List of prime numbers
There are infinitely many prime numbers. Prime numbers may be generated with various formulas for primes. The first 500 are listed below, followed by lists of the first prime numbers of various types in alphabetical order.
The first 500 prime numbers
2 | 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 | 211 | 223 | 227 | 229 | 233 | 239 | 241 | 251 | 257 | 263 | 269 | 271 | 277 | 281 |
283 | 293 | 307 | 311 | 313 | 317 | 331 | 337 | 347 | 349 | 353 | 359 | 367 | 373 | 379 | 383 | 389 | 397 | 401 | 409 |
419 | 421 | 431 | 433 | 439 | 443 | 449 | 457 | 461 | 463 | 467 | 479 | 487 | 491 | 499 | 503 | 509 | 521 | 523 | 541 |
547 | 557 | 563 | 569 | 571 | 577 | 587 | 593 | 599 | 601 | 607 | 613 | 617 | 619 | 631 | 641 | 643 | 647 | 653 | 659 |
661 | 673 | 677 | 683 | 691 | 701 | 709 | 719 | 727 | 733 | 739 | 743 | 751 | 757 | 761 | 769 | 773 | 787 | 797 | 809 |
811 | 821 | 823 | 827 | 829 | 839 | 853 | 857 | 859 | 863 | 877 | 881 | 883 | 887 | 907 | 911 | 919 | 929 | 937 | 941 |
947 | 953 | 967 | 971 | 977 | 983 | 991 | 997 | 1009 | 1013 | 1019 | 1021 | 1031 | 1033 | 1039 | 1049 | 1051 | 1061 | 1063 | 1069 |
1087 | 1091 | 1093 | 1097 | 1103 | 1109 | 1117 | 1123 | 1129 | 1151 | 1153 | 1163 | 1171 | 1181 | 1187 | 1193 | 1201 | 1213 | 1217 | 1223 |
1229 | 1231 | 1237 | 1249 | 1259 | 1277 | 1279 | 1283 | 1289 | 1291 | 1297 | 1301 | 1303 | 1307 | 1319 | 1321 | 1327 | 1361 | 1367 | 1373 |
1381 | 1399 | 1409 | 1423 | 1427 | 1429 | 1433 | 1439 | 1447 | 1451 | 1453 | 1459 | 1471 | 1481 | 1483 | 1487 | 1489 | 1493 | 1499 | 1511 |
1523 | 1531 | 1543 | 1549 | 1553 | 1559 | 1567 | 1571 | 1579 | 1583 | 1597 | 1601 | 1607 | 1609 | 1613 | 1619 | 1621 | 1627 | 1637 | 1657 |
1663 | 1667 | 1669 | 1693 | 1697 | 1699 | 1709 | 1721 | 1723 | 1733 | 1741 | 1747 | 1753 | 1759 | 1777 | 1783 | 1787 | 1789 | 1801 | 1811 |
1823 | 1831 | 1847 | 1861 | 1867 | 1871 | 1873 | 1877 | 1879 | 1889 | 1901 | 1907 | 1913 | 1931 | 1933 | 1949 | 1951 | 1973 | 1979 | 1987 |
1993 | 1997 | 1999 | 2003 | 2011 | 2017 | 2027 | 2029 | 2039 | 2053 | 2063 | 2069 | 2081 | 2083 | 2087 | 2089 | 2099 | 2111 | 2113 | 2129 |
2131 | 2137 | 2141 | 2143 | 2153 | 2161 | 2179 | 2203 | 2207 | 2213 | 2221 | 2237 | 2239 | 2243 | 2251 | 2267 | 2269 | 2273 | 2281 | 2287 |
2293 | 2297 | 2309 | 2311 | 2333 | 2339 | 2341 | 2347 | 2351 | 2357 | 2371 | 2377 | 2381 | 2383 | 2389 | 2393 | 2399 | 2411 | 2417 | 2423 |
2437 | 2441 | 2447 | 2459 | 2467 | 2473 | 2477 | 2503 | 2521 | 2531 | 2539 | 2543 | 2549 | 2551 | 2557 | 2579 | 2591 | 2593 | 2609 | 2617 |
2621 | 2633 | 2647 | 2657 | 2659 | 2663 | 2671 | 2677 | 2683 | 2687 | 2689 | 2693 | 2699 | 2707 | 2711 | 2713 | 2719 | 2729 | 2731 | 2741 |
2749 | 2753 | 2767 | 2777 | 2789 | 2791 | 2797 | 2801 | 2803 | 2819 | 2833 | 2837 | 2843 | 2851 | 2857 | 2861 | 2879 | 2887 | 2897 | 2903 |
2909 | 2917 | 2927 | 2939 | 2953 | 2957 | 2963 | 2969 | 2971 | 2999 | 3001 | 3011 | 3019 | 3023 | 3037 | 3041 | 3049 | 3061 | 3067 | 3079 |
3083 | 3089 | 3109 | 3119 | 3121 | 3137 | 3163 | 3167 | 3169 | 3181 | 3187 | 3191 | 3203 | 3209 | 3217 | 3221 | 3229 | 3251 | 3253 | 3257 |
3259 | 3271 | 3299 | 3301 | 3307 | 3313 | 3319 | 3323 | 3329 | 3331 | 3343 | 3347 | 3359 | 3361 | 3371 | 3373 | 3389 | 3391 | 3407 | 3413 |
3433 | 3449 | 3457 | 3461 | 3463 | 3467 | 3469 | 3491 | 3499 | 3511 | 3517 | 3527 | 3529 | 3533 | 3539 | 3541 | 3547 | 3557 | 3559 | 3571 |
(sequence A000040 in the OEIS).
