List of prime numbers
A prime number (or a 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 not prime nor composite.
The first 1000 prime numbers
The following table lists the first 1000 primes, with 20 columns of consecutive primes in each of the 50 rows.[1]
(sequence A000040 in the OEIS).
The Goldbach conjecture verification project reports that it has computed all primes below 4×1018.[2] That means 95,676,260,903,887,607 primes[3] (nearly 1017), but they were not stored. There are known formulae to evaluate the prime-counting function (the number of primes below a given value) faster than computing the primes. This has been used to compute that there are 1,925,320,391,606,803,968,923 primes (roughly 2×1021) below 1023. A different computation found that there are 18,435,599,767,349,200,867,866 primes (roughly 2×1022) below 1024, if the Riemann hypothesis is true.[4]
Lists of primes by type
Below are listed the first prime numbers of many named forms and types. More details are in the article for the name. n is a natural number (including 0) in the definitions.
Additive primes
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 ( A046704)
Annihilating primes
Let d(p) be the shadow of the sequence f(n) = seq1-1(n) (that 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 ( A072456)
Bell number 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. ( A051131)
Carol primes
Of the form (2n−1)2 − 2.
7, 47, 223, 3967, 16127, 1046527, 16769023, 1073676287, 68718952447, 274876858367, 4398042316799, 1125899839733759, 18014398241046527, 1298074214633706835075030044377087 ( A091516)
Centered decagonal primes
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 ( A090562)
Centered heptagonal primes
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 A069099)
Centered square primes
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 ( A027862)
Centered triangular primes
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 ( A125602)
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 ( A109611)
Circular primes
A circular prime number is a number that remains prime on any cyclic rotation of its digits (in base 10).
2, 3, 5, 7, 11, 13, 17, 31, 37, 71, 73, 79, 97, 113, 131, 197, 199, 311, 337, 373, 719, 733, 919, 971, 991, 1193, 1931, 3119, 3779, 7793, 7937, 9311, 9377, 11939, 19391, 19937, 37199, 39119, 71993, 91193, 93719, 93911, 99371, 193939, 199933, 319993, 331999, 391939, 393919, 919393, 933199, 939193, 939391, 993319, 999331 ( A068652)
Some sources only list the smallest prime in each cycle, for example, listing 13, but omitting 31 (OEIS really calls this sequence circular primes, but not the above sequence):
2, 3, 5, 7, 11, 13, 17, 37, 79, 113, 197, 199, 337, 1193, 3779, 11939, 19937, 193939, 199933, 1111111111111111111, 11111111111111111111111 ( A016114)
All repunit primes are circular.
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) ( A023200, A046132)
Cuban primes
Of the form 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 ( A002407)
Of the form 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 ( A002648)
Cullen primes
Of the form n×2n + 1.
3, 393050634124102232869567034555427371542904833 ( A050920)
Dihedral primes
Primes that remain prime when read upside down or mirrored in a seven-segment display.
2, 5, 11, 101, 181, 1181, 1811, 18181, 108881, 110881, 118081, 120121, 121021, 121151, 150151, 151051, 151121, 180181, 180811, 181081 ( A134996)
Double factorial primes
Of the form n!! + 1. Values of n:
0, 1, 2, 518, 33416, 37310, 52608 ( A080778)
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 ( A007749)
Double Mersenne primes
A subset of Mersenne primes of the form 22p−1 − 1 for prime p.
7, 127, 2147483647, 170141183460469231731687303715884105727 (primes in A077586)
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
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 ( A003627)
Emirps
Primes that become a different prime when their decimal digits are reversed. The name "emirp" is obtained by reversing the word "prime".
13, 17, 31, 37, 71, 73, 79, 97, 107, 113, 149, 157, 167, 179, 199, 311, 337, 347, 359, 389, 701, 709, 733, 739, 743, 751, 761, 769, 907, 937, 941, 953, 967, 971, 983, 991 ( A006567)
Euclid primes
Of the form pn# + 1 (a subset of primorial primes).
