View source for List of prime numbers

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{{short description|List of prime numbers and notable types of prime numbers}}
{{Use dmy dates|date=January 2020}}
{{Dynamic list}}
This is a list of articles about [[prime number]]s. A prime number (or ''prime'') is a [[natural number]] greater than 1 that has no positive [[divisor]]s 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 number|composite]].

== The first 1000 prime numbers ==
The following table lists the first 1000 primes, with 20 columns of consecutive primes in each of the 50 rows.<ref>{{Cite book | last = Lehmer | first = D. N. | author-link = Derrick Norman Lehmer | title = List of prime numbers from 1 to 10,006,721 | publisher = Carnegie Institution of Washington | volume = 165 | year = 1982 | location = Washington D.C. | id = OL16553580M | ol = 16553580M }}</ref>
{| class="wikitable"
!
! 1 !! 2 !! 3 !! 4 !! 5 !! 6 !! 7 !! 8 !! 9 !! 10 !! 11 !! 12 !! 13 !! 14 !! 15 !! 16 !! 17 !! 18 !! 19 !! 20
|- style="text-align: center;" 
! 1–20
| [[2]] || [[3]] || [[5]] || [[7]] || [[11 (number)|11]] || [[13 (number)|13]] ||[[17 (number)|17]] || [[19 (number)|19]] || [[23 (number)|23]] || [[29 (number)|29]] || [[31 (number)|31]] || [[37 (number)|37]] || [[41 (number)|41]] || [[43 (number)|43]] || [[47 (number)|47]] || [[53 (number)|53]] || [[59 (number)|59]] || [[61 (number)|61]] || [[67 (number)|67]] || [[71 (number)|71]]
|- style="text-align: center;" 
! 21–40
| [[73 (number)|73]] || [[79 (number)|79]] || [[83 (number)|83]] || [[89 (number)|89]] || [[97 (number)|97]] || [[101 (number)|101]]|| [[103 (number)|103]] || [[107 (number)|107]] || [[109 (number)|109]] || [[113 (number)|113]]|| [[127 (number)|127]] || [[131 (number)|131]] || [[137 (number)|137]] || [[139 (number)|139]] || [[149 (number)|149]] || [[151 (number)|151]] || [[157 (number)|157]] || [[163 (number)|163]] || [[167 (number)|167]] || [[173 (number)|173]]
|- style="text-align: center;" 
! 41–60
| [[179 (number)|179]] || [[181 (number)|181]] || [[191 (number)|191]] || [[193 (number)|193]] || [[197 (number)|197]] || [[199 (number)|199]] || [[211 (number)|211]] || [[223 (number)|223]] || [[227 (number)|227]] || [[229 (number)|229]]|| [[233 (number)|233]] || [[239 (number)|239]] || [[241 (number)|241]] || [[251 (number)|251]] || [[257 (number)|257]] || [[263 (number)|263]] || [[269 (number)|269]] || [[271 (number)|271]] || [[277 (number)|277]] || [[281 (number)|281]]
|- style="text-align: center;" 
! 61–80
| [[283 (number)|283]] || [[293 (number)|293]] || [[307 (number)|307]] || [[311 (number)|311]] || [[313 (number)|313]] || [[317 (number)|317]] || [[331 (number)|331]] || [[337 (number)|337]] || [[347 (number)|347]] || [[349 (number)|349]]|| [[353 (number)|353]] || [[359 (number)|359]] || [[367 (number)|367]] || [[373 (number)|373]] || [[379 (number)|379]] || [[383 (number)|383]] || [[389 (number)|389]] || [[397 (number)|397]] || [[401 (number)|401]] || [[409 (number)|409]]
|- style="text-align: center;" 
! 81–100
| [[419 (number)|419]] || [[421 (number)|421]] || [[431 (number)|431]] || [[433 (number)|433]] || [[439 (number)|439]] || [[443 (number)|443]] || [[449 (number)|449]] || [[457 (number)|457]] || [[461 (number)|461]] || [[463 (number)|463]]|| [[467 (number)|467]] || [[479 (number)|479]] || [[487 (number)|487]] || [[491 (number)|491]] || [[499 (number)|499]] || [[503 (number)|503]] || [[509 (number)|509]] || [[521 (number)|521]] || [[523 (number)|523]] || [[541 (number)|541]]
|- style="text-align: center;" 
! 101–120
| [[547 (number)|547]] || [[557 (number)|557]] || [[563 (number)|563]] || [[569 (number)|569]] || [[571 (number)|571]] || [[577 (number)|577]] || [[587 (number)|587]] || [[593 (number)|593]] || [[599 (number)|599]] || [[601 (number)|601]]|| [[607 (number)|607]] || [[613 (number)|613]] || [[617 (number)|617]] || [[619 (number)|619]] || [[631 (number)|631]] || [[641 (number)|641]] || [[643 (number)|643]] || [[647 (number)|647]]
