List of prime numbers - wikipedia List of prime numbers This is a dynamic list and may never be able to satisfy particular standards for completeness. You can help by expanding it with reliably sourced entries. 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 neither prime nor composite. Contents 1 The first 1000 prime numbers 2 Lists of primes by type 2.1 Additive primes 2.2 Annihilating primes 2.3 Bell primes 2.4 Carol primes 2.5 Centered decagonal primes 2.6 Centered heptagonal primes 2.7 Centered square primes 2.8 Centered triangular primes 2.9 Chen primes 2.10 Circular primes 2.11 Class 1 + primes 2.12 Cousin primes 2.13 Cuban primes 2.14 Cullen primes 2.15 Dihedral primes 2.16 Double factorial primes 2.17 Double Mersenne primes 2.18 Eisenstein primes without imaginary part 2.19 Emirps 2.20 Euclid primes 2.21 Euler irregular primes 2.21. 1 Euler ( p, p − 3 ) irregular primes 2.22 Even primes 2.23 Factorial primes 2.24 Fermat primes 2.25 Fibonacci primes 2.26 Fortunate primes 2.27 Gaussian primes 2.28 Generalized Fermat primes 2.29 Generalized repunit primes 2.30 Genocchi primes 2.31 Gilda 's primes 2.32 Good primes 2.33 Happy primes 2.34 Harmonic primes 2.35 Higgs primes for squares 2.36 Highly cototient primes 2.37 Irregular primes 2.37. 1 ( p, p − 5 ) irregular primes 2.37. 2 ( p, p − 9 ) irregular primes 2.38 Isolated primes 2.39 Kynea primes 2.40 Left - truncatable primes 2.41 Leyland primes 2.42 Long primes 2.43 Lucas primes 2.44 Lucky primes 2.45 Markov primes 2.46 Mersenne primes 2.47 Mersenne prime exponents 2.48 Mills primes 2.49 Minimal primes 2.50 Motzkin primes 2.51 Newman - Shanks - Williams primes 2.52 Non-generous primes 2.53 Odd primes 2.54 Padovan primes 2.55 Palindromic primes 2.56 Palindromic wing primes 2.57 Partition primes 2.58 Pell primes 2.59 Permutable primes 2.60 Perrin primes 2.61 Pierpont primes 2.62 Pillai primes 2.63 Primes of the form n + 1 2.64 Primeval primes 2.65 Primorial primes 2.66 Proth primes 2.67 Pythagorean primes 2.68 Prime quadruplets 2.69 Primes of binary quadratic form 2.70 Quartan primes 2.71 Ramanujan primes 2.72 Regular primes 2.73 Repunit primes 2.74 Primes in residue classes 2.75 Right - truncatable primes 2.76 Safe primes 2.77 Self primes in base 10 2.78 Sexy primes 2.79 Smarandache primes 2.80 Smarandache -- Wellin primes 2.81 Solinas primes 2.82 Sophie Germain primes 2.83 Star primes 2.84 Stern primes 2.85 Super-primes 2.86 Supersingular primes 2.87 Swinging primes 2.88 Thabit primes 2.89 Prime triplets 2.90 Twin primes 2.91 Two - sided primes 2.92 Ulam primes 2.93 Unique primes 2.94 Wagstaff primes 2.95 Wall - Sun - Sun primes 2.96 Wedderburn - Etherington primes 2.97 Weakly prime numbers 2.98 Wieferich primes 2.99 Wilson primes 2.100 Wolstenholme primes 2.101 Woodall primes 3 See also 4 References 5 External links 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. 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1 -- 20 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 21 -- 40 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 41 -- 60 179 181 191 193 197 199 211 223 227 229 233 239 241 251 257 263 269 271 277 281 61 -- 80 283 293 307 311 313 317 331 337 347 349 353 359 367 373 379 383 389 397 401 409 81 -- 100 419 421 431 433 439 443 449 457 461 463 467 479 487 491 499 503 509 521 523 541 101 -- 120 547 557 563 569 571 577 587 593 599 601 607 613 617 619 631 641 643 647 653 659 121 -- 140 661 673 677 683 691 701 709 719 727 733 739 743 751 757 761 769 773 787 797 809 141 -- 160 811 821 823 827 829 839 853 857 859 863 877 881 883 887 907 911 