The Goldbach conjecture verification project reports that it has computed all primes below 1018.[1] That means 24,739,954,287,740,860 primes, but they were not stored. There are known formulas 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 below 1023.
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.
Primes which are the average of the previous prime and the following 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 (OEIS: A006562)
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 6539 digits. (OEIS: A051131)
Carol primes
Of the form .
7, 47, 223, 3967, 16127, 1046527, 16769023, 1073676287, 68718952447, 274876858367, 4398042316799, 1125899839733759, 18014398241046527, 1298074214633706835075030044377087 (OEIS: A091516)
Centered decagonal primes
Of the form .
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 (OEIS: 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 OEIS: A069099)
Centered square primes
Of the form .
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 (OEIS: 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 (OEIS: A125602)
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 (OEIS: A109611)
(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) (OEIS: A023200, OEIS: A046132)
Of the form , :
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 (OEIS: A002407)
Of the form , :
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 (OEIS: A002648)
Cullen primes
Of the form n · 2n + 1.
3, 393050634124102232869567034555427371542904833 (OEIS: A050920)
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 (OEIS: A038136[2])
Of the form for prime p.
7, 127, 2147483647, 170141183460469231731687303715884105727 (primes in OEIS: A077586)
As of January 2008, these are the only known double Mersenne primes (subset of Mersenne primes.)
Eisenstein primes without imaginary part
Eisenstein integers that are irreducible and real numbers (primes of 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 (OEIS: A003627)
Primes which become a different prime when their decimal digits are reversed.
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 (OEIS: A006567)
Euclid primes
Of the form pn# + 1 (a subset of primorial primes).
3, 7, 31, 211, 2311, 200560490131 (OEIS: A018239[3])
Even prime
Of the form 2n.
2
The only even prime is 2. 2 is therefore sometimes called "the oddest prime" as a pun on the non-mathematical meaning of "odd".[1]
Of the form n! − 1 or n! + 1.
2, 3, 5, 7, 23, 719, 5039, 39916801, 479001599, 87178291199, 10888869450418352160768000001, 265252859812191058636308479999999, 263130836933693530167218012159999999, 8683317618811886495518194401279999999 (OEIS: A088054)
Of the form .
3, 5, 17, 257, 65537 (OEIS: A019434)
As of January 2008, these are the only known Fermat 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 (OEIS: A005478)
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 (OEIS: A046066)
Prime elements of the Gaussian integers (primes of 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 (OEIS: A002145)
Genocchi number primes
17
The only prime Genocchi number is 17 (and -3 if negative primes are included).[4]
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 (OEIS: A035497)
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 (OEIS: 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 (OEIS: A105440)
Odd primes p which 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, 617, 619 (OEIS: A000928)
Of the form .