|| [[653 (number)|653]] || [[659 (number)|659]]
|- style="text-align: center;" 
! 121–140
| [[661 (number)|661]] || [[673 (number)|673]] || [[677 (number)|677]] || [[683 (number)|683]] || [[691 (number)|691]] || [[701 (number)|701]] || [[709 (number)|709]] || [[719 (number)|719]] || [[727 (number)|727]] || [[733 (number)|733]]|| [[739 (number)|739]] || [[743 (number)|743]] || [[751 (number)|751]] || [[757 (number)|757]] || [[761 (number)|761]] || [[769 (number)|769]] || [[773 (number)|773]] || [[787 (number)|787]] || [[797 (number)|797]] || [[809 (number)|809]]
|- style="text-align: center;" 
! 141–160
| [[811 (number)|811]] || [[821 (number)|821]] || [[823 (number)|823]] || [[827 (number)|827]] || [[829 (number)|829]] || [[839 (number)|839]] || [[853 (number)|853]] || [[857 (number)|857]] || [[859 (number)|859]] || [[863 (number)|863]]|| [[877 (number)|877]] || [[881 (number)|881]] || [[883 (number)|883]] || [[887 (number)|887]] || [[907 (number)|907]] || [[911 (number)|911]] || [[919 (number)|919]] || [[929 (number)|929]] || [[937 (number)|937]] || [[941 (number)|941]]
|- style="text-align: center;" 
! 161–180
| [[947 (number)|947]] || [[953 (number)|953]] || [[967 (number)|967]] || [[971 (number)|971]] || [[977 (number)|977]] || [[983 (number)|983]] || [[991 (number)|991]] || [[997 (number)|997]] || [[1009 (number)|1009]] || [[1013 (number)|1013]]|| [[1019 (number)|1019]] || [[1021 (number)|1021]] || [[1031 (number)|1031]] || [[1033 (number)|1033]] || [[1039 (number)|1039]] || [[1049 (number)|1049]] || [[1051 (number)|1051]] || [[1061 (number)|1061]] || [[1063 (number)|1063]] || [[1069 (number)|1069]]
|- style="text-align: center;" 
! 181–200
| [[1087 (number)|1087]] || [[1091 (number)|1091]] || [[1093 (number)|1093]] || [[1097 (number)|1097]] || [[1103 (number)|1103]] || [[1109 (number)|1109]] || [[1117 (number)|1117]] || [[1123 (number)|1123]] || [[1129 (number)|1129]] || [[1151 (number)|1151]]|| [[1153 (number)|1153]] || [[1163 (number)|1163]] || [[1171 (number)|1171]] || [[1181 (number)|1181]] || [[1187 (number)|1187]] || [[1193 (number)|1193]] || [[1201 (number)|1201]] || [[1213 (number)|1213]] || [[1217 (number)|1217]] || [[1223 (number)|1223]]
|- style="text-align: center;" 
! 201–220
| [[1229 (number)|1229]] || [[1231 (number)|1231]] || [[1237 (number)|1237]] || [[1249 (number)|1249]] || [[1259 (number)|1259]] || [[1277 (number)|1277]] || [[1279 (number)|1279]] || [[1283 (number)|1283]] || [[1289 (number)|1289]] || [[1291 (number)|1291]]|| [[1297 (number)|1297]] || [[1301 (number)|1301]] || [[1303 (number)|1303]] || [[1307 (number)|1307]] || [[1319 (number)|1319]] || [[1321 (number)|1321]] || [[1327 (number)|1327]] || [[1361 (number)|1361]] || [[1367 (number)|1367]] || [[1373 (number)|1373]]
|- style="text-align: center;" 
! 221–240
| [[1381 (number)|1381]] || [[1399 (number)|1399]] || [[1409 (number)|1409]] || [[1423 (number)|1423]] || [[1427 (number)|1427]] || [[1429 (number)|1429]] || [[1433 (number)|1433]] || [[1439 (number)|1439]] || [[1447 (number)|1447]] || [[1451 (number)|1451]]|| [[1453 (number)|1453]] || [[1459 (number)|1459]] || [[1471 (number)|1471]] || [[1481 (number)|1481]] || [[1483 (number)|1483]] || [[1487 (number)|1487]] || [[1489 (number)|1489]] || [[1493 (number)|1493]] || [[1499 (number)|1499]] || [[1511 (number)|1511]]
|- style="text-align: center;" 
! 241–260
| [[1523 (number)|1523]] || [[1531 (number)|1531]] || [[1543 (number)|1543]] || [[1549 (number)|1549]] || [[1553 (number)|1553]] || [[1559 (number)|1559]] || [[1567 (number)|1567]] || [[1571 (number)|1571]] || [[1579 (number)|1579]] || [[1583 (number)|1583]]|| [[1597 (number)|1597]] || [[1601 (number)|1601]] || [[1607 (number)|1607]] || [[1609 (number)|1609]] || [[1613 (number)|1613]] || [[1619 (number)|1619]] || [[1621 (number)|1621]] || [[1627 (number)|1627]] || [[1637 (number)|1637]] || [[1657 (number)|1657]]
