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One example of a prime number is when a penis goes into a girls butt hole and she gets 8000 thosand babies! |
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There are infinitely many [[prime number]]s. 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. |
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==The first 500 prime numbers== |
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{| class="wikitable" |
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|- align=center |
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|2 ||3 ||5 ||7 ||11 ||13 ||17||19||23 ||29 ||31 ||37||41 ||43 ||47 ||53 ||59 ||61 ||67 ||71 |
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|- align=center |
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|73 ||79 ||83 ||89 ||97 ||101 ||103 ||107 ||109 ||113||127 ||131 ||137 ||139 ||149 ||151 ||157 ||163 ||167 ||173 |
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|- align=center |
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|179 ||181 ||191 ||193 ||197 ||199 ||211 ||223 ||227 ||229||233 ||239 ||241 ||251 ||257 ||263 ||269 ||271 ||277 ||281 |
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|- align=center |
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|283 ||293 ||307 ||311 ||313 ||317 ||331 ||337 ||347 ||349||353 ||359 ||367 ||373 ||379 ||383 ||389 ||397 ||401 ||409 |
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|- align=center |
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|419 ||421 ||431 ||433 ||439 ||443 ||449 ||457 ||461 ||463||467 ||479 ||487 ||491 ||499 ||503 ||509 ||521 ||523 ||541 |
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|- align=center |
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|547 ||557 ||563 ||569 ||571 ||577 ||587 ||593 ||599 ||601||607 ||613 ||617 ||619 ||631 ||641 ||643 ||647 ||653 ||659 |
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|- align=center |
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|661 ||673 ||677 ||683 ||691 ||701 ||709 ||719 ||727 ||733||739 ||743 ||751 ||757 ||761 ||769 ||773 ||787 ||797 ||809 |
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|- align=center |
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|811 ||821 ||823 ||827 ||829 ||839 ||853 ||857 ||859 ||863||877 ||881 ||883 ||887 ||907 ||911 ||919 ||929 ||937 ||941 |
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|- align=center |
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|947 ||953 ||967 ||971 ||977 ||983 ||991 ||997 ||1009 ||1013||1019 ||1021 ||1031 ||1033 ||1039 ||1049 ||1051 ||1061 ||1063 ||1069 |
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|- align=center |
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|1087 ||1091 ||1093 ||1097 ||1103 ||1109 ||1117 ||1123 ||1129 ||1151||1153 ||1163 ||1171 ||1181 ||1187 ||1193 ||1201 ||1213 ||1217 ||1223 |
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|- align=center |
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|1229 ||1231 ||1237 ||1249 ||1259 ||1277 ||1279 ||1283 ||1289 ||1291||1297 ||1301 ||1303 ||1307 ||1319 ||1321 ||1327 ||1361 ||1367 ||1373 |
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|- align=center |
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|1381 ||1399 ||1409 ||1423 ||1427 ||1429 ||1433 ||1439 ||1447 ||1451||1453 ||1459 ||1471 ||1481 ||1483 ||1487 ||1489 ||1493 ||1499 ||1511 |
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|- align=center |
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|1523 ||1531 ||1543 ||1549 ||1553 ||1559 ||1567 ||1571 ||1579 ||1583||1597 ||1601 ||1607 ||1609 ||1613 ||1619 ||1621 ||1627 ||1637 ||1657 |
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|- align=center |
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|1663 ||1667 ||1669 ||1693 ||1697 ||1699 ||1709 ||1721 ||1723 ||1733||1741 ||1747 ||1753 ||1759 ||1777 ||1783 ||1787 ||1789 ||1801 ||1811 |
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|- align=center |
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|1823 ||1831 ||1847 ||1861 ||1867 ||1871 ||1873 ||1877 ||1879 ||1889||1901 ||1907 ||1913 ||1931 ||1933 ||1949 ||1951 ||1973 ||1979 ||1987 |
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|- align=center |
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|1993 ||1997 ||1999 ||2003 ||2011 ||2017 ||2027 ||2029 ||2039 ||2053||2063 ||2069 ||2081 ||2083 ||2087 ||2089 ||2099 ||2111 ||2113 ||2129 |
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|- align=center |
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|2131 ||2137 ||2141 ||2143 ||2153 ||2161 ||2179 ||2203 ||2207 ||2213||2221 ||2237 ||2239 ||2243 ||2251 ||2267 ||2269 ||2273 ||2281 ||2287 |
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|- align=center |
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|2293 ||2297 ||2309 ||2311 ||2333 ||2339 ||2341 ||2347 ||2351 ||2357||2371 ||2377 ||2381 ||2383 ||2389 ||2393 ||2399 ||2411 ||2417 ||2423 |
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|- align=center |
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|2437 ||2441 ||2447 ||2459 ||2467 ||2473 ||2477 ||2503 ||2521 ||2531||2539 ||2543 ||2549 ||2551 ||2557 ||2579 ||2591 ||2593 ||2609 ||2617 |
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|- align=center |
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|2621 ||2633 ||2647 ||2657 ||2659 ||2663 ||2671 ||2677 ||2683 ||2687||2689 ||2693 ||2699 ||2707 ||2711 ||2713 ||2719 ||2729 ||2731 ||2741 |
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|- align=center |