919 929 937 941 161 -- 180 947 953 967 971 977 983 991 997 1009 1013 1019 1021 1031 1033 1039 1049 1051 1061 1063 1069 181 -- 200 1087 1091 1093 1097 1103 1109 1117 1123 1129 1151 1153 1163 1171 1181 1187 1193 1201 1213 1217 1223 201 -- 220 1229 1231 1237 1249 1259 1277 1279 1283 1289 1291 1297 1301 1303 1307 1319 1321 1327 1361 1367 1373 221 -- 240 1381 1399 1409 1423 1427 1429 1433 1439 1447 1451 1453 1459 1471 1481 1483 1487 1489 1493 1499 1511 241 -- 260 1523 1531 1543 1549 1553 1559 1567 1571 1579 1583 1597 1601 1607 1609 1613 1619 1621 1627 1637 1657 261 -- 280 1663 1667 1669 1693 1697 1699 1709 1721 1723 1733 1741 1747 1753 1759 1777 1783 1787 1789 1801 1811 281 -- 300 1823 1831 1847 1861 1867 1871 1873 1877 1879 1889 1901 1907 1913 1931 1933 1949 1951 1973 1979 301 -- 320 1999 2003 2011 2017 2027 2029 2039 2053 2063 2069 2081 2083 2087 2089 2099 2111 2113 2129 321 -- 340 2131 2137 2141 2143 2153 2161 2179 2203 2207 2213 2221 2237 2239 2243 2251 2267 2269 2273 2281 2287 341 -- 360 2293 2297 2309 2311 2333 2339 2341 2347 2351 2357 2371 2377 2381 2383 2389 2393 2399 2411 2417 2423 361 -- 380 2437 2441 2447 2459 2467 2473 2477 2503 2521 2531 2539 2543 2549 2551 2557 2579 2591 2593 2609 2617 381 -- 400 2621 2633 2647 2657 2659 2663 2671 2677 2683 2687 2689 2693 2699 2707 2711 2713 2719 2729 2731 2741 401 -- 420 2749 2753 2767 2777 2789 2791 2797 2801 2803 2819 2833 2837 2843 2851 2857 2861 2879 2887 2897 2903 421 -- 440 2909 2917 2927 2939 2953 2957 2963 2969 2971 2999 3001 3011 3019 3023 3037 3041 3049 3061 3067 3079 441 -- 460 3083 3089 3109 3119 3121 3137 3163 3167 3169 3181 3187 3191 3203 3209 3217 3221 3229 3251 3253 3257 461 -- 480 3259 3271 3299 3301 3307 3313 3319 3323 3329 3331 3343 3347 3359 3361 3371 3373 3389 3391 3407 3413 481 -- 500 3433 3449 3457 3461 3463 3467 3469 3491 3499 3511 3517 3527 3529 3533 3539 3541 3547 3557 3559 3571 501 -- 520 3581 3583 3593 3607 3613 3617 3623 3631 3637 3643 3659 3671 3673 3677 3691 3697 3701 3709 3719 3727 521 -- 540 3733 3739 3761 3767 3769 3779 3793 3797 3803 3821 3823 3833 3847 3851 3853 3863 3877 3881 3889 3907 541 -- 560 3911 3917 3919 3923 3929 3931 3943 3947 3967 3989 4001 4003 4007 4013 4019 4021 4027 4049 4051 4057 561 -- 580 4073 4079 4091 4093 4099 4111 4127 4129 4133 4139 4153 4157 4159 4177 4201 4211 4217 4219 4229 4231 581 -- 600 4241 4243 4253 4259 4261 4271 4273 4283 4289 4297 4327 4337 4339 4349 4357 4363 4373 4391 4397 4409 601 -- 620 4421 4423 4441 4447 4451 4457 4463 4481 4483 4493 4507 4513 4517 4519 4523 4547 4549 4561 4567 4583 621 -- 640 4591 4597 4603 4621 4637 4639 4643 4649 4651 4657 4663 4673 4679 4691 4703 4721 4723 4729 4733 4751 641 -- 660 4759 4783 4787 4789 4793 4799 4801 4813 4817 4831 4861 4871 4877 4889 4903 4909 4919 4931 4933 4937 661 -- 680 4943 4951 4957 4967 4969 4973 4987 4993 4999 5003 5009 5011 5021 5023 5039 5051 5059 5077 5081 5087 681 -- 700 5099 5101 5107 5113 5119 5147 5153 5167 5171 5179 5189 5197 5209 5227 5231 5233 5237 5261 5273 5279 701 -- 720 5281 5297 5303 5309 5323 5333 5347 5351 5381 5387 5393 5399 5407 5413 5417 5419 5431 5437 5441 5443 721 -- 740 5449 5471 5477 5479 5483 5501 5503 5507 5519 5521 5527 5531 5557 5563 5569 5573 5581 5591 5623 5639 741 -- 760 5641 5647 5651 5653 5657 5659 5669 5683 5689 5693 5701 5711 5717 5737 5741 5743 5749 5779 5783 5791 761 -- 780 5801 5807 5813 5821 5827 5839 5843 5849 5851 5857 5861 5867 5869 5879 5881 5897 5903 5923 5927 5939 781 -- 800 5953 5981 5987 6007 6011 6029 6037 6043 6047 6053 6067 6073 6079 6089 6091 6101 6113 6121 6131 6133 801 -- 820 6143 6151 6163 6173 6197 6199 