|- style="text-align: center;" 
! 261–280
| [[1663 (number)|1663]] || [[1667 (number)|1667]] || [[1669 (number)|1669]] || [[1693 (number)|1693]] || [[1697 (number)|1697]] || [[1699 (number)|1699]] || [[1709 (number)|1709]] || [[1721 (number)|1721]] || [[1723 (number)|1723]] || [[1733 (number)|1733]]|| [[1741 (number)|1741]] || [[1747 (number)|1747]] || [[1753 (number)|1753]] || [[1759 (number)|1759]] || [[1777 (number)|1777]] || [[1783 (number)|1783]] || [[1787 (number)|1787]] || [[1789 (number)|1789]] || [[1801 (number)|1801]] || [[1811 (number)|1811]]
|- style="text-align: center;" 
! 281–300
| [[1823 (number)|1823]] || [[1831 (number)|1831]] || [[1847 (number)|1847]] || [[1861 (number)|1861]] || [[1867 (number)|1867]] || [[1871 (number)|1871]] || [[1873 (number)|1873]] || [[1877 (number)|1877]] || [[1879 (number)|1879]] || [[1889 (number)|1889]]|| [[1901 (number)|1901]] || [[1907 (number)|1907]] || [[1913 (number)|1913]] || [[1931 (number)|1931]] || [[1933 (number)|1933]] || [[1949 (number)|1949]] || [[1951 (number)|1951]] || [[1973 (number)|1973]] || [[1000 (number)#1900 to 1999|1979]] || [[1987 (number)|1987]]
|- style="text-align: center;" 
! 301–320
| [[1993 (number)|1993]] || [[1997 (number)|1997]] || [[1999 (number)|1999]] || [[2003 (number)|2003]] || [[2011 (number)|2011]] || [[2017 (number)|2017]] || [[2027 (number)|2027]] || [[2029 (number)|2029]] || [[2039 (number)|2039]] || [[2053 (number)|2053]]|| [[2063 (number)|2063]] || [[2069 (number)|2069]] || [[2081 (number)|2081]] || [[2083 (number)|2083]] || [[2087 (number)|2087]] || [[2089 (number)|2089]] || [[2099 (number)|2099]] || [[2111 (number)|2111]] || [[2113 (number)|2113]] || [[2129 (number)|2129]]
|- style="text-align: center;" 
! 321–340
| [[2131 (number)|2131]] || [[2137 (number)|2137]] || [[2141 (number)|2141]] || [[2143 (number)|2143]] || [[2153 (number)|2153]] || [[2161 (number)|2161]] || [[2179 (number)|2179]] || [[2203 (number)|2203]] || [[2207 (number)|2207]] || [[2213 (number)|2213]]|| [[2221 (number)|2221]] || [[2237 (number)|2237]] || [[2239 (number)|2239]] || [[2243 (number)|2243]] || [[2251 (number)|2251]] || [[2267 (number)|2267]] || [[2269 (number)|2269]] || [[2273 (number)|2273]] || [[2281 (number)|2281]] || [[2287 (number)|2287]]
|- style="text-align: center;" 
! 341–360
| [[2293 (number)|2293]] || [[2297 (number)|2297]] || [[2309 (number)|2309]] || [[2311 (number)|2311]] || [[2333 (number)|2333]] || [[2339 (number)|2339]] || [[2341 (number)|2341]] || [[2347 (number)|2347]] || [[2351 (number)|2351]] || [[2357 (number)|2357]]|| [[2371 (number)|2371]] || [[2377 (number)|2377]] || [[2381 (number)|2381]] || [[2383 (number)|2383]] || [[2389 (number)|2389]] || [[2393 (number)|2393]] || [[2399 (number)|2399]] || [[2411 (number)|2411]] || [[2417 (number)|2417]] || [[2423 (number)|2423]]
|- style="text-align: center;" 
! 361–380
| [[2437 (number)|2437]] || [[2441 (number)|2441]] || [[2447 (number)|2447]] || [[2459 (number)|2459]] || [[2467 (number)|2467]] || [[2473 (number)|2473]] || [[2477 (number)|2477]] || [[2503 (number)|2503]] || [[2521 (number)|2521]] || [[2531 (number)|2531]]|| [[2539 (number)|2539]] || [[2543 (number)|2543]] || [[2549 (number)|2549]] || [[2551 (number)|2551]] || [[2557 (number)|2557]] || [[2579 (number)|2579]] || [[2591 (number)|2591]] || [[2593 (number)|2593]] || [[2609 (number)|2609]] || [[2617 (number)|2617]]
|- style="text-align: center;" 
! 381–400
