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 following table lists the first 1000 primes, with 20 columns of consecutive primes in each of the 50 rows.[1]
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | ||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1–20 | 71 | ||||||||||||||||||||
| 21–40 | 101 | 113 | 173 | ||||||||||||||||||
| 41–60 | 229 | 281 | |||||||||||||||||||
| 61–80 | 349 | 409 | |||||||||||||||||||
| 81–100 | 463 | 541 | |||||||||||||||||||
| 101–120 | 601 | 647 | 659 | ||||||||||||||||||
| 121–140 | 733 | 809 | |||||||||||||||||||
| 141–160 | 863 | 941 | |||||||||||||||||||
| 161–180 | 1013 | 1069 | |||||||||||||||||||
| 181–200 | 1151 | 1223 | |||||||||||||||||||
| 201–220 | 1291 | 1373 | |||||||||||||||||||
| 221–240 | 1451 | 1511 | |||||||||||||||||||
| 241–260 | 1583 | 1657 | |||||||||||||||||||
| 261–280 | 1733 | 1811 | |||||||||||||||||||
| 281–300 | 1889 | 1987 | |||||||||||||||||||
| 301–320 | 2053 | 2129 | |||||||||||||||||||
| 321–340 | 2213 | 2287 | |||||||||||||||||||
| 341–360 | 2357 | 2423 | |||||||||||||||||||
| 361–380 | 2531 | 2617 | |||||||||||||||||||
| 381–400 | 2687 | 2741 | |||||||||||||||||||
| 401–420 | 2819 | 2903 | |||||||||||||||||||
| 421–440 | 2999 | 3079 | |||||||||||||||||||
| 441–460 | 3181 | 3257 | |||||||||||||||||||
| 461–480 | 3331 | 3413 | |||||||||||||||||||
| 481–500 | 3457 | 3511 | 3571 | ||||||||||||||||||
| 501–520 | 3727 | ||||||||||||||||||||
| 521–540 | 3907 | ||||||||||||||||||||
| 541–560 | 4057 | ||||||||||||||||||||
| 561–580 | 4231 | ||||||||||||||||||||
| 581–600 | 4409 | ||||||||||||||||||||
| 601–620 | 4583 | ||||||||||||||||||||
| 621–640 | 4751 | ||||||||||||||||||||
| 641–660 | 4937 | ||||||||||||||||||||
| 661–680 | 5087 | ||||||||||||||||||||
| 681–700 | 5279 | ||||||||||||||||||||
| 701–720 | 5443 | ||||||||||||||||||||
| 721–740 | 5639 | ||||||||||||||||||||
| 741–760 | 5791 | ||||||||||||||||||||
| 761–780 | 5939 | ||||||||||||||||||||
| 781–800 | 6133 | ||||||||||||||||||||
| 801–820 | 6301 | ||||||||||||||||||||
| 821–840 | 6473 | ||||||||||||||||||||
| 841–860 | 6673 | ||||||||||||||||||||
| 861–880 | 6833 | ||||||||||||||||||||
| 881–900 | 6997 | ||||||||||||||||||||
| 901–920 | 7207 | ||||||||||||||||||||
| 921–940 | 7411 | ||||||||||||||||||||
| 941–960 | 7561 | ||||||||||||||||||||
| 961–980 | 7723 | ||||||||||||||||||||
| 981–1000 | 7919 |
.
The Goldbach conjecture verification project reports that it has computed all primes smaller than 4×10.[2] 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 smaller than 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) smaller than 10. A different computation found that there are 18,435,599,767,349,200,867,866 primes (roughly 2) smaller than 10, if the Riemann hypothesis is true.[3]
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.
Primes with equal-sized prime gaps after and before them, so that they are equal to the arithmetic mean of the nearest primes after and before.
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.
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
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.
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.
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)
Of the form
\tfrac{x3-y3}{x-y}
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{x3-y3}{x-y}
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
Of the form n×2 + 1.
3, 393050634124102232869567034555427371542904833
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 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
Primes that become a different prime when their decimal digits are 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
Of the form p# + 1 (a subset of primorial primes).