6203 6211 6217 6221 6229 6247 6257 6263 6269 6271 6277 6287 6299 6301 821 -- 840 6311 6317 6323 6329 6337 6343 6353 6359 6361 6367 6373 6379 6389 6397 6421 6427 6449 6451 6469 6473 841 -- 860 6481 6491 6521 6529 6547 6551 6553 6563 6569 6571 6577 6581 6599 6607 6619 6637 6653 6659 6661 6673 861 -- 880 6679 6689 6691 6701 6703 6709 6719 6733 6737 6761 6763 6779 6781 6791 6793 6803 6823 6827 6829 6833 881 -- 900 6841 6857 6863 6869 6871 6883 6899 6907 6911 6917 6947 6949 6959 6961 6967 6971 6977 6983 6991 6997 901 -- 920 7001 7013 7019 7027 7039 7043 7057 7069 7079 7103 7109 7121 7127 7129 7151 7159 7177 7187 7193 7207 921 -- 940 7211 7213 7219 7229 7237 7243 7247 7253 7283 7297 7307 7309 7321 7331 7333 7349 7351 7369 7393 7411 941 -- 960 7417 7433 7451 7457 7459 7477 7481 7487 7489 7499 7507 7517 7523 7529 7537 7541 7547 7549 7559 7561 961 -- 980 7573 7577 7583 7589 7591 7603 7607 7621 7639 7643 7649 7669 7673 7681 7687 7691 7699 7703 7717 7723 981 -- 1000 7727 7741 7753 7757 7759 7789 7793 7817 7823 7829 7841 7853 7867 7873 7877 7879 7883 7901 7907 7919 ( sequence A000040 in the OEIS ). The Goldbach conjecture verification project reports that it has computed all primes below 4 × 10. That means 95,676,260,903,887,607 primes ( nearly 10 ), but they were not stored. There are known formulae to evaluate the prime - counting function ( the number of primes below a given value ) faster than computing the primes. This has been used to compute that there are 1,925,320,391,606,803,968,923 primes ( roughly 2 × 10 ) below 10. A different computation found that there are 18,435,599,767,349,200,867,866 primes ( roughly 2 × 10 ) below 10, if the Riemann hypothesis is true. Lists of primes by type Below are listed the first prime numbers of many named forms and types. More details are in the article for the name. n is a natural number ( including 0 ) in the definitions. Additive primes Primes such that the sum of digits with 10 as the base 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 ) = seq ( 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. 3, 7, 11, 17, 47, 53, 61, 67, 73, 79, 89, 101, 139, 151, 157, 191, 199 ( A072456 ) Bell primes Primes that are the number of partitions of a set with n members. 2, 5, 877, 27644437, 35742549198872617291353508656626642567, 359334085968622831041960188598043661065388726959079837. The next term has 6,539 digits. ( A051131 ) Carol primes Of the form ( 2 − 1 ) − 2. 7, 47, 223, 3967, 16127, 1046527, 16769023, 1073676287, 68718952447, 274876858367, 4398042316799, 1125899839733759, 18014398241046527, 1298074214633706835075030044377087 ( A091516 ) Centered decagonal primes Of the form 5 ( n + 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 ( 7n − 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 ( A144974 ) Centered square primes Of the form n + ( n + 1 ). 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 ( 3n + 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. Class 1 + primes Of the form 2 3 − 1 for some integers u, v ≥ 0. 2, 3, 5, 7, 11, 17, 23, 31, 47, 53, 71, 107, 127, 191, 383, 431, 647, 863, 971, 1151, 2591, 4373, 6143, 6911, 8191, 8747, 13121, 15551, 23327, 27647, 62207, 73727, 131071, 139967, 165887 ( A005105 ) Cousin primes See also : § Twin primes, § Prime triplets, and § Prime quadruplets 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 3 − y 3 x − y, ( displaystyle ( \tfrac ( x ^ ( 3 ) - y ^ ( 3 ) ) ( x-y ) ), ) 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 3 − y 3 x − y, ( displaystyle ( \tfrac ( x ^ ( 3 ) - y ^ ( 3 ) ) ( x-y ) ), ) 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 × 2 + 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 2 − 1 for prime p. 7, 127, 2147483647, 170141183460469231731687303715884105727 ( primes in A077586 ) As of June 2017, 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 p # + 1 ( a subset of primorial primes ). 