| [[2621 (number)|2621]] || [[2633 (number)|2633]] || [[2647 (number)|2647]] || [[2657 (number)|2657]] || [[2659 (number)|2659]] || [[2663 (number)|2663]] || [[2671 (number)|2671]] || [[2677 (number)|2677]] || [[2683 (number)|2683]] || [[2687 (number)|2687]]|| [[2689 (number)|2689]] || [[2693 (number)|2693]] || [[2699 (number)|2699]] || [[2707 (number)|2707]] || [[2711 (number)|2711]] || [[2713 (number)|2713]] || [[2719 (number)|2719]] || [[2729 (number)|2729]] || [[2731 (number)|2731]] || [[2741 (number)|2741]]
|- style="text-align: center;" 
! 401–420
| [[2749 (number)|2749]] || [[2753 (number)|2753]] || [[2767 (number)|2767]] || [[2777 (number)|2777]] || [[2789 (number)|2789]] || [[2791 (number)|2791]] || [[2797 (number)|2797]] || [[2801 (number)|2801]] || [[2803 (number)|2803]] || [[2819 (number)|2819]]|| [[2833 (number)|2833]] || [[2837 (number)|2837]] || [[2843 (number)|2843]] || [[2851 (number)|2851]] || [[2857 (number)|2857]] || [[2861 (number)|2861]] || [[2879 (number)|2879]] || [[2887 (number)|2887]] || [[2897 (number)|2897]] || [[2903 (number)|2903]]
|- style="text-align: center;" 
! 421–440
| [[2909 (number)|2909]] || [[2917 (number)|2917]] || [[2927 (number)|2927]] || [[2939 (number)|2939]] || [[2953 (number)|2953]] || [[2957 (number)|2957]] || [[2963 (number)|2963]] || [[2969 (number)|2969]] || [[2971 (number)|2971]] || [[2999 (number)|2999]]|| [[3001 (number)|3001]] || [[3011 (number)|3011]] || [[3019 (number)|3019]] || [[3023 (number)|3023]] || [[3037 (number)|3037]] || [[3041 (number)|3041]] || [[3049 (number)|3049]] || [[3061 (number)|3061]] || [[3067 (number)|3067]] || [[3079 (number)|3079]]
|- style="text-align: center;" 
! 441–460
| [[3083 (number)|3083]] || [[3089 (number)|3089]] || [[3109 (number)|3109]] || [[3119 (number)|3119]] || [[3121 (number)|3121]] || [[3137 (number)|3137]] || [[3163 (number)|3163]] || [[3167 (number)|3167]] || [[3169 (number)|3169]] || [[3181 (number)|3181]]|| [[3187 (number)|3187]] || [[3191 (number)|3191]] || [[3203 (number)|3203]] || [[3209 (number)|3209]] || [[3217 (number)|3217]] || [[3221 (number)|3221]] || [[3229 (number)|3229]] || [[3251 (number)|3251]] || [[3253 (number)|3253]] || [[3257 (number)|3257]]
|- style="text-align: center;" 
! 461–480
| [[3259 (number)|3259]] || [[3271 (number)|3271]] || [[3299 (number)|3299]] || [[3301 (number)|3301]] || [[3307 (number)|3307]] || [[3313 (number)|3313]] || [[3319 (number)|3319]] || [[3323 (number)|3323]] || [[3329 (number)|3329]] || [[3331 (number)|3331]]|| [[3343 (number)|3343]] || [[3347 (number)|3347]] || [[3359 (number)|3359]] || [[3361 (number)|3361]] || [[3371 (number)|3371]] || [[3373 (number)|3373]] || [[3389 (number)|3389]] || [[3391 (number)|3391]] || [[3407 (number)|3407]] || [[3413 (number)|3413]]
|- style="text-align: center;" 
! 481–500
| [[3433 (number)|3433]] || [[3449 (number)|3449]] || [[3457 (number)|3457]]|| [[3461 (number)|3461]] || [[3463 (number)|3463]] || [[3467 (number)|3467]] || [[3469 (number)|3469]] || [[3491 (number)|3491]] || [[3499 (number)|3499]] || [[3511 (number)|3511]]|| [[3517 (number)|3517]] || [[3527 (number)|3527]] || [[3529 (number)|3529]] || [[3533 (number)|3533]] || [[3539 (number)|3539]] || [[3541 (number)|3541]] || [[3547 (number)|3547]] || [[3557 (number)|3557]] || [[3559 (number)|3559]] || [[3571 (number)|3571]]
|- style="text-align: center;" 
! 501–520
| [[3581 (number)|3581]] || [[3583 (number)|3583]] || [[3593 (number)|3593]] || [[3607 (number)|3607]] || [[3613 (number)|3613]] || [[3617 (number)|3617]] || [[3623 (number)|3623]] || [[3631 (number)|3631]] || [[3637 (number)|3637]] || [[3643 (number)|3643]] || [[3659 (number)|3659]] || [[3671 (number)|3671]] || [[3673 (number)|3673]] || [[3677 (number)|3677]] || [[3691 (number)|3691]] || [[3697 (number)|3697]] || [[3701 (number)|3701]] || [[3709 (number)|3709]] || [[3719 (number)|3719]] || [[3727 (number)|3727]]
|- style="text-align: center;" 
! 521–540