3, 7, 31, 211, 2311, 200560490131 ( A018239[6])
Euler irregular primes
A prime p that divides Euler number E2n for some 0≤2n≤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 ( A120337)
Euler (p, p−3) irregular primes
Primes p such that (p, p−3) is an Euler irregular pair.
Even prime
Of the form 2n.
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
2, 3, 5, 7, 23, 719, 5039, 39916801, 479001599, 87178291199, 10888869450418352160768000001, 265252859812191058636308479999999, 263130836933693530167218012159999999, 8683317618811886495518194401279999999 ( A088054)
Fermat primes
Of the form 22n + 1.
3, 5, 17, 257, 65537 ( A019434)
As of 2013 these are the only known Fermat primes, and conjecturally the only Fermat primes.
Fibonacci primes
Primes in the Fibonacci sequence F0 = 0, F1 = 1, Fn = Fn−1 + Fn−2.
2, 3, 5, 13, 89, 233, 1597, 28657, 514229, 433494437, 2971215073, 99194853094755497, 1066340417491710595814572169, 19134702400093278081449423917 ( A005478)
Fortunate primes
Fortunate numbers that are prime (it has been conjectured they all are).
3, 5, 7, 13, 17, 19, 23, 37, 47, 59, 61, 67, 71, 79, 89, 101, 103, 107, 109, 127, 151, 157, 163, 167, 191, 197, 199, 223, 229, 233, 239, 271, 277, 283, 293, 307, 311, 313, 331, 353, 373, 379, 383, 397 ( A046066)
Gaussian primes
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 ( A002145)
Generalized Fermat primes base 10
Of the form 102n + 1.
As of April 2011, these are the only known generalized Fermat primes in base 10.[8]
Genocchi number primes
The only positive prime Genocchi number is 17.[9]
Gilda's primes
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 ( A046850; another entry A135995 is erroneous)
Good primes
Primes pn for which pn2 > pn−i 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 ( A028388)
Happy primes
Happy numbers that are prime.
7, 13, 19, 23, 31, 79, 97, 103, 109, 139, 167, 193, 239, 263, 293, 313, 331, 367, 379, 383, 397, 409, 487, 563, 617, 653, 673, 683, 709, 739, 761, 863, 881, 907, 937, 1009, 1033, 1039, 1093 ( A035497)
Harmonic primes
Primes p for which there are no solutions to Hk ≡ 0 (mod p) and Hk ≡ −ωp (mod p) for 1 ≤ k ≤ p−2, where Hk denotes the k-th harmonic number and ωp denotes the Wolstenholme quotient.[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 ( A092101)
Higgs primes for squares
Primes p for which p−1 divides the square of the product of all earlier terms.
2, 3, 5, 7, 11, 13, 19, 23, 29, 31, 37, 43, 47, 53, 59, 61, 67, 71, 79, 101, 107, 127, 131, 139, 149, 151, 157, 173, 181, 191, 197, 199, 211, 223, 229, 263, 269, 277, 283, 311, 317, 331, 347, 349 ( A007459)
Highly cototient number 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 ( A105440)
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 ( A000928)
(p, p−5) irregular primes
Primes p such that (p, p−5) is an irregular pair.[12]
(p, p−9) irregular primes
Primes p such that (p, p−9) is an irregular pair.[12]
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 ( A007510)
Kynea primes
Of the form (2n + 1)2 − 2.
2, 7, 23, 79, 1087, 66047, 263167, 16785407, 1073807359, 17180131327, 68720001023, 4398050705407, 70368760954879, 18014398777917439, 18446744082299486207 ( A091514)
Left-truncatable primes
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 ( A024785)
Leyland primes
Of the form xy + yx, with 1 < x ≤ y.
17, 593, 32993, 2097593, 8589935681, 59604644783353249, 523347633027360537213687137, 43143988327398957279342419750374600193 ( A094133)
Long primes
Primes p for which, in a given base b, gives a cyclic number. They are also called full reptend primes. Primes p for base 10:
7, 17, 19, 23, 29, 47, 59, 61, 97, 109, 113, 131, 149, 167, 179, 181, 193, 223, 229, 233, 257, 263, 269, 313, 337, 367, 379, 383, 389, 419, 433, 461, 487, 491, 499, 503, 509, 541, 571, 577, 593 ( A001913)
Lucas primes
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 ( A005479)
Lucky primes
Lucky numbers that are prime.