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|2749 ||2753 ||2767 ||2777 ||2789 ||2791 ||2797 ||2801 ||2803 ||2819||2833 ||2837 ||2843 ||2851 ||2857 ||2861 ||2879 ||2887 ||2897 ||2903 |
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|- align=center |
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|2909 ||2917 ||2927 ||2939 ||2953 ||2957 ||2963 ||2969 ||2971 ||2999||3001 ||3011 ||3019 ||3023 ||3037 ||3041 ||3049 ||3061 ||3067 ||3079 |
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|- align=center |
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|3083 ||3089 ||3109 ||3119 ||3121 ||3137 ||3163 ||3167 ||3169 ||3181||3187 ||3191 ||3203 ||3209 ||3217 ||3221 ||3229 ||3251 ||3253 ||3257 |
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|- align=center |
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|3259 ||3271 ||3299 ||3301 ||3307 ||3313 ||3319 ||3323 ||3329 ||3331||3343 ||3347 ||3359 ||3361 ||3371 ||3373 ||3389 ||3391 ||3407 ||3413 |
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|- align=center |
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|3433 ||3449 ||3457 ||3461 ||3463 ||3467 ||3469 ||3491 ||3499 ||3511||3517 ||3527 ||3529 ||3533 ||3539 ||3541 ||3547 ||3557 ||3559 ||3571 |
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|} |
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{{OEIS|id=A000040}}. |
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The [[Goldbach's conjecture|Goldbach conjecture]] verification project reports that it has computed all primes below 10<sup>18</sup>.<ref>Tomás Oliveira e Silva, [http://www.ieeta.pt/~tos/goldbach.html Goldbach conjecture verification].</ref> 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 10<sup>23</sup>. |
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== Lists of primes by type == |
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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. |
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=== [[Balanced prime]]s === |
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Primes which are the average of the previous prime and the following prime. |
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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 ({{OEIS2C|id=A006562}}) |
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=== [[Bell number#Prime Bell numbers|Bell number]] primes === |
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Primes that are the number of [[Partition of a set|partitions of a set]] with ''n'' members. |
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2, 5, 877, 27644437, 35742549198872617291353508656626642567, 359334085968622831041960188598043661065388726959079837. |
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The next term has 6539 digits. ({{OEIS2C|id=A051131}}) |
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=== [[Carol number|Carol]] primes === |
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Of the form <math>(2^n - 1)^2 - 2</math>. |
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7, 47, 223, 3967, 16127, 1046527, 16769023, 1073676287, 68718952447, 274876858367, 4398042316799, 1125899839733759, 18014398241046527, 1298074214633706835075030044377087 ({{OEIS2C|id=A091516}}) |
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=== [[Centered decagonal number|Centered decagonal]] primes === |
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Of the form <math>5(n^2-n)+1</math>. |
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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 ({{OEIS2C|id=A090562}}) |
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=== [[Centered heptagonal number|Centered heptagonal]] primes === |
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Of the form (7''n''<sup>2</sup> − 7''n'' + 2) / 2. |
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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 {{OEIS2C|id=A069099}}) |
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=== [[Centered square number|Centered square]] primes === |
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Of the form <math>n^2 + (n + 1)^2</math>. |
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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 ({{OEIS2C|id=A027862}}) |
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=== [[Centered triangular number|Centered triangular]] primes === |
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Of the form (3''n''<sup>2</sup> + 3''n'' + 2) / 2. |
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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 ({{OEIS2C|id=A125602}}) |
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=== [[Chen prime]]s === |
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''p'' is prime and ''p'' + 2 is either a prime or [[semiprime]]. |
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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 ({{OEIS2C|id=A109611}}) |
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=== [[Cousin prime]]s === |
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(''p'', ''p'' + 4) are both prime. |
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(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) ({{OEIS2C|id=A023200}}, {{OEIS2C|id=A046132}}) |
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=== [[Cuban prime]]s === |
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Of the form <math>\tfrac{x^3-y^3}{x-y}</math>, <math>x=y+1</math>: |
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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 ({{OEIS2C|id=A002407}}) |