3, 7, 31, 211, 2311, 200560490131 ([4])
A prime
p
E2n
0\leq2n\leqp-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
Primes
p
(p,p-3)
2, 3, 5, 7, 23, 719, 5039, 39916801, 479001599, 87178291199, 10888869450418352160768000001, 265252859812191058636308479999999, 263130836933693530167218012159999999, 8683317618811886495518194401279999999
Of the form 2 + 1.
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.[5]
Of the form a + 1 for fixed integer a.
a = 8: (does not exist)
a = 12: 13
a = 14: 197
a = 18: 19
a = 20: 401, 160001
a = 22: 23
a = 24: 577, 331777
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 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
Prime elements of the Gaussian integers; equivalently, primes of the form 4n + 3.
3, 7, 11, 19, 23, 31, 43, 47, 59, 67, 71, 79, 83, 103, 107, 127, 131, 139, 151, 163, 167, 179, 191, 199, 211, 223, 227, 239, 251, 263, 271, 283, 307, 311, 331, 347, 359, 367, 379, 383, 419, 431, 439, 443, 463, 467, 479, 487, 491, 499, 503
Primes p for which p > p p for all 1 ≤ i ≤ n−1, where p is the nth prime.
5, 11, 17, 29, 37, 41, 53, 59, 67, 71, 97, 101, 127, 149, 179, 191, 223, 227, 251, 257, 269, 307
Happy 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
Primes p for which there are no solutions to H ≡ 0 (mod p) and H ≡ −ω (mod p) for 1 ≤ k ≤ p−2, where H denotes the k-th harmonic number and ω denotes the Wolstenholme quotient.[6]
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
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
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
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
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
(See Wolstenholme prime)
Primes p such that (p, p−5) is an irregular pair.[7]
Primes p such that (p, p − 9) is an irregular pair.
67, 877
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
Of the form x + y, with 1 < x < y.
17, 593, 32993, 2097593, 8589935681, 59604644783353249, 523347633027360537213687137, 43143988327398957279342419750374600193
Primes p for which, in a given base b,
| bp-1-1 | |
| p |
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
Primes in the Lucas number sequence L = 2, L = 1,L = L + L.
2,[8] 3, 7, 11, 29, 47, 199, 521, 2207, 3571, 9349, 3010349, 54018521, 370248451, 6643838879, 119218851371, 5600748293801, 688846502588399, 32361122672259149
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
Of the form 2 − 1.
3, 7, 31, 127, 8191, 131071, 524287, 2147483647, 2305843009213693951, 618970019642690137449562111, 162259276829213363391578010288127, 170141183460469231731687303715884105727
, there are 51 known Mersenne primes. The 13th, 14th, and 51st have respectively 157, 183, and 24,862,048 digits.
, this class of prime numbers also contains the largest known prime: M82589933, the 51st known Mersenne prime.
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.
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
A subset of Mersenne primes of the form 2 − 1 for prime p.
7, 127, 2147483647, 170141183460469231731687303715884105727 (primes in)
Of the form (a − 1) / (a − 1) for fixed integer a.
For a = 2, these are the Mersenne primes, while for a = 10 they are the repunit primes. For other small a, they are given below:
a = 3: 13, 1093, 797161, 3754733257489862401973357979128773, 6957596529882152968992225251835887181478451547013
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
Many generalizations of Mersenne primes have been defined. This include the following:
Of the form ⌊θ⌋, where θ is Mills' constant. This form is prime for all positive integers n.
2, 11, 1361, 2521008887, 16022236204009818131831320183
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 numbers that are prime.
7, 41, 239, 9369319, 63018038201, 489133282872437279, 19175002942688032928599
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.[9]
2, 40487, 6692367337
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
Primes of the form
| a(10m-1) | |
| 9 |
\pmb x
| ||||
| 10 |
0\lea\pmb<10
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 function values that are prime.
2, 3, 5, 7, 11, 101, 17977, 10619863, 6620830889, 80630964769, 228204732751, 1171432692373, 1398341745571, 10963707205259, 15285151248481, 10657331232548839, 790738119649411319, 18987964267331664557
Primes in the Pell number sequence P = 0, P = 1,P = 2P + P.
2, 5, 29, 5741, 33461, 44560482149, 1746860020068409, 68480406462161287469, 13558774610046711780701, 4125636888562548868221559797461449
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
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
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
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
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
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
Of the form p# ± 1.