3, 7, 31, 211, 2311, 200560490131 ( A018239 ) Euler irregular primes A prime p ( displaystyle p ) that divides Euler number E 2 n ( displaystyle E_ ( 2n ) ) for some 0 ≤ 2 n ≤ p − 3 ( displaystyle 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 ( A120337 ) Euler ( p, p − 3 ) irregular primes Primes p ( displaystyle p ) such that ( p, p − 3 ) ( displaystyle ( p, p - 3 ) ) is an Euler irregular pair. 149, 241, 2946901 ( A198245 ) Even primes 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 ''. Factorial primes Of the form n ! − 1 or n ! + 1. 2, 3, 5, 7, 23, 719, 5039, 39916801, 479001599, 87178291199, 10888869450418352160768000001, 265252859812191058636308479999999, 263130836933693530167218012159999999, 8683317618811886495518194401279999999 ( A088054 ) Fermat primes Of the form 2 + 1. 3, 5, 17, 257, 65537 ( A019434 ) As of April 2017 these are the only known Fermat primes, and conjecturally the only Fermat primes. Fibonacci primes Primes in the Fibonacci sequence F = 0, F = 1, F = F + F. 2, 3, 5, 13, 89, 233, 1597, 28657, 514229, 433494437, 2971215073, 99194853094755497, 1066340417491710595814572169, 19134702400093278081449423917 ( 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 Of the form a + 1 for fixed integer a. a = 2 : 3, 5, 17, 257, 65537 ( A019434 ) a = 4 : 5, 17, 257, 65537 a = 6 : 7, 37, 1297 a = 8 : ( does not exist ) a = 10 : 11, 101 a = 12 : 13 a = 14 : 197 a = 16 : 17, 257, 65537 a = 18 : 19 a = 20 : 401, 160001 a = 22 : 23 a = 24 : 577, 331777 As of April 2017 these are the only known generalized Fermat primes for a ≤ 24. Generalized repunit primes Of the form ( a − 1 ) / ( a − 1 ) for fixed integer a. a = 2 : 3, 7, 31, 127, 8191, 131071, 524287, 2147483647, 2305843009213693951, 618970019642690137449562111, 162259276829213363391578010288127 ( A000668 ) a = 3 : 13, 1093, 797161, 3754733257489862401973357979128773, 6957596529882152968992225251835887181478451547013 ( A076481 ) a = 4 : 5 ( the only prime for a = 4 ) a = 5 : 31, 19531, 12207031, 305175781, 177635683940025046467781066894531, 14693679385278593849609206715278070972733319459651094018859396328480215743184089660644531 ( A086122 ) a = 6 : 7, 43, 55987, 7369130657357778596659, 3546245297457217493590449191748546458005595187661976371 ( A165210 ) a = 7 : 2801, 16148168401, 85053461164796801949539541639542805770666392330682673302530819774105141531698707146930307290253537320447270457 a = 8 : 73 ( the only prime for a = 8 ) a = 9 : ( does not exist ) a = 10 : 11, 1111111111111111111, 11111111111111111111111 ( A004022 ) a = 11 : 50544702849929377, 6115909044841454629, 1051153199500053598403188407217590190707671147285551702341089650185945215953 a = 12 : 13, 157, 22621, 29043636306420266077, 43570062353753446053455610056679740005056966111842089407838902783209959981593077811330507328327968191581 Genocchi primes 17 The only positive prime Genocchi number is 17. If negative primes are allowed, then the only other one is - 3. 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. 29, 683, 997, 2207, 30571351 ( A046850 ; another entry A135995 is erroneous ) Good primes 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 ( 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 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. 