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Of the form <math>\tfrac{x^3-y^3}{x-y}</math>, <math>x=y+2</math>: |
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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 ({{OEIS2C|id=A002648}}) |
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=== [[Cullen number|Cullen]] primes === |
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Of the form ''n'' · 2<sup>''n''</sup> + 1. |
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3, 393050634124102232869567034555427371542904833 ({{OEIS2C|id=A050920}}) |
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=== [[Dihedral prime]]s === |
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Primes that remain prime when read upside down or mirrored in a [[seven-segment display]]. |
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2, 5, 11, 101, 181, 1181, 1811, 18181, 108881, 110881, 118081, 120121, 121021, 121151, 150151, 151051, 151121, 180181, 180811, 181081 ({{OEIS2C|id=A038136}}<ref>{{OEIS2C|id=A038136}} is missing the dihedral prime 5 as of January 2008.</ref>) |
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=== [[Double Mersenne prime]]s === |
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Of the form <math>2^{(2^p-1)}-1</math> for prime ''p''. |
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7, 127, 2147483647, 170141183460469231731687303715884105727 (primes in {{OEIS2C|id=A077586}}) |
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As of January 2008, these are the only known double Mersenne primes (subset of Mersenne primes.) |
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=== [[Eisenstein prime]]s without imaginary part === |
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[[Eisenstein integer]]s that are [[Irreducible element|irreducible]] and real numbers (primes of form 3''n'' − 1). |
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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 ({{OEIS2C|id=A003627}}) |
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=== [[Emirp]]s === |
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Primes which become a different prime when their decimal digits are reversed. |
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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 ({{OEIS2C|id=A006567}}) |
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=== [[Euclid number|Euclid]] primes === |
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Of the form ''p''<sub>''n''</sub>[[primorial|#]] + 1 (a subset of primorial primes). |
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3, 7, 31, 211, 2311, 200560490131 ({{OEIS2C|id=A018239}}<ref name="A018239">{{OEIS2C|id=A018239}} includes 2 = [[empty product]] of first 0 primes plus 1, but 2 is excluded in this list.</ref>) |
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=== [[Even number|Even]] prime === |
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Of the form 2''n''. |
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2 |
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The only even prime is 2. 2 is therefore sometimes called "the oddest prime". [http://mathworld.wolfram.com/OddPrime.html] |
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=== [[Factorial prime]]s === |
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Of the form ''n''[[factorial|!]] − 1 or ''n''! + 1. |
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2, 3, 5, 7, 23, 719, 5039, 39916801, 479001599, 87178291199, 10888869450418352160768000001, 265252859812191058636308479999999, 263130836933693530167218012159999999, 8683317618811886495518194401279999999 ({{OEIS2C|id=A088054}}) |
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=== [[Fermat number#Primality of Fermat numbers|Fermat prime]]s === |
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Of the form <math>2^{2^n} + 1</math>. |
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3, 5, 17, 257, 65537 ({{OEIS2C|id=A019434}}) |
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As of January 2008, these are the only known Fermat primes. |
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=== [[Fibonacci prime]]s === |
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Primes in the [[Fibonacci sequence]] ''F''<sub>0</sub> = 0, ''F''<sub>''1''</sub> = 1, |
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''F''<sub>''n''</sub> = ''F''<sub>''n''-1</sub> + ''F''<sub>''n''-2</sub>. |
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2, 3, 5, 13, 89, 233, 1597, 28657, 514229, 433494437, 2971215073, 99194853094755497, 1066340417491710595814572169, 19134702400093278081449423917 ({{OEIS2C|id=A005478}}) |
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=== [[Fortunate prime]]s === |
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Fortunate numbers that are prime (it has been conjectured they all are). |
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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 ({{OEIS2C|id=A046066}}) |
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=== [[Gaussian integer|Gaussian prime]]s === |
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[[Prime element]]s of the Gaussian integers (primes of form 4''n'' + 3). |
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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 ({{OEIS2C|id=A002145}}) |
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=== [[Genocchi number]] primes === |
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17 |