| [[3733 (number)|3733]] || [[3739 (number)|3739]] || [[3761 (number)|3761]] || [[3767 (number)|3767]] || [[3769 (number)|3769]] || [[3779 (number)|3779]] || [[3793 (number)|3793]] || [[3797 (number)|3797]] || [[3803 (number)|3803]] || [[3821 (number)|3821]] || [[3823 (number)|3823]] || [[3833 (number)|3833]] || [[3847 (number)|3847]] || [[3851 (number)|3851]] || [[3853 (number)|3853]] || [[3863 (number)|3863]] || [[3877 (number)|3877]] || [[3881 (number)|3881]] || [[3889 (number)|3889]] || [[3907 (number)|3907]]
|- style="text-align: center;" 
! 541–560
| [[3911 (number)|3911]] || [[3917 (number)|3917]] || [[3919 (number)|3919]] || [[3923 (number)|3923]] || [[3929 (number)|3929]] || [[3931 (number)|3931]] || [[3943 (number)|3943]] || [[3947 (number)|3947]] || [[3967 (number)|3967]] || [[3989 (number)|3989]] || [[4000 (number)#4001 to 4099|4001]] || [[4000 (number)#4001 to 4099|4003]] || [[4000 (number)#4001 to 4099|4007]] || [[4000 (number)#4001 to 4099|4013]] || [[4000 (number)#4001 to 4099|4019]] || [[4000 (number)#4001 to 4099|4021]] || [[4000 (number)#4001 to 4099|4027]] || [[4000 (number)#4001 to 4099|4049]] || [[4000 (number)#4001 to 4099|4051]] || [[4000 (number)#4001 to 4099|4057]]
|- style="text-align: center;" 
! 561–580
| [[4000 (number)#4001 to 4099|4073]] || [[4000 (number)#4001 to 4099|4079]] || [[4000 (number)#4001 to 4099|4091]] || [[4000 (number)#4001 to 4099|4093]] || [[4000 (number)#4001 to 4099|4099]] || [[4000 (number)#4100 to 4199|4111]] || [[4000 (number)#4100 to 4199|4127]] || [[4000 (number)#4100 to 4199|4129]] || [[4000 (number)#4100 to 4199|4133]] || [[4000 (number)#4100 to 4199|4139]] || [[4000 (number)#4100 to 4199|4153]] || [[4000 (number)#4100 to 4199|4157]] || [[4000 (number)#4100 to 4199|4159]] || [[4000 (number)#4100 to 4199|4177]] || [[4000 (number)#4200 to 4299|4201]] || [[4000 (number)#4200 to 4299|4211]] || [[4000 (number)#4200 to 4299|4217]] || [[4000 (number)#4200 to 4299|4219]] || [[4000 (number)#4200 to 4299|4229]] || [[4000 (number)#4200 to 4299|4231]]
|- style="text-align: center;" 
! 581–600
| [[4000 (number)#4200 to 4299|4241]] || [[4000 (number)#4200 to 4299|4243]] || [[4000 (number)#4200 to 4299|4253]] || [[4000 (number)#4200 to 4299|4259]] || [[4000 (number)#4200 to 4299|4261]] || [[4000 (number)#4200 to 4299|4271]] || [[4000 (number)#4200 to 4299|4273]] || [[4000 (number)#4200 to 4299|4283]] || [[4000 (number)#4200 to 4299|4289]] || [[4000 (number)#4200 to 4299|4297]] || [[4000 (number)#4300 to 4399|4327]] || [[4000 (number)#4300 to 4399|4337]] || [[4000 (number)#4300 to 4399|4339]] || [[4000 (number)#4300 to 4399|4349]] || [[4000 (number)#4300 to 4399|4357]] || [[4000 (number)#4300 to 4399|4363]] || [[4000 (number)#4300 to 4399|4373]] || [[4000 (number)#4300 to 4399|4391]] || [[4000 (number)#4300 to 4399|4397]] || [[4000 (number)#4400 to 4499|4409]]
|- style="text-align: center;" 
! 601–620
| [[4000 (number)#4400 to 4499|4421]] || [[4000 (number)#4400 to 4499|4423]] || [[4000 (number)#4400 to 4499|4441]] || [[4000 (number)#4400 to 4499|4447]] || [[4000 (number)#4400 to 4499|4451]] || [[4000 (number)#4400 to 4499|4457]] || [[4000 (number)#4400 to 4499|4463]] || [[4000 (number)#4400 to 4499|4481]] || [[4000 (number)#4400 to 4499|4483]] || [[4000 (number)#4400 to 4499|4493]] || [[4000 (number)#4500 to 4599|4507]] || [[4000 (number)#4500 to 4599|4513]] || [[4000 (number)#4500 to 4599|4517]] || [[4000 (number)#4500 to 4599|4519]] || [[4000 (number)#4500 to 4599|4523]] || [[4000 (number)#4500 to 4599|4547]] || [[4000 (number)#4500 to 4599|4549]] || [[4000 (number)#4500 to 4599|4561]] || [[4000 (number)#4500 to 4599|4567]] || [[4000 (number)#4500 to 4599|4583]]
|- style="text-align: center;" 
! 621–640