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 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. 37 ( P, p − 9 ) irregular primes Primes p such that ( p, p − 9 ) is an irregular pair. 67, 877 ( A212557 ) 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 ( 2 + 1 ) − 2. 2, 7, 23, 79, 1087, 66047, 263167, 16785407, 1073807359, 17180131327, 68720001023, 4398050705407, 70368760954879, 18014398777917439, 18446744082299486207 ( A091514 ) Left - truncatable primes See also : § Right - truncatable primes, and § Two - sided 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 x + y, with 1 < x ≤ y. 17, 593, 32993, 2097593, 8589935681, 59604644783353249, 523347633027360537213687137, 43143988327398957279342419750374600193 ( A094133 ) Long primes Primes p for which, in a given base b, b p − 1 − 1 p ( displaystyle ( \frac ( b ^ ( p - 1 ) - 1 ) ( p ) ) ) 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 L = 2, L = 1, L = L + L. 2, 3, 7, 11, 29, 47, 199, 521, 2207, 3571, 9349, 3010349, 54018521, 370248451, 6643838879, 119218851371, 5600748293801, 688846502588399, 32361122672259149 ( 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 x + y + p = 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 2 − 1. 3, 7, 31, 127, 8191, 131071, 524287, 2147483647, 2305843009213693951, 618970019642690137449562111, 162259276829213363391578010288127, 170141183460469231731687303715884105727 ( A000668 ) As of 2018, there are 50 known Mersenne primes. The 13th, 14th, and 50th have respectively 157, 183, and 23,249,425 digits. 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 ( A000043 ) As of January 2018 five more are known to be in the sequence, but it is not known whether they are the next : 42643801, 43112609, 57885161, 74207281, 77232917 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 ( 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 p. 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 a ( 10 m − 1 ) 9 ± b × 10 m − 1 2 ( displaystyle ( \frac ( a ( \big ( ) 10 ^ ( m ) - 1 ( \big ) ) ) ( 9 ) ) pm b \times 10 ^ ( \frac ( m - 1 ) ( 2 ) ) ) with 0 ≤ a ± b < 10 ( displaystyle 0 leq a pm b < 10 ). This means all digits except the middle digit are equal. 101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, 797, 919, 929, 11311, 11411, 33533, 77377, 77477, 77977, 1114111, 1117111, 3331333, 3337333, 7772777, 7774777, 7778777, 111181111, 111191111, 777767777, 77777677777, 99999199999 ( 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 P = 0, P = 1, P = 2P + P. 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 2 3 + 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 n + 1 Of the form n + 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 p # ± 1. 3, 5, 7, 29, 31, 211, 2309, 2311, 30029, 200560490131, 304250263527209, 23768741896345550770650537601358309 ( union of A057705 and A018239 ) 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 ( 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 See also : § Cousin primes, § Twin primes, and § Prime triplets 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 x + xy + 2y, 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 x + y, where x, y > 0. 2, 17, 97, 257, 337, 641, 881 ( A002645 ) Ramanujan primes 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 ( 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 ( A040117 ) 12n + 7 : 7, 19, 31, 43, 67, 79, 103, 127, 139, 151, 163, 199, 211, 223, 271 ( A068229 ) 12n + 11 : 11, 23, 47, 59, 71, 83, 107, 131, 167, 179, 191, 227, 239, 251, 263 ( A068231 )... 