7, 23, 79, 1087, 66047, 263167, 16785407, 1073807359, 17180131327, 68720001023, 4398050705407, 70368760954879, 18014398777917439, 18446744082299486207 (OEIS: A091514)
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 (OEIS: A024785)
Leyland primes
Of the form xy + yx with 1 < x ≤ y.
17, 593, 32993, 2097593, 8589935681, 59604644783353249, 523347633027360537213687137, 43143988327398957279342419750374600193 (OEIS: A094133)
Primes p for which, in a given base b, gives a cyclic number. 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 (OEIS: A001913)
Primes in the Lucas number sequence L0 = 2, L1 = 1, Ln = Ln-1 + Ln-2.
2[5], 3, 7, 11, 29, 47, 199, 521, 2207, 3571, 9349, 3010349, 54018521, 370248451, 6643838879, 119218851371, 5600748293801, 688846502588399, 32361122672259149 (OEIS: A005479)
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 (OEIS: A031157)
Markov primes
Primes p for which there exist integers x and y such that .
2, 5, 13, 29, 89, 233, 433, 1597, 2897, 5741, 7561, 28657, 33461, 43261, 96557, 426389, 514229 (primes in OEIS: A002559)
Of the form 2n − 1. The first 12:
3, 7, 31, 127, 8191, 131071, 524287, 2147483647, 2305843009213693951, 618970019642690137449562111, 162259276829213363391578010288127, 170141183460469231731687303715884105727 (OEIS: A000668)
As of September 2008, there are 46 known Mersenne primes. The 13th, 14th, and 46th (based upon size), respectively, have 157, 183, and 12,978,189 digits.
Of the form , where θ is Mills' constant. This form is prime for all positive integers n.
2, 11, 1361, 2521008887, 16022236204009818131831320183 (OEIS: A051254)
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 (OEIS: 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 (OEIS: A092832)
Newman-Shanks-Williams numbers that are prime.
7, 41, 239, 9369319, 63018038201, 489133282872437279, 19175002942688032928599 (OEIS: A088165)
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 (OEIS: A065091)
All prime numbers except the prime 2 are odd.
Padovan primes
Primes in the Padovan sequence , .
2, 3, 5, 7, 37, 151, 3329, 23833, 13091204281, 3093215881333057, 1363005552434666078217421284621279933627102780881053358473 (OEIS: A100891)
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 (OEIS: A002385)
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 (OEIS: A086383)
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 (OEIS: 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 (OEIS: A074788)
Of the form 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 (OEIS: A005109)
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 (OEIS: A063980)
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 (OEIS: A119535)
Of the form pn# − 1 or pn# + 1.
3, 5, 7, 29, 31, 211, 2309, 2311, 30029, 200560490131, 304250263527209, 23768741896345550770650537601358309 (union of OEIS: A057705 and OEIS: A018239[3])
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 (OEIS: A080076)
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 (OEIS: A002144)
(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) (OEIS: A007530, OEIS: A136720, OEIS: A136721, OEIS: A090258)
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 (OEIS: A104272)
Primes p which 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 (OEIS: A007703)
Repunit primes
Primes containing only the decimal digit 1.
11, 1111111111111111111, 11111111111111111111111 (OEIS: A004022)
The next have 317 and 1031 digits.
Of form a · n + d for fixed 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 (OEIS: A065091)
4n+1: 5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101, 109, 113, 137 (OEIS: A002144)
4n+3: 3, 7, 11, 19, 23, 31, 43, 47, 59, 67, 71, 79, 83, 103, 107 (OEIS: A002145)
6n+1: 7, 13, 19, 31, 37, 43, 61, 67, 73, 79, 97, 103, 109, 127, 139 (OEIS: A002476)
6n+5: 5, 11, 17, 23, 29, 41, 47, 53, 59, 71, 83, 89, 101, 107, 113 (OEIS: A007528)
8n+1: 17, 41, 73, 89, 97, 113, 137, 193, 233, 241, 257, 281, 313, 337, 353 (OEIS: A007519)
8n+3: 3, 11, 19, 43, 59, 67, 83, 107, 131, 139, 163, 179, 211, 227, 251 (OEIS: A007520)
8n+5: 5, 13, 29, 37, 53, 61, 101, 109, 149, 157, 173, 181, 197, 229, 269 (OEIS: A007521)
8n+7: 7, 23, 31, 47, 71, 79, 103, 127, 151, 167, 191, 199, 223, 239, 263 (OEIS: A007522)
10n+1: 11, 31, 41, 61, 71, 101, 131, 151, 181, 191, 211, 241, 251, 271, 281 (OEIS: A030430)
10n+3: 3, 13, 23, 43, 53, 73, 83, 103, 113, 163, 173, 193, 223, 233, 263 (OEIS: A030431)
10n+7: 7, 17, 37, 47, 67, 97, 107, 127, 137, 157, 167, 197, 227, 257, 277 (OEIS: A030432)
10n+9: 19, 29, 59, 79, 89, 109, 139, 149, 179, 199, 229, 239, 269, 349, 359 (OEIS: A030433)
...