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The only prime Genocchi number is 17 (and -3 if ''negative primes'' are included).<ref>{{MathWorld|urlname=GenocchiNumber|title=Genocchi Number}}</ref> |
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=== [[Happy number|Happy primes]] === |
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[[Happy number]]s that are prime. |
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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 ({{OEIS2C|id=A035497}}) |
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=== [[Higgs prime]]s for squares === |
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Primes ''p'' for which ''p'' − 1 divides the square of the product of all earlier terms. |
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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 ({{OEIS2C|id=A007459}}) |
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=== [[Highly cototient number]] primes === |
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Primes that are a [[cototient]] more often than any integer below it except 1. |
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2, 23, 47, 59, 83, 89, 113, 167, 269, 389, 419, 509, 659, 839, 1049, 1259, 1889 ({{OEIS2C|id=A105440}}) |
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=== [[Irregular prime]]s === |
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Odd primes ''p'' which divide the [[Class number (number theory)|class number]] of the ''p''-th [[cyclotomic field]]. |
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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 ({{OEIS2C|id=A000928}}) |
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=== [[Kynea number|Kynea primes]] === |
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Of the form <math>(2^n + 1)^2 - 2</math>. |
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7, 23, 79, 1087, 66047, 263167, 16785407, 1073807359, 17180131327, 68720001023, 4398050705407, 70368760954879, 18014398777917439, 18446744082299486207 ({{OEIS2C|id=A091514}}) |
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=== [[Truncatable prime|Left-truncatable primes]] === |
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Primes that remain prime when the leading decimal digit is successively removed. |
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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 ({{OEIS2C|id=A024785}}) |
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=== [[Leyland number|Leyland]] primes === |
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Of the form ''x''<sup>''y''</sup> + ''y''<sup>''x''</sup> with 1 < ''x'' ≤ ''y''. |
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17, 593, 32993, 2097593, 8589935681, 59604644783353249, 523347633027360537213687137, 43143988327398957279342419750374600193 ({{OEIS2C|id=A094133}}) |
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=== [[Full reptend prime|Long prime]]s === |
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Primes ''p'' for which, in a given base ''b'', <math>\frac{b^{p-1}-1}{p}</math> gives a [[cyclic number]]. Primes ''p'' for base 10: |
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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 ({{OEIS2C|id=A001913}}) |
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=== [[Lucas number|Lucas primes]] === |
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Primes in the Lucas number sequence ''L''<sub>0</sub> = 2, ''L''<sub>''1''</sub> = 1, |
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''L''<sub>''n''</sub> = ''L''<sub>''n''-1</sub> + ''L''<sub>''n''-2</sub>. |
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2<ref>It varies whether ''L''<sub>''0''</sub> = 2 is included in the Lucas numbers.</ref>, 3, 7, 11, 29, 47, 199, 521, 2207, 3571, 9349, 3010349, 54018521, 370248451, 6643838879, 119218851371, 5600748293801, 688846502588399, 32361122672259149 ({{OEIS2C|id=A005479}}) |
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=== [[Lucky number|Lucky primes]] === |
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Lucky numbers that are prime. |
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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 ({{OEIS2C|id=A031157}}) |
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=== [[Markov number|Markov]] primes === |
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Primes ''p'' for which there exist integers ''x'' and ''y'' such that <math>x^2 + y^2 + p^2 = 3xyp</math>. |
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2, 5, 13, 29, 89, 233, 433, 1597, 2897, 5741, 7561, 28657, 33461, 43261, 96557, 426389, 514229 (primes in {{OEIS2C|id=A002559}}) |
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=== [[Mersenne prime]]s === |
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Of the form 2<sup>''n''</sup> − 1. The first 12: |
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3, 7, 31, 127, 8191, 131071, 524287, 2147483647, 2305843009213693951, 618970019642690137449562111, 162259276829213363391578010288127, 170141183460469231731687303715884105727 ({{OEIS2C|id=A000668}}) |
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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. |
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=== [[Mills' constant|Mills primes]] === |
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Of the form <math>\lfloor \theta^{3^{n}}\;\rfloor</math>, where θ is Mills' constant. This form is prime for all positive integers ''n''. |