10n+d ( d = 1, 3, 7, 9 ) are primes ending in the decimal digit d. Right - truncatable primes See also : § Left - truncatable primes, and § Two - sided 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 can not 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 primes Primes that are the concatenation of the first n number written in decimal. As of April 2017, no Smarandache primes are known, and if exist, the smallest one must be with n > 200000. Smarandache -- Wellin primes Primes that are the concatenation of the first n primes written in decimal. 2, 23, 2357 ( A069151 ) 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 ± 2 ± 1, where 0 < b < a. 3, 5, 7, 11, 13 ( A165255 ) 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 n ≀ ± 1 ( displaystyle n wr pm 1 ), where n ≀ ( displaystyle n wr ) denotes the swinging factorial, which is defined in terms of the double swinging factorial as n ≀ = ( n − 1 ) ≀ ≀ n ≀ ≀ ( displaystyle n wr = ( n - 1 ) wr wr n wr wr ) and n ≀ ≀ = ( 1 n ⩽ 0 ( n − 2 ) ≀ ≀ n ( n odd ) ( 4 / n ) ( n even ) n > 0 ( displaystyle n wr wr = ( \begin ( cases ) 1 qquad qquad qquad qquad qquad qquad quad \n leqslant 0 ( n - 2 ) wr wr n ^ ( ( \big ( ) ( \text ( n odd ) ) ( \big ) ) ) ( 4 / n ) ^ ( ( \big ( ) ( \text ( n even ) ) ( \big ) ) ) quad n > 0 end ( cases ) ) ) 2, 3, 5, 7, 19, 29, 31, 71, 139, 251, 631, 3433, 12011 ( A163074 ) Thabit primes Of the form 3 × 2 − 1. 2, 5, 11, 23, 47, 191, 383, 6143, 786431, 51539607551, 824633720831, 26388279066623, 108086391056891903, 55340232221128654847, 226673591177742970257407 ( A007505 ) The primes of the form 3 × 2 + 1 are related. 7, 13, 97, 193, 769, 12289, 786433, 3221225473, 206158430209, 6597069766657 ( A039687 ) Prime triplets See also : § Cousin primes, § Twin primes, and § Prime quadruplets 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 See also : § Cousin primes, § Prime triplets, and § Prime quadruplets 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 See also : § Right - truncatable primes, and § Left - truncatable 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 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 ( 2 + 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 p divides the Fibonacci number F p − ( p 5 ) ( displaystyle F_ ( p - left ( ( \frac ( p ) ( 5 ) ) \right ) ) ), where the Legendre symbol ( p 5 ) ( displaystyle left ( ( \frac ( p ) ( 5 ) ) \right ) ) is defined as ( p 5 ) = ( 1 if p ≡ ± 1 ( mod 5 ) − 1 if p ≡ ± 2 ( mod 5 ). ( displaystyle left ( ( \frac ( p ) ( 5 ) ) \right ) = ( \begin ( cases ) 1& ( \textrm ( if ) ) ; p equiv pm 1 ( pmod ( 5 ) ) - 1& ( \textrm ( if ) ) ; p equiv pm 2 ( pmod ( 5 ) ). end ( cases ) ) ) As of 2015, no Wall - Sun - Sun primes are known. Wedderburn - Etherington 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 a ≡ 1 ( mod p ) for fixed integer a > 1. 2 ≡ 1 ( mod p ) : 1093, 3511 ( A001220 ) 3 ≡ 1 ( mod p ) : 11, 1006003 ( A014127 ) 4 ≡ 1 ( mod p ) : 1093, 3511 5 ≡ 1 ( mod p ) : 2, 20771, 40487, 53471161, 1645333507, 6692367337, 188748146801 ( A123692 ) 6 ≡ 1 ( mod p ) : 66161, 534851, 3152573 ( A212583 ) 7 ≡ 1 ( mod p ) : 5, 491531 ( A123693 ) 8 ≡ 1 ( mod p ) : 3, 1093, 3511 9 ≡ 1 ( mod p ) : 2, 11, 1006003 10 ≡ 1 ( mod p ) : 3, 487, 56598313 ( A045616 ) 11 ≡ 1 ( mod p ) : 71 12 ≡ 1 ( mod p ) : 2693, 123653 ( A111027 ) 13 ≡ 1 ( mod p ) : 2, 863, 1747591 ( A128667 ) 14 ≡ 1 ( mod p ) : 29, 353, 7596952219 ( A234810 ) 15 ≡ 1 ( mod p ) : 29131, 119327070011 ( A242741 ) 16 ≡ 1 ( mod p ) : 1093, 3511 17 ≡ 1 ( mod p ) : 2, 3, 46021, 48947 ( A128668 ) 18 ≡ 1 ( mod p ) : 5, 7, 37, 331, 33923, 1284043 ( A244260 ) 19 ≡ 1 ( mod p ) : 3, 7, 13, 43, 137, 63061489 ( A090968 ) 20 ≡ 1 ( mod p ) : 281, 46457, 9377747, 122959073 ( A242982 ) 21 ≡ 1 ( mod p ) : 2 22 ≡ 1 ( mod p ) : 13, 673, 1595813, 492366587, 9809862296159 ( A298951 ) 23 ≡ 1 ( mod p ) : 13, 2481757, 13703077, 15546404183, 2549536629329 ( A128669 ) 24 ≡ 1 ( mod p ) : 5, 25633 25 ≡ 1 ( mod p ) : 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 p divides ( p − 1 ) ! + 1. 