| [[4000 (number)#4500 to 4599|4591]] || [[4000 (number)#4500 to 4599|4597]] || [[4000 (number)#4600 to 4699|4603]] || [[4000 (number)#4600 to 4699|4621]] || [[4000 (number)#4600 to 4699|4637]] || [[4000 (number)#4600 to 4699|4639]] || [[4000 (number)#4600 to 4699|4643]] || [[4000 (number)#4600 to 4699|4649]] || [[4000 (number)#4600 to 4699|4651]] || [[4000 (number)#4600 to 4699|4657]] || [[4000 (number)#4600 to 4699|4663]] || [[4000 (number)#4600 to 4699|4673]] || [[4000 (number)#4600 to 4699|4679]] || [[4000 (number)#4600 to 4699|4691]] || [[4000 (number)#4700 to 4799|4703]] || [[4000 (number)#4700 to 4799|4721]] || [[4000 (number)#4700 to 4799|4723]] || [[4000 (number)#4700 to 4799|4729]] || [[4000 (number)#4700 to 4799|4733]] || [[4000 (number)#4700 to 4799|4751]]
|- style="text-align: center;" 
! 641–660
| [[4000 (number)#4700 to 4799|4759]] || [[4000 (number)#4700 to 4799|4783]] || [[4000 (number)#4700 to 4799|4787]] || [[4000 (number)#4700 to 4799|4789]] || [[4000 (number)#4700 to 4799|4793]] || [[4000 (number)#4700 to 4799|4799]] || [[4000 (number)#4800 to 4899|4801]] || [[4000 (number)#4800 to 4899|4813]] || [[4000 (number)#4800 to 4899|4817]] || [[4000 (number)#4800 to 4899|4831]] || [[4000 (number)#4800 to 4899|4861]] || [[4000 (number)#4800 to 4899|4871]] || [[4000 (number)#4800 to 4899|4877]] || [[4000 (number)#4800 to 4899|4889]] || [[4000 (number)#4900 to 4999|4903]] || [[4000 (number)#4900 to 4999|4909]] || [[4000 (number)#4900 to 4999|4919]] || [[4000 (number)#4900 to 4999|4931]] || [[4000 (number)#4900 to 4999|4933]] || [[4000 (number)#4900 to 4999|4937]]
|- style="text-align: center;" 
! 661–680
| [[4000 (number)#4900 to 4999|4943]] || [[4000 (number)#4900 to 4999|4951]] || [[4000 (number)#4900 to 4999|4957]] || [[4000 (number)#4900 to 4999|4967]] || [[4000 (number)#4900 to 4999|4969]] || [[4000 (number)#4900 to 4999|4973]] || [[4000 (number)#4900 to 4999|4987]] || [[4000 (number)#4900 to 4999|4993]] || [[4000 (number)#4900 to 4999|4999]] || [[5000 (number)#5001 to 5099|5003]] || [[5000 (number)#5001 to 5099|5009]] || [[5000 (number)#5001 to 5099|5011]] || [[5000 (number)#5001 to 5099|5021]] || [[5000 (number)#5001 to 5099|5023]] || [[5000 (number)#5001 to 5099|5039]] || [[5000 (number)#5001 to 5099|5051]] || [[5000 (number)#5001 to 5099|5059]] || [[5000 (number)#5001 to 5099|5077]] || [[5000 (number)#5001 to 5099|5081]] || [[5000 (number)#5001 to 5099|5087]]
|- style="text-align: center;" 
! 681–700
| [[5000 (number)#5001 to 5099|5099]] || [[5000 (number)#5100 to 5199|5101]] || [[5000 (number)#5100 to 5199|5107]] || [[5000 (number)#5100 to 5199|5113]] || [[5000 (number)#5100 to 5199|5119]] || [[5000 (number)#5100 to 5199|5147]] || [[5000 (number)#5100 to 5199|5153]] || [[5000 (number)#5100 to 5199|5167]] || [[5000 (number)#5100 to 5199|5171]] || [[5000 (number)#5100 to 5199|5179]] || [[5000 (number)#5100 to 5199|5189]] || [[5000 (number)#5100 to 5199|5197]] || [[5000 (number)#5200 to 5299|5209]] || [[5000 (number)#5200 to 5299|5227]] || [[5000 (number)#5200 to 5299|5231]] || [[5000 (number)#5200 to 5299|5233]] || [[5000 (number)#5200 to 5299|5237]] || [[5000 (number)#5200 to 5299|5261]] || [[5000 (number)#5200 to 5299|5273]] || [[5000 (number)#5200 to 5299|5279]]
|- style="text-align: center;" 
! 701–720
| [[5000 (number)#5200 to 5299|5281]] || [[5000 (number)#5200 to 5299|5297]] || [[5000 (number)#5300 to 5399|5303]] || [[5000 (number)#5300 to 5399|5309]] || [[5000 (number)#5300 to 5399|5323]] || [[5000 (number)#5300 to 5399|5333]] || [[5000 (number)#5300 to 5399|5347]] || [[5000 (number)#5300 to 5399|5351]] || [[5000 (number)#5300 to 5399|5381]] || [[5000 (number)#5300 to 5399|5387]] || [[5000 (number)#5300 to 5399|5393]] || [[5000 (number)#5300 to 5399|5399]] || [[5000 (number)#5400 to 5499|5407]] || [[5000 (number)#5400 to 5499|5413]] || [[5000 (number)#5400 to 5499|5417]] || [[5000 (number)#5400 to 5499|5419]] || [[5000 (number)#5400 to 5499|5431]] || [[5000 (number)#5400 to 5499|5437]] || [[5000 (number)#5400 to 5499|5441]] || [[5000 (number)#5400 to 5499|5443]]