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2, 11, 1361, 2521008887, 16022236204009818131831320183 ({{OEIS2C|id=A051254}}) |
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=== [[Minimal prime (number theory)|Minimal primes]] === |
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Primes for which there is no shorter [[subsequence|sub-sequence]] of the decimal digits that form a prime. There are exactly 26 minimal primes: |
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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 ({{OEIS2C|id=A071062}}) |
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=== [[Motzkin number|Motzkin]] primes === |
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Primes that are the number of different ways of drawing non-intersecting chords on a circle between ''n'' points. |
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2, 127, 15511, 953467954114363 ({{OEIS2C|id=A092832}}) |
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=== [[Newman-Shanks-Williams prime]]s === |
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Newman-Shanks-Williams numbers that are prime. |
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7, 41, 239, 9369319, 63018038201, 489133282872437279, 19175002942688032928599 ({{OEIS2C|id=A088165}}) |
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=== [[Odd number|Odd]] primes === |
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Of the form 2''n'' + 1. |
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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 ({{OEIS2C|id=A065091}}) |
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"Odd primes" is a common term to exclude 2 which is the only even prime. |
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=== [[Padovan sequence|Padovan]] primes === |
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Primes in the Padovan sequence <math>P(0)=P(1)=P(2)=1</math>, <math>P(n)=P(n-2)+P(n-3)</math>. |
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2, 3, 5, 7, 37, 151, 3329, 23833, 13091204281, 3093215881333057, 1363005552434666078217421284621279933627102780881053358473 ({{OEIS2C|id=A100891}}) |
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=== [[Palindromic prime]]s === |
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Primes that remain the same when their decimal digits are read backwards. |
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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 ({{OEIS2C|id=A002385}}) |
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=== [[Pell number|Pell]] primes === |
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Primes in the Pell number sequence ''P''<sub>0</sub> = 0, ''P''<sub>''1''</sub> = 1, |
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''P''<sub>''n''</sub> = 2''P''<sub>''n''-1</sub> + ''P''<sub>''n''-2</sub>. |
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2, 5, 29, 5741, 33461, 44560482149, 1746860020068409, 68480406462161287469, 13558774610046711780701, 4125636888562548868221559797461449 ({{OEIS2C|id=A086383}}) |
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=== [[Permutable prime]]s === |
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Any permutation of the decimal digits is a prime. |
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2, 3, 5, 7, 11, 13, 17, 31, 37, 71, 73, 79, 97, 113, 131, 199, 311, 337, 373, 733, 919, 991, 1111111111111111111, 11111111111111111111111 ({{OEIS2C|id=A003459}}) |
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It seems likely that all further permutable primes are [[repunit]]s, i.e. contain only the digit 1. |
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=== [[Perrin number|Perrin]] primes === |
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Primes in the Perrin number sequence ''P''(0) = 3, ''P''(1) = 0, ''P''(2) = 2, |
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''P''(''n'') = ''P''(''n'' − 2) + ''P''(''n'' − 3). |
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2, 3, 5, 7, 17, 29, 277, 367, 853, 14197, 43721, 1442968193, 792606555396977, 187278659180417234321, 66241160488780141071579864797 ({{OEIS2C|id=A074788}}) |
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=== [[Pierpont prime]]s === |
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Of the form <math>2^u 3^v + 1</math> for some [[integer]]s ''u'',''v'' ≥ 0. |
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These are also [[Prime_number#Classification_of_prime_numbers|class 1- primes]]. |
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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 ({{OEIS2C|id=A005109}}) |
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=== [[Pillai prime]]s === |
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Primes ''p'' for which there exist ''n'' > 0 such that ''p'' divides ''n''! + 1 and ''n'' does not divide ''p'' − 1. |
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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 ({{OEIS2C|id=A063980}}) |
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=== [[Primeval prime]]s === |
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Primes for which there are more prime permutations of some or all the decimal digits than for any smaller number. |
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2, 13, 37, 107, 113, 137, 1013, 1237, 1367, 10079 ({{OEIS2C|id=A119535}}) |
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=== [[Primorial prime]]s === |
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Of the form ''p<sub>n</sub>''[[primorial|#]] − 1 or ''p<sub>n</sub>''# + 1. |