5, 13, 563 ( A007540 ) As of 2015, these are the only known Wilson primes. Wolstenholme primes Primes p for which the binomial coefficient ( 2 p − 1 p − 1 ) ≡ 1 ( mod p 4 ). ( displaystyle ( ( 2p - 1 ) choose ( p - 1 ) ) equiv 1 ( pmod ( p ^ ( 4 ) ) ). ) 16843, 2124679 ( A088164 ) As of 2015, these are the only known Wolstenholme primes. Woodall primes Of the form n × 2 − 1. 7, 23, 383, 32212254719, 2833419889721787128217599, 195845982777569926302400511, 4776913109852041418248056622882488319 ( A050918 ) 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 References Jump up ^ Lehmer, D.N. ( 1982 ). List of prime numbers from 1 to 10,006,721. 165. Washington D.C. : Carnegie Institution of Washington. OL16553580M. Archived from the original on 20 April 2013. Jump up ^ Tomás Oliveira e Silva, Goldbach conjecture verification Archived 24 May 2011 at the Wayback Machine... Retrieved 16 July 2013 Jump up ^ ( sequence A080127 in the OEIS ) Jump up ^ Jens Franke ( 29 July 2010 ). `` Conditional Calculation of pi ( 10 ) ''. Archived from the original on 24 August 2014. Retrieved 17 May 2011. Jump up ^ L. Halbeisen, N. Hungerbühler, Number theoretic aspects of a combinatorial function Archived 15 April 2012 at the Wayback Machine. ^ Jump up to : A018239 includes 2 = empty product of first 0 primes plus 1, but 2 is excluded in this list. Jump up ^ W., Weisstein, Eric. `` Odd Prime ''. mathworld.wolfram.com. Archived from the original on 21 September 2017. Retrieved 3 May 2018. Jump up ^ Weisstein, Eric W. `` Genocchi Number ''. MathWorld. Jump up ^ Russo, F., A Set of New Samarandache Functions, Sequences and Conjectures in Number Theory ( PDF ), pp. 73 -- 74, archived ( PDF ) from the original on 18 July 2011 Jump up ^ 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. Archived from the original on 27 January 2016. ^ Jump up to : Johnson, W. ( 1975 ). `` Irregular Primes and Cyclotomic Invariants '' ( PDF ). Mathematics of Computation. AMS. 29 ( 129 ) : 113 -- 120. doi : 10.2307 / 2005468. Archived from the original ( PDF ) on 20 December 2010. Jump up ^ It varies whether L = 2 is included in the Lucas numbers. Jump up ^ Caldwell, C. ; Dubner, H. ( 1996 -- 97 ). `` The near repdigit primes A n − k − 1 B 1 A k ( displaystyle A_ ( n-k - 1 ) B_ ( 1 ) A_ ( k ) ), especially 9 n − k − 1 8 1 9 k ( displaystyle 9_ ( n-k - 1 ) 8_ ( 1 ) 9_ ( k ) ) ``. Journal of Recreational Mathematics. 28 ( 1 ) : 1 -- 9. Jump up ^ Lal, M. ( 1967 ). `` Primes of the Form n + 1 '' ( PDF ). Mathematics of Computation. AMS. 21 : 245 -- 247. doi : 10.1090 / S0025 - 5718 - 1967 - 0222007 - 9. ISSN 1088 - 6842. Archived ( PDF ) from the original on 13 January 2015. Jump up ^ Bohman, J. ( 1973 ). `` New primes of the form n + 1 ''. BIT Numerical Mathematics. Springer. 13 ( 3 ) : 370 -- 372. doi : 10.1007 / BF01951947. ISSN 1572 - 9125. Jump up ^ Luschny, Swinging factorial Archived 28 November 2012 at the Wayback Machine. Jump up ^ Ribenboim, P. The new book of prime number records. New York : Springer - Verlag. p. 347. ISBN 0 - 387 - 94457 - 5. Jump up ^ `` Mirimanoff 's Congruence : Other Congruences ''. Retrieved 26 January 2011. Jump up ^ Gallot, Y. ; Moree, P. ; Zudilin, W. ( 2011 ). `` The Erdös - Moser equation 1 + 2 +... + ( m − 1 ) = m revisited using continued fractions ''. Mathematics of Computation. American Mathematical Society. 