|- style="text-align: center;" 
! 721–740
| [[5000 (number)#5400 to 5499|5449]] || [[5000 (number)#5400 to 5499|5471]] || [[5000 (number)#5400 to 5499|5477]] || [[5000 (number)#5400 to 5499|5479]] || [[5000 (number)#5400 to 5499|5483]] || [[5000 (number)#5500 to 5599|5501]] || [[5000 (number)#5500 to 5599|5503]] || [[5000 (number)#5500 to 5599|5507]] || [[5000 (number)#5500 to 5599|5519]] || [[5000 (number)#5500 to 5599|5521]] || [[5000 (number)#5500 to 5599|5527]] || [[5000 (number)#5500 to 5599|5531]] || [[5000 (number)#5500 to 5599|5557]] || [[5000 (number)#5500 to 5599|5563]] || [[5000 (number)#5500 to 5599|5569]] || [[5000 (number)#5500 to 5599|5573]] || [[5000 (number)#5500 to 5599|5581]] || [[5000 (number)#5500 to 5599|5591]] || [[5000 (number)#5600 to 5699|5623]] || [[5000 (number)#5600 to 5699|5639]]
|- style="text-align: center;" 
! 741–760
| [[5000 (number)#5600 to 5699|5641]] || [[5000 (number)#5600 to 5699|5647]] || [[5000 (number)#5600 to 5699|5651]] || [[5000 (number)#5600 to 5699|5653]] || [[5000 (number)#5600 to 5699|5657]] || [[5000 (number)#5600 to 5699|5659]] || [[5000 (number)#5600 to 5699|5669]] || [[5000 (number)#5600 to 5699|5683]] || [[5000 (number)#5600 to 5699|5689]] || [[5000 (number)#5600 to 5699|5693]] || [[5000 (number)#5700 to 5799|5701]] || [[5000 (number)#5700 to 5799|5711]] || [[5000 (number)#5700 to 5799|5717]] || [[5000 (number)#5700 to 5799|5737]] || [[5000 (number)#5700 to 5799|5741]] || [[5000 (number)#5700 to 5799|5743]] || [[5000 (number)#5700 to 5799|5749]] || [[5000 (number)#5700 to 5799|5779]] || [[5000 (number)#5700 to 5799|5783]] || [[5000 (number)#5700 to 5799|5791]]
|- style="text-align: center;" 
! 761–780
| [[5000 (number)#5800 to 5899|5801]] || [[5000 (number)#5800 to 5899|5807]] || [[5000 (number)#5800 to 5899|5813]] || [[5000 (number)#5800 to 5899|5821]] || [[5000 (number)#5800 to 5899|5827]] || [[5000 (number)#5800 to 5899|5839]] || [[5000 (number)#5800 to 5899|5843]] || [[5000 (number)#5800 to 5899|5849]] || [[5000 (number)#5800 to 5899|5851]] || [[5000 (number)#5800 to 5899|5857]] || [[5000 (number)#5800 to 5899|5861]] || [[5000 (number)#5800 to 5899|5867]] || [[5000 (number)#5800 to 5899|5869]] || [[5000 (number)#5800 to 5899|5879]] || [[5000 (number)#5800 to 5899|5881]] || [[5000 (number)#5800 to 5899|5897]] || [[5000 (number)#5900 to 5999|5903]] || [[5000 (number)#5900 to 5999|5923]] || [[5000 (number)#5900 to 5999|5927]] || [[5000 (number)#5900 to 5999|5939]]
|- style="text-align: center;" 
! 781–800
| [[5000 (number)#5900 to 5999|5953]] || [[5000 (number)#5900 to 5999|5981]] || [[5000 (number)#5900 to 5999|5987]] || [[6000 (number)#6001 to 6099|6007]] || [[6000 (number)#6001 to 6099|6011]] || [[6000 (number)#6001 to 6099|6029]] || [[6000 (number)#6001 to 6099|6037]] || [[6000 (number)#6001 to 6099|6043]] || [[6000 (number)#6001 to 6099|6047]] || [[6000 (number)#6001 to 6099|6053]] || [[6000 (number)#6001 to 6099|6067]] || [[6000 (number)#6001 to 6099|6073]] || [[6000 (number)#6001 to 6099|6079]] || [[6000 (number)#6001 to 6099|6089]] || [[6000 (number)#6001 to 6099|6091]] || [[6000 (number)#6100 to 6199|6101]] || [[6000 (number)#6100 to 6199|6113]] || [[6000 (number)#6100 to 6199|6121]] || [[6000 (number)#6100 to 6199|6131]] || [[6000 (number)#6100 to 6199|6133]]
|- style="text-align: center;" 
! 801–820
| [[6000 (number)#6100 to 6199|6143]] || [[6000 (number)#6100 to 6199|6151]] || [[6000 (number)#6100 to 6199|6163]] || [[6000 (number)#6100 to 6199|6173]] || [[6000 (number)#6100 to 6199|6197]] || [[6000 (number)#6100 to 6199|6199]] || [[6000 (number)#6200 to 6299|6203]] || [[6000 (number)#6200 to 6299|6211]] || [[6000 (number)#6200 to 6299|6217]] || [[6000 (number)#6200 to 6299|6221]] || [[6000 (number)#6200 to 6299|6229]] || [[6000 (number)#6200 to 6299|6247]] || [[6000 (number)#6200 to 6299|6257]] || [[6000 (number)#6200 to 6299|6263]] || [[6000 (number)#6200 to 6299|6269]] || [[6000 (number)#6200 to 6299|6271]] || [[6000 (number)#6200 to 6299|6277]] || [[6000 (number)#6200 to 6299|6287]] || [[6000 (number)#6200 to 6299|6299]] || [[6000 (number)#6300 to 6399|6301]]