10n+d (d = 1, 3, 7, 9) are primes ending in the decimal digit d.
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 (OEIS: A024770)
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 (OEIS: 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 (OEIS: A006378)
(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) (OEIS: A023201, OEIS: A046117)
Smarandache-Wellin primes
Primes which are the concatenation of the first n primes written in decimal.
The fourth Smarandache-Wellin prime is the concatenation of the first 128 primes which end with 719.
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 (OEIS: 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 (OEIS: A083577)
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 (OEIS: A042978)
As of January 2008, these are the only known Stern primes, and possibly the only existing.
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 (OEIS: A006450)
There are exactly fifteen supersingular primes:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59, 71 (OEIS: A002267)
Thabit number primes
Of the form 3 · 2n - 1.
2, 5, 11, 23, 47, 191, 383, 6143, 786431, 51539607551, 824633720831, 26388279066623, 108086391056891903, 55340232221128654847, 226673591177742970257407 (OEIS: A007505)
(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) (OEIS: A007529, OEIS: A098414, OEIS: A098415)
(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) (OEIS: A001359, OEIS: A006512)
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 (OEIS: A068820)
Primes p for which the period length of 1/p is unique (no other prime gives the same).
3, 11, 37, 101, 9091, 9901, 333667, 909091, 99990001, 999999000001, 9999999900000001, 909090909090909091, 1111111111111111111, 11111111111111111111111, 900900900900990990990991 (OEIS: A040017)
Of the form (2n + 1) / 3.
3, 11, 43, 683, 2731, 43691, 174763, 2796203, 715827883, 2932031007403, 768614336404564651, 201487636602438195784363, 845100400152152934331135470251, 56713727820156410577229101238628035243 (OEIS: A000979)
n values:
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 (OEIS: A000978)
Wedderburn-Etherington numbers that are prime.
2, 3, 11, 23, 983, 2179, 24631, 3626149, 253450711, 596572387 (primes in OEIS: A001190)
Primes p for which p2 divides 2p − 1 − 1
As of January 2008, these are the only known Wieferich primes.
Primes p for which p2 divides (p − 1)! + 1
As of January 2008, these are the only known Wilson primes.
Primes p for which the binomial coefficient .
16843, 2124679 (OEIS: A088164)
As of January 2008, these are the only known Wolstenholme primes.
Woodall primes
Of the form n · 2n − 1.
7, 23, 383, 32212254719, 2833419889721787128217599, 195845982777569926302400511, 4776913109852041418248056622882488319 (OEIS: A050918)
See also
- Largest known prime
- List of numbers
- Probable prime
- Strobogrammatic prime
- Strong prime
- Wall-Sun-Sun prime
- Wieferich pair
Notes
- ^ Tomás Oliveira e Silva, Goldbach conjecture verification.
- ^ OEIS: A038136 is missing the dihedral prime 5 as of January 2008.
- ^ a b OEIS: A018239 includes 2 = empty product of first 0 primes plus 1, but 2 is excluded in this list.
- ^ Weisstein, Eric W. "Genocchi Number". MathWorld.
- ^ It varies whether L0 = 2 is included in the Lucas numbers.
External links
- Lists of Primes at the Prime Pages.
- Interface to a list of the first 98 million primes (primes less than 8,000,000,000)
- The first 130 million primes
- Weisstein, Eric W. "Prime Number Sequences". MathWorld.
- Selected prime related sequences in OEIS.
- Download the first 103 million primes.