List of 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 (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. == 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 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. (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 Cuban primes are primes p of the form p = k^3 - (k - 1)^3, where k 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 p of the form p = (k^3 - (k - 2)^3)/2, where k 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 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 p that divide an Euler number E_{2n}, 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 Euler (p, p - 3) irregular primes are primes p that divide the (p + 3)rd Euler number. 149, 241, 2946901 () Factorial primes Factorial primes are of the form n! ± 1. 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. 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  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 (primes in ) ==== Generalized repunit primes ==== 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 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 \frac{a \big( 10^m-1 \big)}{9} \pm b \times 10^{\frac{ m-1 }{2}} with 0 \le a \pm b . 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(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 ≥ 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 n4 + 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 (union of and • 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 (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_{p - \left(\frac\right)}, where the Legendre symbol \left(\frac\right) is defined as :\left(\frac{p}{5}\right) = \begin{cases} 1 &\textrm{if}\;p \equiv \pm1 \pmod 5\\ -1 &\textrm{if}\;p \equiv \pm2 \pmod 5. \end{cases} , no Wall-Sun-Sun primes have been found below 2^{64} (about 18\cdot 10^{18}). Wieferich primes Primes p such that for fixed integer a > 1. 2p − 1 ≡ 1 (mod p2): 1093, 3511 () 3p − 1 ≡ 1 (mod p2): 11, 1006003 () 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 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. 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 {{2p-1}\choose{p-1}} \equiv 1 \pmod{p^4}. 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 ==