7, 11, 101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, 797, 919, 929, 10301, 10501, 10601, 11311, 11411, 12421, 12721, 12821, 13331, 13831, 13931, 14341, 14741
Palindromic wing primes
Primes of the form with. This means all digits except the middle digit are equal.101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, 797, 919, 929, 11311, 11411, 33533, 77377, 77477, 77977, 1114111, 1117111, 3331333, 3337333, 7772777, 7774777, 7778777, 111181111, 111191111, 777767777, 77777677777, 99999199999
Partition primes
Partition 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 = 2P + 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 = 3, P = 0, P = 2,P = P + P.
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'' ≥ 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 n4 + 1.2, 17, 257, 1297, 65537, 160001, 331777, 614657, 1336337, 4477457, 5308417, 8503057, 9834497, 29986577, 40960001, 45212177, 59969537, 65610001, 126247697, 193877777, 303595777, 384160001, 406586897, 562448657, 655360001
Primeval 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# ± 1.3, 5, 7, 29, 31, 211, 2309, 2311, 30029, 200560490131, 304250263527209, 23768741896345550770650537601358309
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 4n + 1.5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101, 109, 113, 137, 149, 157, 173, 181, 193, 197, 229, 233, 241, 257, 269, 277, 281, 293, 313, 317, 337, 349, 353, 373, 389, 397, 401, 409, 421, 433, 449
Prime quadruplets
Where are all prime.,,,,,,,,,,,
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 ≥ R.2, 11, 17, 29, 41, 47, 59, 67, 71, 97, 101, 107, 127, 149, 151, 167, 179, 181, 227, 229, 233, 239, 241, 263, 269, 281, 307, 311, 347, 349, 367, 373, 401, 409, 419, 431, 433, 439, 461, 487, 491
Regular 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, 11111111111111111111111
The next have 317, 1031, 49081, 86453, 109297, and 270343 digits, respectively.
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 the Pythagorean primes, 4n+3 are the integer Gaussian primes, and 6n+5 are the Eisenstein primes. The classes 10n+''d are primes ending in the decimal digit d''.
If a and d are relatively prime, the arithmetic progression contains infinitely many primes.
2n+1: 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53
4n+1: 5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101, 109, 113, 137
4n+3: 3, 7, 11, 19, 23, 31, 43, 47, 59, 67, 71, 79, 83, 103, 107
6n+1: 7, 13, 19, 31, 37, 43, 61, 67, 73, 79, 97, 103, 109, 127, 139
6n+5: 5, 11, 17, 23, 29, 41, 47, 53, 59, 71, 83, 89, 101, 107, 113
8n+1: 17, 41, 73, 89, 97, 113, 137, 193, 233, 241, 257, 281, 313, 337, 353
8n+3: 3, 11, 19, 43, 59, 67, 83, 107, 131, 139, 163, 179, 211, 227, 251
8n+5: 5, 13, 29, 37, 53, 61, 101, 109, 149, 157, 173, 181, 197, 229, 269
8n+7: 7, 23, 31, 47, 71, 79, 103, 127, 151, 167, 191, 199, 223, 239, 263
10n+1: 11, 31, 41, 61, 71, 101, 131, 151, 181, 191, 211, 241, 251, 271, 281
10n+3: 3, 13, 23, 43, 53, 73, 83, 103, 113, 163, 173, 193, 223, 233, 263
10n+7: 7, 17, 37, 47, 67, 97, 107, 127, 137, 157, 167, 197, 227, 257, 277
10n+9: 19, 29, 59, 79, 89, 109, 139, 149, 179, 199, 229, 239, 269, 349, 359
12n+1: 13, 37, 61, 73, 97, 109, 157, 181, 193, 229, 241, 277, 313, 337, 349
12n+5: 5, 17, 29, 41, 53, 89, 101, 113, 137, 149, 173, 197, 233, 257, 269
12n+7: 7, 19, 31, 43, 67, 79, 103, 127, 139, 151, 163, 199, 211, 223, 271
12n+11: 11, 23, 47, 59, 71, 83, 107, 131, 167, 179, 191, 227, 239, 251, 263
Safe primes
Where p and / 2 are both prime.5, 7, 11, 23, 47, 59, 83, 107, 167, 179, 227, 263, 347, 359, 383, 467, 479, 503, 563, 587, 719, 839, 863, 887, 983, 1019, 1187, 1283, 1307, 1319, 1367, 1439, 1487, 1523, 1619, 1823, 1907
Self primes 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 are both prime.,,,,,,,,,,,,,,,,,,,,,,,,,
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 − c·2 − c·2 − ... − c.- 3, 5, 7, 11, 13 2 − 5, the largest prime that fits into 32 bits of memory.2 − 59, the largest prime that fits into 64 bits of memory.