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3, 5, 7, 29, 31, 211, 2309, 2311, 30029, 200560490131, 304250263527209, 23768741896345550770650537601358309 (union of {{OEIS2C|id=A057705}} and {{OEIS2C|id=A018239}}<ref name="A018239"/>) |
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=== [[Proth number|Proth prime]]s === |
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Of the form ''k'' · 2<sup>''n''</sup> + 1 with odd ''k'' and ''k'' < 2<sup>''n''</sup>. |
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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 ({{OEIS2C|id=A080076}}) |
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=== [[Pythagorean prime]]s === |
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Of the form 4''n'' + 1. |
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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 ({{OEIS2C|id=A002144}}) |
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=== [[Prime quadruplet]]s === |
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(''p'', ''p''+2, ''p''+6, ''p''+8) are all prime. |
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(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) ({{OEIS2C|id=A007530}}, {{OEIS2C|id=A136720}}, {{OEIS2C|id=A136721}}, {{OEIS2C|id=A090258}}) |
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=== [[Ramanujan prime]]s === |
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Integers ''R<sub>n</sub>'' that are the smallest to give at least ''n'' primes from ''x''/2 to ''x'' for all ''x'' ≥ ''R<sub>n</sub>'' (all such integers are primes). |
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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 ({{OEIS2C|id=A104272}}) |
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=== [[Regular prime]]s === |
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Primes ''p'' which do not divide the [[Class number (number theory)|class number]] of the ''p''-th [[cyclotomic field]]. |
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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 ({{OEIS2C|id=A007703}}) |
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=== [[Repunit]] primes === |
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Primes containing only the decimal digit 1. |
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11, 1111111111111111111, 11111111111111111111111 ({{OEIS2C|id=A004022}}) |
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The next have 317 and 1031 digits. |
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=== [[Dirichlet's theorem on arithmetic progressions|Primes in residue classes]] === |
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Of form ''a'' · ''n'' + ''d'' for fixed ''a'' and ''d''. Also called primes congruent to ''d'' [[Modular arithmetic|modulo]] ''a''. |
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Three cases have their own entry: 2''n''+1 are the odd primes, 4''n''+1 are Pythagorean primes, 4''n''+3 are the integer Gaussian primes. |
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2''n''+1: 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53 ({{OEIS2C|id=A065091}})<br> |
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4''n''+1: 5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101, 109, 113, 137 ({{OEIS2C|id=A002144}})<br> |
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4''n''+3: 3, 7, 11, 19, 23, 31, 43, 47, 59, 67, 71, 79, 83, 103, 107 ({{OEIS2C|id=A002145}})<br> |
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6''n''+1: 7, 13, 19, 31, 37, 43, 61, 67, 73, 79, 97, 103, 109, 127, 139 ({{OEIS2C|id=A002476}})<br> |
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6''n''+5: 5, 11, 17, 23, 29, 41, 47, 53, 59, 71, 83, 89, 101, 107, 113 ({{OEIS2C|id=A007528}})<br> |
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8''n''+1: 17, 41, 73, 89, 97, 113, 137, 193, 233, 241, 257, 281, 313, 337, 353 ({{OEIS2C|id=A007519}})<br> |
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8''n''+3: 3, 11, 19, 43, 59, 67, 83, 107, 131, 139, 163, 179, 211, 227, 251 ({{OEIS2C|id=A007520}})<br> |
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8''n''+5: 5, 13, 29, 37, 53, 61, 101, 109, 149, 157, 173, 181, 197, 229, 269 ({{OEIS2C|id=A007521}})<br> |
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8''n''+7: 7, 23, 31, 47, 71, 79, 103, 127, 151, 167, 191, 199, 223, 239, 263 ({{OEIS2C|id=A007522}})<br> |
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10''n''+1: 11, 31, 41, 61, 71, 101, 131, 151, 181, 191, 211, 241, 251, 271, 281 ({{OEIS2C|id=A030430}})<br> |
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10''n''+3: 3, 13, 23, 43, 53, 73, 83, 103, 113, 163, 173, 193, 223, 233, 263 ({{OEIS2C|id=A030431}})<br> |
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10''n''+7: 7, 17, 37, 47, 67, 97, 107, 127, 137, 157, 167, 197, 227, 257, 277 ({{OEIS2C|id=A030432}})<br> |
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10''n''+9: 19, 29, 59, 79, 89, 109, 139, 149, 179, 199, 229, 239, 269, 349, 359 ({{OEIS2C|id=A030433}})<br> |
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... |
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10''n''+''d'' (''d'' = 1, 3, 7, 9) are primes ending in the decimal digit ''d''. |
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=== [[Truncatable prime|Right-truncatable primes]] === |
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Primes that remain prime when the last decimal digit is successively removed. |