3, 7, 13, 31, 37, 43, 67, 73, 79, 127, 151, 163, 193, 211, 223, 241, 283, 307, 331, 349, 367, 409, 421, 433, 463, 487, 541, 577, 601, 613, 619, 631, 643, 673, 727, 739, 769, 787, 823, 883, 937, 991, 997 ( A031157)
Markov primes
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 A002559)
Mersenne primes
Of the form 2n − 1.
3, 7, 31, 127, 8191, 131071, 524287, 2147483647, 2305843009213693951, 618970019642690137449562111, 162259276829213363391578010288127, 170141183460469231731687303715884105727 ( A000668)
As of 2016, there are 49 known Mersenne primes. The 13th, 14th, and 49th have respectively 157, 183, and 22,338,618 digits.
Mersenne prime exponents
Primes p such that 2p − 1 is prime.
2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583, 25964951, 30402457, 32582657 ( A000043)
As of January 2016 five more are known to be in the sequence, but it is not known whether they are the next:
37156667, 42643801, 43112609, 57885161, 74207281
Mills primes
Of the form ⌊θ3n⌋, where θ is Mills' constant. This form is prime for all positive integers n.
2, 11, 1361, 2521008887, 16022236204009818131831320183 ( A051254)
Minimal primes
Primes for which there is no shorter sub-sequence of the decimal digits that form a prime. There are exactly 26 minimal primes:
2, 3, 5, 7, 11, 19, 41, 61, 89, 409, 449, 499, 881, 991, 6469, 6949, 9001, 9049, 9649, 9949, 60649, 666649, 946669, 60000049, 66000049, 66600049 ( A071062)
Motzkin primes
Primes that are the number of different ways of drawing non-intersecting chords on a circle between n points.
2, 127, 15511, 953467954114363 ( A092832)
Newman–Shanks–Williams primes
Newman–Shanks–Williams numbers that are prime.
7, 41, 239, 9369319, 63018038201, 489133282872437279, 19175002942688032928599 ( A088165)
Non-generous primes
Primes p for which the least positive primitive root is not a primitive root of p2.
2, 40487, 6692367337 ( A055578)
Odd primes
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... ( A065091)
All prime numbers except 2 are odd.
Padovan primes
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 ( A100891)
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 ( A002385)
Palindromic wing primes
Primes of the form with .[14] This means all digits except the middle digit are equal.
101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, 797, 919, 929, 11311, 11411, 33533, 77377, 77477, 77977, 1114111, 1117111, 3331333, 3337333, 7772777, 7774777, 7778777, 111181111, 111191111, 777767777, 77777677777, 99999199999 ( A077798)
Partition primes
Partition function values that are prime.
2, 3, 5, 7, 11, 101, 17977, 10619863, 6620830889, 80630964769, 228204732751, 1171432692373, 1398341745571, 10963707205259, 15285151248481, 10657331232548839, 790738119649411319, 18987964267331664557 ( A049575)
Pell primes
Primes in the Pell number sequence P0 = 0, P1 = 1, Pn = 2Pn−1 + Pn−2.
2, 5, 29, 5741, 33461, 44560482149, 1746860020068409, 68480406462161287469, 13558774610046711780701, 4125636888562548868221559797461449 ( A086383)
Permutable primes
Any permutation of the decimal digits is a prime.
2, 3, 5, 7, 11, 13, 17, 31, 37, 71, 73, 79, 97, 113, 131, 199, 311, 337, 373, 733, 919, 991, 1111111111111111111, 11111111111111111111111 ( A003459)
It seems likely that all further permutable primes are repunits, i.e. contain only the digit 1.
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 ( A074788)
Pierpont primes
Of the form 2u3v + 1 for some integers u,v ≥ 0.