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
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 prime-numbered indexes in the sequence of prime numbers.3, 5, 11, 17, 31, 41, 59, 67, 83, 109, 127, 157, 179, 191, 211, 241, 277, 283, 331, 353, 367, 401, 431, 461, 509, 547, 563, 587, 599, 617, 709, 739, 773, 797, 859, 877, 919, 967, 991
Supersingular primes
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 or are all prime.,,,,,,,,,,,,,,,,,,,
Truncatable prime
Left-truncatable
Primes that remain prime when the leading decimal digit is successively removed.2, 3, 5, 7, 13, 17, 23, 37, 43, 47, 53, 67, 73, 83, 97, 113, 137, 167, 173, 197, 223, 283, 313, 317, 337, 347, 353, 367, 373, 383, 397, 443, 467, 523, 547, 613, 617, 643, 647, 653, 673, 683
Right-truncatable
Primes that remain prime when the least significant decimal digit is successively removed.2, 3, 5, 7, 23, 29, 31, 37, 53, 59, 71, 73, 79, 233, 239, 293, 311, 313, 317, 373, 379, 593, 599, 719, 733, 739, 797, 2333, 2339, 2393, 2399, 2939, 3119, 3137, 3733, 3739, 3793, 3797
Two-sided
Primes that are both left-truncatable and right-truncatable. There are exactly fifteen two-sided primes:2, 3, 5, 7, 23, 37, 53, 73, 313, 317, 373, 797, 3137, 3797, 739397
Twin primes
Where are both prime.,,,,,,,,,,,,,,,,,,,,,,,
Unique primes
The list of primes p for which the period length of the decimal expansion of 1/p is unique.3, 11, 37, 101, 9091, 9901, 333667, 909091, 99990001, 999999000001, 9999999900000001, 909090909090909091, 1111111111111111111, 11111111111111111111111, 900900900900990990990991
Wagstaff primes
Of the form / 3.3, 11, 43, 683, 2731, 43691, 174763, 2796203, 715827883, 2932031007403, 768614336404564651, 201487636602438195784363, 845100400152152934331135470251, 56713727820156410577229101238628035243
Values of n:
3, 5, 7, 11, 13, 17, 19, 23, 31, 43, 61, 79, 101, 127, 167, 191, 199, 313, 347, 701, 1709, 2617, 3539, 5807, 10501, 10691, 11279, 12391, 14479, 42737, 83339, 95369, 117239, 127031, 138937, 141079, 267017, 269987, 374321
Wall–Sun–Sun primes
A prime p > 5, if p divides the Fibonacci number, where the Legendre symbol is defined as, no Wall-Sun-Sun primes have been found below .
Wieferich primes
Primes p such that for fixed integer a > 1.2p − 1 ≡ 1 : 1093, 3511
3p − 1 ≡ 1 : 11, 1006003
4p − 1 ≡ 1 : 1093, 3511
5p − 1 ≡ 1 : 2, 20771, 40487, 53471161, 1645333507, 6692367337, 188748146801
6p − 1 ≡ 1 : 66161, 534851, 3152573
7p − 1 ≡ 1 : 5, 491531
8p − 1 ≡ 1 : 3, 1093, 3511
9p − 1 ≡ 1 : 2, 11, 1006003
10p − 1 ≡ 1 : 3, 487, 56598313
11p − 1 ≡ 1 : 71
12p − 1 ≡ 1 : 2693, 123653
13p − 1 ≡ 1 : 2, 863, 1747591
14p − 1 ≡ 1 : 29, 353, 7596952219
15p − 1 ≡ 1 : 29131, 119327070011
16p − 1 ≡ 1 : 1093, 3511
17p − 1 ≡ 1 : 2, 3, 46021, 48947
18p − 1 ≡ 1 : 5, 7, 37, 331, 33923, 1284043
19p − 1 ≡ 1 : 3, 7, 13, 43, 137, 63061489
20p − 1 ≡ 1 : 281, 46457, 9377747, 122959073
21p − 1 ≡ 1 : 2
22p − 1 ≡ 1 : 13, 673, 1595813, 492366587, 9809862296159
23p − 1 ≡ 1 : 13, 2481757, 13703077, 15546404183, 2549536629329
24p − 1 ≡ 1 : 5, 25633
25p − 1 ≡ 1 : 2, 20771, 40487, 53471161, 1645333507, 6692367337, 188748146801
, these are all known Wieferich primes with a ≤ 25.
Wilson primes
Primes p for which p divides ! + 1.5, 13, 563
, these are the only known Wilson primes.
Wolstenholme primes
Primes p for which the binomial coefficient16843, 2124679
, these are the only known Wolstenholme primes.
Woodall primes
Of the form n×2 − 1.7, 23, 383, 32212254719, 2833419889721787128217599, 195845982777569926302400511, 4776913109852041418248056622882488319