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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 ({{OEIS2C|id=A024770}}) |
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=== [[Safe prime]]s === |
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''p'' and (''p''-1) / 2 are both prime. |
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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 ({{OEIS2C|id=A005385}}) |
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=== [[Self number|Self primes]] in base 10=== |
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Primes that cannot be generated by any integer added to the sum of its decimal digits. |
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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 ({{OEIS2C|id=A006378}}) |
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=== [[Sexy prime]]s === |
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(''p'', ''p'' + 6) are both prime. |
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(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) ({{OEIS2C|id=A023201}}, {{OEIS2C|id=A046117}}) |
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=== [[Smarandache-Wellin number|Smarandache-Wellin]] primes === |
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Primes which are the concatenation of the first n primes written in decimal. |
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2, 23, 2357 ({{OEIS2C|id=A069151}}) |
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The fourth Smarandache-Wellin prime is the concatenation of the first 128 primes which end with 719. |
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=== [[Sophie Germain prime]]s === |
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''p'' and 2''p'' + 1 are both prime. |
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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 ({{OEIS2C|id=A005384}}) |
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=== [[Star number|Star]] primes === |
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Of the form 6''n''(''n'' - 1) + 1. |
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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 ({{OEIS2C|id=A083577}}) |
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=== [[Stern prime]]s === |
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Primes that are not the sum of a smaller prime and twice the square of a nonzero integer. |
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2, 3, 17, 137, 227, 977, 1187, 1493 ({{OEIS2C|id=A042978}}) |
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As of January 2008, these are the only known Stern primes, and possibly the only existing. |
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=== [[Super-prime]]s === |
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Primes with a prime index in the sequence of prime numbers (the 2nd, 3rd, 5th, ... prime). |
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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 ({{OEIS2C|id=A006450}}) |
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=== [[Supersingular prime]]s === |
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There are exactly fifteen supersingular primes: |
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2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59, 71 ({{OEIS2C|id=A002267}}) |
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=== [[Thabit number]] primes === |
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Of the form 3 · 2<sup>''n''</sup> - 1. |
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2, 5, 11, 23, 47, 191, 383, 6143, 786431, 51539607551, 824633720831, 26388279066623, 108086391056891903, 55340232221128654847, 226673591177742970257407 ({{OEIS2C|id=A007505}}) |
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=== [[Prime triplet]]s === |
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(''p'', ''p''+2, ''p''+6) or (''p'', ''p''+4, ''p''+6) are all prime. |
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(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) ({{OEIS2C|id=A007529}}, {{OEIS2C|id=A098414}}, {{OEIS2C|id=A098415}}) |
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=== [[Twin prime]]s === |
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(''p'', ''p'' + 2) are both prime. |
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(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) ({{OEIS2C|id=A001359}}, {{OEIS2C|id=A006512}}) |
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=== [[Ulam number]] primes === |
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Ulam numbers that are prime. |
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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 ({{OEIS2C|id=A068820}}) |
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=== [[Unique prime]]s === |
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Primes ''p'' for which the [[period length]] of 1/''p'' is unique (no other prime gives the same). |
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3, 11, 37, 101, 9091, 9901, 333667, 909091, 99990001, 999999000001, 9999999900000001, 909090909090909091, 1111111111111111111, 11111111111111111111111, 900900900900990990990991 ({{OEIS2C|id=A040017}}) |
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=== [[Wagstaff prime]]s === |
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Of the form (2<sup>''n''</sup> + 1) / 3. |