3, 5, 7, 29, 31, 211, 2309, 2311, 30029, 200560490131, 304250263527209, 23768741896345550770650537601358309 (union of and [4])
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
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
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)
Of the form x + y, where x,y > 0.
Integers R that are the smallest to give at least n primes from x/2 to x for all x ≥ 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
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
Primes containing only the decimal digit 1.
11, 1111111111111111111 (19 digits), 11111111111111111111111 (23 digits)
The next have 317, 1031, 49081, 86453, 109297, 270343 digits
Of the form an + d for fixed integers a and d. Also called primes congruent to d modulo a.
The primes of the form 2n+1 are the odd primes, including all primes other than 2. Some sequences have alternate names: 4n+1 are Pythagorean primes, 4n+3 are the integer Gaussian primes, and 6n+5 are the Eisenstein primes (with 2 omitted). The classes 10n+d (d = 1, 3, 7, 9) are primes ending in the decimal digit d.
2n+1: 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53
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
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
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
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)
Primes that are the concatenation of the first n primes written in decimal.
The fourth Smarandache-Wellin prime is the 355-digit concatenation of the first 128 primes that end with 719.
Of the form 2 ± 2 ± 1, where 0 < b < a.
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
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.
Primes with prime-numbered indexes 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
There are exactly fifteen supersingular primes:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59, 71
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
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)
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
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
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
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)
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
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
| F | ||||||
|
{{5}}\right)}
| \left( | {p |
| \left( | p |
| 5 |
\right)=\begin{cases}1&rm{if} p\equiv\pm1\pmod5\ -1&rm{if} p\equiv\pm2\pmod5.\end{cases}
, no Wall-Sun-Sun primes are known.
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
Primes p such that for fixed integer a > 1.
2p − 1 ≡ 1 (mod p2): 1093, 3511
3p − 1 ≡ 1 (mod p2): 11, 1006003 [13] [14] [15]
4p − 1 ≡ 1 (mod p2): 1093, 3511
5p − 1 ≡ 1 (mod p2): 2, 20771, 40487, 53471161, 1645333507, 6692367337, 188748146801
6p − 1 ≡ 1 (mod p2): 66161, 534851, 3152573
7p − 1 ≡ 1 (mod p2): 5, 491531
8p − 1 ≡ 1 (mod p2): 3, 1093, 3511
9p − 1 ≡ 1 (mod p2): 2, 11, 1006003
10p − 1 ≡ 1 (mod p2): 3, 487, 56598313
11p − 1 ≡ 1 (mod p2): 71[16]
12p − 1 ≡ 1 (mod p2): 2693, 123653
13p − 1 ≡ 1 (mod p2): 2, 863, 1747591
14p − 1 ≡ 1 (mod p2): 29, 353, 7596952219
15p − 1 ≡ 1 (mod p2): 29131, 119327070011
16p − 1 ≡ 1 (mod p2): 1093, 3511
17p − 1 ≡ 1 (mod p2): 2, 3, 46021, 48947
18p − 1 ≡ 1 (mod p2): 5, 7, 37, 331, 33923, 1284043
19p − 1 ≡ 1 (mod p2): 3, 7, 13, 43, 137, 63061489
20p − 1 ≡ 1 (mod p2): 281, 46457, 9377747, 122959073
21p − 1 ≡ 1 (mod p2): 2
22p − 1 ≡ 1 (mod p2): 13, 673, 1595813, 492366587, 9809862296159
23p − 1 ≡ 1 (mod p2): 13, 2481757, 13703077, 15546404183, 2549536629329
24p − 1 ≡ 1 (mod p2): 5, 25633
25p − 1 ≡ 1 (mod p2): 2, 20771, 40487, 53471161, 1645333507, 6692367337, 188748146801
, these are all known Wieferich primes with a ≤ 25.
Primes p for which p divides (p−1)! + 1.
, these are the only known Wilson primes.
{{2p-1}\choose{p-1}}\equiv1\pmod{p4}.
16843, 2124679
, these are the only known Wolstenholme primes.
Of the form n×2 − 1.
7, 23, 383, 32212254719, 2833419889721787128217599, 195845982777569926302400511, 4776913109852041418248056622882488319
p
(rm{mod}p)
(rm{mod}p2)
An-k-1B1Ak
9n-k-1819k