These are also class 1- primes.
2, 3, 5, 7, 13, 17, 19, 37, 73, 97, 109, 163, 193, 257, 433, 487, 577, 769, 1153, 1297, 1459, 2593, 2917, 3457, 3889, 10369, 12289, 17497, 18433, 39367, 52489, 65537, 139969, 147457 ( A005109)
Pillai primes
Primes p for which there exist n > 0 such that p divides n!+ 1 and n does not divide p−1.
23, 29, 59, 61, 67, 71, 79, 83, 109, 137, 139, 149, 193, 227, 233, 239, 251, 257, 269, 271, 277, 293, 307, 311, 317, 359, 379, 383, 389, 397, 401, 419, 431, 449, 461, 463, 467, 479, 499 ( A063980)
Primes 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 ( A037896)
Primeval primes
Primes for which there are more prime permutations of some or all the decimal digits than for any smaller number.
2, 13, 37, 107, 113, 137, 1013, 1237, 1367, 10079 ( A119535)
Primorial primes
Of the form pn# ± 1.
3, 5, 7, 29, 31, 211, 2309, 2311, 30029, 200560490131, 304250263527209, 23768741896345550770650537601358309 (union of A057705 and A018239[6])
Proth primes
Of the form k×2n + 1, with odd k and k < 2n.
3, 5, 13, 17, 41, 97, 113, 193, 241, 257, 353, 449, 577, 641, 673, 769, 929, 1153, 1217, 1409, 1601, 2113, 2689, 2753, 3137, 3329, 3457, 4481, 4993, 6529, 7297, 7681, 7937, 9473, 9601, 9857 ( A080076)
Pythagorean primes
Of the form 4n + 1.
5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101, 109, 113, 137, 149, 157, 173, 181, 193, 197, 229, 233, 241, 257, 269, 277, 281, 293, 313, 317, 337, 349, 353, 373, 389, 397, 401, 409, 421, 433, 449 ( A002144)
Prime quadruplets
Where (p, p+2, p+6, p+8) are all prime.
(5, 7, 11, 13), (11, 13, 17, 19), (101, 103, 107, 109), (191, 193, 197, 199), (821, 823, 827, 829), (1481, 1483, 1487, 1489), (1871, 1873, 1877, 1879), (2081, 2083, 2087, 2089), (3251, 3253, 3257, 3259), (3461, 3463, 3467, 3469), (5651, 5653, 5657, 5659), (9431, 9433, 9437, 9439) ( A007530, A136720, A136721, A090258)
Primes of binary quadratic form
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 ( A106856)
Quartan primes
Of the form x4 + y4, where x,y > 0.
2, 17, 97, 257, 337, 641, 881 ( A002645)
Ramanujan primes
Integers Rn that are the smallest to give at least n primes from x/2 to x for all x ≥ Rn (all such integers are primes).
2, 11, 17, 29, 41, 47, 59, 67, 71, 97, 101, 107, 127, 149, 151, 167, 179, 181, 227, 229, 233, 239, 241, 263, 269, 281, 307, 311, 347, 349, 367, 373, 401, 409, 419, 431, 433, 439, 461, 487, 491 ( A104272)
Regular primes
Primes p that do not divide the class number of the p-th cyclotomic field.
3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 43, 47, 53, 61, 71, 73, 79, 83, 89, 97, 107, 109, 113, 127, 137, 139, 151, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 239, 241, 251, 269, 277, 281 ( A007703)
Repunit primes
Primes containing only the decimal digit 1.
11, 1111111111111111111, 11111111111111111111111 ( A004022)
The next have 317 and 1,031 digits.