80 : 1221 -- 1237. arXiv : 0907.1356. doi : 10.1090 / S0025 - 5718 - 2010 - 02439 - 1. ^ Jump up to : Ribenboim, P. ( 2006 ). Die Welt der Primzahlen. Berlin : Springer. p. 240. ISBN 3 - 540 - 34283 - 4. Archived from the original on 6 November 2012. 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 ) ( in German ) ( Lists Wieferich primes in all bases up to 1052 ) Padilla, Tony. `` New Largest Known Prime Number ''. Numberphile. Brady Haran. Prime number classes By formula Fermat ( 2 + 1 ) Mersenne ( 2 − 1 ) Double Mersenne ( 2 − 1 ) Wagstaff ( 2 + 1 ) / 3 Proth ( k 2 + 1 ) Factorial ( n ! ± 1 ) Primorial ( p # ± 1 ) Euclid ( p # + 1 ) Pythagorean ( 4n + 1 ) Pierpont ( 2 3 + 1 ) Quartan ( x + y ) Solinas ( 2 ± 2 ± 1 ) Cullen ( n 2 + 1 ) Woodall ( n 2 − 1 ) Cuban ( x − y ) / ( x − y ) Carol ( 2 − 1 ) − 2 Kynea ( 2 + 1 ) − 2 Leyland ( x + y ) Thabit ( 3 2 − 1 ) Mills ( ⌊ A ⌋ ) By integer sequence Fibonacci Lucas Pell Newman -- Shanks -- Williams Perrin Partitions Bell Motzkin By property Wieferich ( pair ) Wall -- Sun -- Sun Wolstenholme Wilson Lucky Fortunate Ramanujan Pillai Regular Strong Stern Supersingular ( elliptic curve ) Supersingular ( moonshine theory ) Good Super Higgs Highly cototient Base - dependent Happy Dihedral Palindromic Emirp Repunit ( 10 − 1 ) / 9 Permutable Circular Truncatable Strobogrammatic Minimal Weakly Full reptend Unique Primeval Self Smarandache -- Wellin Patterns Twin ( p, p + 2 ) Bi-twin chain ( n − 1, n + 1, 2n − 1, 2n + 1,... ) Triplet ( p, p + 2 or p + 4, p + 6 ) Quadruplet ( p, p + 2, p + 6, p + 8 ) k − Tuple Cousin ( p, p + 4 ) Sexy ( p, p + 6 ) Chen Sophie Germain ( p, 2p + 1 ) Cunningham chain ( p, 2p ± 1,... ) Safe ( p, ( p − 1 ) / 2 ) Arithmetic progression ( p + a n, n = 0, 1,... ) Balanced ( consecutive p − n, p, p + n ) By size Titanic ( 1,000 + digits ) Gigantic ( 10,000 + digits ) Mega ( 1,000,000 + digits ) Largest known Complex numbers Eisenstein prime Gaussian prime Composite numbers Pseudoprime Almost prime Semiprime Interprime Related topics Probable prime Industrial - grade prime Illegal prime Formula for primes Prime gap First 50 primes 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211 223 227 229 List of prime numbers Retrieved from `` https://en.wikipedia.org/w/index.php?title=List_of_prime_numbers&oldid=841579987 '' Categories : Prime numbers Classes of prime numbers Mathematics - related lists Hidden categories : Webarchive template wayback links Dynamic lists Articles containing potentially dated statements from April 2017 All articles containing potentially dated statements Articles containing potentially dated statements from 2018 Articles containing potentially dated statements from January 2018 Articles containing potentially dated statements from 2011 Articles containing potentially dated statements from 2015 Articles with German - language external links Use dmy dates from May 2011 Talk Contents About Wikipedia Wikidata Aragonés Bosanski Čeština Cymraeg Emiliàn e rumagnòl Español Français Galego 한국어 Italiano Magyar Nederlands 日本 語 Norsk nynorsk Oʻzbekcha / ўзбекча Português Русский Simple English Slovenčina Српски / srpski Svenska தமிழ் ไทย Türkçe Українська Tiếng Việt 中文 19 more Edit links This page was last edited on 16 May 2018, at 18 : 07. 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Question: what is the smallest 4 digit prime number
Answer: 1009