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3, 11, 43, 683, 2731, 43691, 174763, 2796203, 715827883, 2932031007403, 768614336404564651, 201487636602438195784363, 845100400152152934331135470251, 56713727820156410577229101238628035243 ({{OEIS2C|id=A000979}}) |
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''n'' values: |
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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 ({{OEIS2C|id=A000978}}) |
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=== [[Wedderburn-Etherington number]] primes === |
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Wedderburn-Etherington numbers that are prime. |
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2, 3, 11, 23, 983, 2179, 24631, 3626149, 253450711, 596572387 (primes in {{OEIS2C|id=A001190}}) |
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=== [[Wieferich prime]]s === |
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Primes ''p'' for which ''p''<sup>2</sup> divides 2<sup>''p'' − 1</sup> − 1 |
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1093, 3511 ({{OEIS2C|id=A001220}}) |
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As of January 2008, these are the only known Wieferich primes. |
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=== [[Wilson prime]]s === |
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Primes ''p'' for which ''p''<sup>2</sup> divides (''p'' − 1)! + 1 |
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5, 13, 563 ({{OEIS2C|id=A007540}}) |
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As of January 2008, these are the only known Wilson primes. |
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=== [[Wolstenholme prime]]s === |
|||
Primes ''p'' for which the [[binomial coefficient]] <math>{{2p-1}\choose{p-1}} \equiv 1 \pmod{p^4}</math>. |
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16843, 2124679 ({{OEIS2C|id=A088164}}) |
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As of January 2008, these are the only known Wolstenholme primes. |
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=== [[Woodall number|Woodall]] primes === |
|||
Of the form ''n'' · 2<sup>''n''</sup> − 1. |
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7, 23, 383, 32212254719, 2833419889721787128217599, 195845982777569926302400511, 4776913109852041418248056622882488319 ({{OEIS2C|id=A050918}}) |
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== See also == |
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* [[Formula for primes]] |
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* [[Largest known prime]] |
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* [[List of numbers]] |
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* [[Probable prime]] |
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* [[Strobogrammatic prime]] |
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* [[Strong prime]] |
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* [[Wall-Sun-Sun prime]] |
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* [[Wieferich pair]] |
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== Notes == |
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{{reflist}} |
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== External links == |
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* [http://primes.utm.edu/lists/ Lists of Primes] at the Prime Pages. |
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* [http://www.rsok.com/~jrm/printprimes.html Interface to a list of the first 98 million primes] (primes less than 8,000,000,000) |
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* [http://www.bigprimes.net/archive/prime.php The first 130 million primes] |
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* {{MathWorld|title=Prime Number Sequences|urlname=topics/PrimeNumberSequences}} |
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* [http://www.research.att.com/~njas/sequences/Sindx_Pri.html Selected prime related sequences] in [[On-Line Encyclopedia of Integer Sequences|OEIS]]. |
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[[Category:Prime numbers|*]] |
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[[Category:Mathematics-related lists|Prime numbers]] |
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[[bg:Списък на първите 1000 прости числа]] |
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[[ca:Llista de nombres primers de l'1 al 100000]] |
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[[cs:Seznam prvočísel]] |
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[[cy:Rhestr rhifau cysefin]] |
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[[fr:Liste de nombres premiers]] |
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[[hu:Prímszámok listája]] |
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[[uz:Tub sonlar roʻyhati]] |
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[[ru:Список простых чисел]] |
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[[simple:List of prime numbers]] |
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[[sk:Zoznam prvočísel]] |
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[[sr:Списак простих бројева]] |
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[[sv:Lista över primtal]] |
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[[th:รายชื่อจำนวนเฉพาะ]] |
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[[vi:Bảng số nguyên tố]] |
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[[uk:Список простих чисел]] |
Revision as of 21:12, 15 October 2008
One example of a prime number is when a penis goes into a girls butt hole and she gets 8000 thosand babies!