Primes in residue classes
Of the form an + d for fixed integers 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 ( A065091)
4n+1: 5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101, 109, 113, 137 ( A002144)
4n+3: 3, 7, 11, 19, 23, 31, 43, 47, 59, 67, 71, 79, 83, 103, 107 ( A002145)
6n+1: 7, 13, 19, 31, 37, 43, 61, 67, 73, 79, 97, 103, 109, 127, 139 ( A002476)
6n+5: 5, 11, 17, 23, 29, 41, 47, 53, 59, 71, 83, 89, 101, 107, 113 ( A007528)
8n+1: 17, 41, 73, 89, 97, 113, 137, 193, 233, 241, 257, 281, 313, 337, 353 ( A007519)
8n+3: 3, 11, 19, 43, 59, 67, 83, 107, 131, 139, 163, 179, 211, 227, 251 ( A007520)
8n+5: 5, 13, 29, 37, 53, 61, 101, 109, 149, 157, 173, 181, 197, 229, 269 ( A007521)
8n+7: 7, 23, 31, 47, 71, 79, 103, 127, 151, 167, 191, 199, 223, 239, 263 ( A007522)
10n+1: 11, 31, 41, 61, 71, 101, 131, 151, 181, 191, 211, 241, 251, 271, 281 ( A030430)
10n+3: 3, 13, 23, 43, 53, 73, 83, 103, 113, 163, 173, 193, 223, 233, 263 ( A030431)
10n+7: 7, 17, 37, 47, 67, 97, 107, 127, 137, 157, 167, 197, 227, 257, 277 ( A030432)
10n+9: 19, 29, 59, 79, 89, 109, 139, 149, 179, 199, 229, 239, 269, 349, 359 ( A030433)
12n+1: 13, 37, 61, 73, 97, 109, 157, 181, 193, 229, 241, 277, 313, 337, 349 ( A068228)
12n+5: 5, 17, 29, 41, 53, 89, 101, 113, 137, 149, 173, 197, 233, 257, 269, 281 ( A040117)
12n+7: 7, 19, 31, 43, 67, 79, 103, 127, 139, 151, 163, 199, 211, 223, 271, 283 ( A068229)
12n+11: 11, 23, 47, 59, 71, 83, 107, 131, 167, 179, 191, 227, 239, 251, 263, 311 ( A068231)
...
10n+d (d = 1, 3, 7, 9) are primes ending in the decimal digit d.
Right-truncatable primes
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 ( A024770)
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 ( A005385)
Self primes in base 10
Primes that cannot be generated by any integer added to the sum of its decimal digits.
3, 5, 7, 31, 53, 97, 211, 233, 277, 367, 389, 457, 479, 547, 569, 613, 659, 727, 839, 883, 929, 1021, 1087, 1109, 1223, 1289, 1447, 1559, 1627, 1693, 1783, 1873 ( A006378)
Sexy primes
Where (p, p+6) are both prime.
(5, 11), (7, 13), (11, 17), (13, 19), (17, 23), (23, 29), (31, 37), (37, 43), (41, 47), (47, 53), (53, 59), (61, 67), (67, 73), (73, 79), (83, 89), (97, 103), (101, 107), (103, 109), (107, 113), (131, 137), (151, 157), (157, 163), (167, 173), (173, 179), (191, 197), (193, 199) ( A023201, A046117)
Smarandache–Wellin primes
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.
Solinas primes
Of the form 2a ± 2b ± 1, where 0 < b < a.
Sophie Germain primes
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 ( A005384)
Star primes
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 ( A083577)
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 ( A042978)
As of 2011, these are the only known Stern primes, and possibly the only existing.
Super-primes
Primes with a prime index in the sequence of prime numbers (the 2nd, 3rd, 5th, ... prime).
3, 5, 11, 17, 31, 41, 59, 67, 83, 109, 127, 157, 179, 191, 211, 241, 277, 283, 331, 353, 367, 401, 431, 461, 509, 547, 563, 587, 599, 617, 709, 739, 773, 797, 859, 877, 919, 967, 991 ( A006450)
Supersingular primes
There are exactly fifteen supersingular primes:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59, 71 ( A002267)
Swinging primes
Primes of the form , where denotes the swinging factorial, which is defined in terms of the double swinging factorial as[17] and
2, 3, 5, 7, 19, 29, 31, 71, 139, 251, 631, 3433, 12011 ( A163074)
Thabit number primes
Of the form 3×2n − 1.
2, 5, 11, 23, 47, 191, 383, 6143, 786431, 51539607551, 824633720831, 26388279066623, 108086391056891903, 55340232221128654847, 226673591177742970257407 ( A007505)
The primes of the form 3×2n + 1 are related.
7, 13, 97, 193, 769, 12289, 786433, 3221225473, 206158430209, 6597069766657 ( A039687)
Prime triplets
Where (p, p+2, p+6) or (p, p+4, p+6) are all prime.
(5, 7, 11), (7, 11, 13), (11, 13, 17), (13, 17, 19), (17, 19, 23), (37, 41, 43), (41, 43, 47), (67, 71, 73), (97, 101, 103), (101, 103, 107), (103, 107, 109), (107, 109, 113), (191, 193, 197), (193, 197, 199), (223, 227, 229), (227, 229, 233), (277, 281, 283), (307, 311, 313), (311, 313, 317), (347, 349, 353) ( A007529, A098414, A098415)
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) ( A001359, A006512)
Two-sided primes
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 ( A020994)
Ulam number primes
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 ( A068820)
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 ( A040017)
Wagstaff primes
Of the form (2n+1) / 3.
3, 11, 43, 683, 2731, 43691, 174763, 2796203, 715827883, 2932031007403, 768614336404564651, 201487636602438195784363, 845100400152152934331135470251, 56713727820156410577229101238628035243 ( A000979)
Values of n:
3, 5, 7, 11, 13, 17, 19, 23, 31, 43, 61, 79, 101, 127, 167, 191, 199, 313, 347, 701, 1709, 2617, 3539, 5807, 10501, 10691, 11279, 12391, 14479, 42737, 83339, 95369, 117239, 127031, 138937, 141079, 267017, 269987, 374321 ( A000978)
Wall–Sun–Sun primes
A prime p > 5, if p2 divides the Fibonacci number , where the Legendre symbol is defined as
As of 2015, no Wall-Sun-Sun primes are known.
Wedderburn-Etherington number primes
Wedderburn-Etherington numbers that are prime.
2, 3, 11, 23, 983, 2179, 24631, 3626149, 253450711, 596572387 ( A136402)
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 ( A050249)
Wieferich primes
Primes p such that ap − 1 ≡ 1 (mod p2) for fixed integer a > 1.
2p − 1 ≡ 1 (mod p2): 1093, 3511 ( A001220)
3p − 1 ≡ 1 (mod p2): 11, 1006003 ( A014127)[18][19][20]
4p − 1 ≡ 1 (mod p2): 1093, 3511
5p − 1 ≡ 1 (mod p2): 2, 20771, 40487, 53471161, 1645333507, 6692367337, 188748146801 ( A123692)
6p − 1 ≡ 1 (mod p2): 66161, 534851, 3152573 ( A212583)
7p − 1 ≡ 1 (mod p2): 5, 491531 ( A123693)
8p − 1 ≡ 1 (mod p2): 3, 1093, 3511
9p − 1 ≡ 1 (mod p2): 2, 11, 1006003
10p − 1 ≡ 1 (mod p2): 3, 487, 56598313 ( A045616)
11p − 1 ≡ 1 (mod p2): 71[21]
12p − 1 ≡ 1 (mod p2): 2693, 123653 ( A111027)
13p − 1 ≡ 1 (mod p2): 2, 863, 1747591 ( A128667)[21]
14p − 1 ≡ 1 (mod p2): 29, 353, 7596952219 ( A234810)
15p − 1 ≡ 1 (mod p2): 29131, 119327070011 ( A242741)
16p − 1 ≡ 1 (mod p2): 1093, 3511
17p − 1 ≡ 1 (mod p2): 2, 3, 46021, 48947 ( A128668)[21]
18p − 1 ≡ 1 (mod p2): 5, 7, 37, 331, 33923, 1284043 ( A244260)
19p − 1 ≡ 1 (mod p2): 3, 7, 13, 43, 137, 63061489 ( A090968)[21]
20p − 1 ≡ 1 (mod p2): 281, 46457, 9377747, 122959073 ( A242982)
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 ( A128669)
24p − 1 ≡ 1 (mod p2): 5, 25633
25p − 1 ≡ 1 (mod p2): 2, 20771, 40487, 53471161, 1645333507, 6692367337, 188748146801
As of 2015, these are all known Wieferich primes with a ≤ 25.
Wilson primes
Primes p for which p2 divides (p−1)! + 1.
As of 2015, these are the only known Wilson primes.
Wolstenholme primes
Primes p for which the binomial coefficient
16843, 2124679 ( A088164)
As of 2015, these are the only known Wolstenholme primes.
Woodall primes
Of the form n×2n − 1.
7, 23, 383, 32212254719, 2833419889721787128217599, 195845982777569926302400511, 4776913109852041418248056622882488319 ( A050918)
References
- ↑ Lehmer, D. N. (1982). List of prime numbers from 1 to 10,006,721. 165. Washington D.C.: Carnegie Institution of Washington. OL16553580M.
- ↑ Tomás Oliveira e Silva, Goldbach conjecture verification. Retrieved 16 July 2013
- ↑ (sequence A080127 in the OEIS)
- ↑ Jens Franke (29 July 2010). "Conditional Calculation of pi(1024)". Retrieved 2011-05-17.
- ↑ L. Halbeisen, N. Hungerbühler, Number theoretic aspects of a combinatorial function
- 1 2 A018239 includes 2 = empty product of first 0 primes plus 1, but 2 is excluded in this list.
- ↑ http://mathworld.wolfram.com/OddPrime.html
- ↑ Caldwell, C.; Honaker, G. L., Jr. "101". Prime Curios!. Retrieved 1 April 2011.
- ↑ Weisstein, Eric W. "Genocchi Number". MathWorld.
- ↑ Russo, F., A Set of New Samarandache Functions, Sequences and Conjectures in Number Theory (PDF), pp. 73–74
- ↑ 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 - 1 2 Johnson, W. (1975). "Irregular Primes and Cyclotomic Invariants" (PDF). Mathematics of Computation. AMS. 29 (129): 113–120. doi:10.2307/2005468.
- ↑ It varies whether L0 = 2 is included in the Lucas numbers.
- ↑ Caldwell, C.; Dubner, H. (1996–97). "The near repdigit primes , especially ". Journal of Recreational Mathematics. 28 (1): 1–9.
- ↑ Lal, M. (1967). "Primes of the Form n4 + 1" (PDF). Mathematics of Computation. AMS. 21: 245–247. doi:10.1090/S0025-5718-1967-0222007-9. ISSN 1088-6842.
- ↑ Bohman, J. (1973). "New primes of the form n4 + 1". BIT Numerical Mathematics. Springer. 13 (3): 370–372. doi:10.1007/BF01951947. ISSN 1572-9125.
- ↑ Luschny, Swinging factorial
- ↑ Ribenboim, P. The new book of prime number records. New York: Springer-Verlag. p. 347. ISBN 0-387-94457-5.
- ↑ "Mirimanoff's Congruence: Other Congruences". Retrieved 26 January 2011.
- ↑ 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.
- 1 2 3 4 Ribenboim, P. (2006). Die Welt der Primzahlen. Berlin: Springer. p. 240. ISBN 3-540-34283-4.
See also
- Illegal prime
- Largest known prime number
- List of numbers
- Prime gap
- Prime number theorem
- Probable prime
- Pseudoprime
- Strobogrammatic prime
- Strong prime
- Wieferich pair
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.
- Interface to a list of the first 98 million primes (primes less than 2,000,000,000)
- Weisstein, Eric W. "Prime Number Sequences". MathWorld.
- Selected prime related sequences in OEIS.
- Fischer, R. Thema: Fermatquotient B^(P−1) == 1 (mod P^2) (German) (Lists Wieferich primes in all bases up to 1052)
- Padilla, Tony. "New Largest Known Prime Number". Numberphile. Brady Haran.