|- style="text-align: center;" 
! 821–840
| [[6000 (number)#6300 to 6399|6311]] || [[6000 (number)#6300 to 6399|6317]] || [[6000 (number)#6300 to 6399|6323]] || [[6000 (number)#6300 to 6399|6329]] || [[6000 (number)#6300 to 6399|6337]] || [[6000 (number)#6300 to 6399|6343]] || [[6000 (number)#6300 to 6399|6353]] || [[6000 (number)#6300 to 6399|6359]] || [[6000 (number)#6300 to 6399|6361]] || [[6000 (number)#6300 to 6399|6367]] || [[6000 (number)#6300 to 6399|6373]] || [[6000 (number)#6300 to 6399|6379]] || [[6000 (number)#6300 to 6399|6389]] || [[6000 (number)#6300 to 6399|6397]] || [[6000 (number)#6400 to 6499|6421]] || [[6000 (number)#6400 to 6499|6427]] || [[6000 (number)#6400 to 6499|6449]] || [[6000 (number)#6400 to 6499|6451]] || [[6000 (number)#6400 to 6499|6469]] || [[6000 (number)#6400 to 6499|6473]]
|- style="text-align: center;" 
! 841–860
| [[6000 (number)#6400 to 6499|6481]] || [[6000 (number)#6400 to 6499|6491]] || [[6000 (number)#6500 to 6599|6521]] || [[6000 (number)#6500 to 6599|6529]] || [[6000 (number)#6500 to 6599|6547]] || [[6000 (number)#6500 to 6599|6551]] || [[6000 (number)#6500 to 6599|6553]] || [[6000 (number)#6500 to 6599|6563]] || [[6000 (number)#6500 to 6599|6569]] || [[6000 (number)#6500 to 6599|6571]] || [[6000 (number)#6500 to 6599|6577]] || [[6000 (number)#6500 to 6599|6581]] || [[6000 (number)#6500 to 6599|6599]] || [[6000 (number)#6600 to 6699|6607]] || [[6000 (number)#6600 to 6699|6619]] || [[6000 (number)#6600 to 6699|6637]] || [[6000 (number)#6600 to 6699|6653]] || [[6000 (number)#6600 to 6699|6659]] || [[6000 (number)#6600 to 6699|6661]] || [[6000 (number)#6600 to 6699|6673]]
|- style="text-align: center;" 
! 861–880
| [[6000 (number)#6600 to 6699|6679]] || [[6000 (number)#6600 to 6699|6689]] || [[6000 (number)#6600 to 6699|6691]] || [[6000 (number)#6700 to 6799|6701]] || [[6000 (number)#6700 to 6799|6703]] || [[6000 (number)#6700 to 6799|6709]] || [[6000 (number)#6700 to 6799|6719]] || [[6000 (number)#6700 to 6799|6733]] || [[6000 (number)#6700 to 6799|6737]] || [[6000 (number)#6700 to 6799|6761]] || [[6000 (number)#6700 to 6799|6763]] || [[6000 (number)#6700 to 6799|6779]] || [[6000 (number)#6700 to 6799|6781]] || [[6000 (number)#6700 to 6799|6791]] || [[6000 (number)#6700 to 6799|6793]] || [[6000 (number)#6800 to 6899|6803]] || [[6000 (number)#6800 to 6899|6823]] || [[6000 (number)#6800 to 6899|6827]] || [[6000 (number)#6800 to 6899|6829]] || [[6000 (number)#6800 to 6899|6833]]
|- style="text-align: center;" 
! 881–900
| [[6000 (number)#6800 to 6899|6841]] || [[6000 (number)#6800 to 6899|6857]] || [[6000 (number)#6800 to 6899|6863]] || [[6000 (number)#6800 to 6899|6869]] || [[6000 (number)#6800 to 6899|6871]] || [[6000 (number)#6800 to 6899|6883]] || [[6000 (number)#6800 to 6899|6899]] || [[6000 (number)#6900 to 6999|6907]] || [[6000 (number)#6900 to 6999|6911]] || [[6000 (number)#6900 to 6999|6917]] || [[6000 (number)#6900 to 6999|6947]] || [[6000 (number)#6900 to 6999|6949]] || [[6000 (number)#6900 to 6999|6959]] || [[6000 (number)#6900 to 6999|6961]] || [[6000 (number)#6900 to 6999|6967]] || [[6000 (number)#6900 to 6999|6971]] || [[6000 (number)#6900 to 6999|6977]] || [[6000 (number)#6900 to 6999|6983]] || [[6000 (number)#6900 to 6999|6991]] || [[6000 (number)#6900 to 6999|6997]]
|- style="text-align: center;" 
! 901–920
| [[7000 (number)#7001 to 7099|7001]] || [[7000 (number)#7001 to 7099|7013]] || [[7000 (number)#7001 to 7099|7019]] || [[7000 (number)#7001 to 7099|7027]] || [[7000 (number)#7001 to 7099|7039]] || [[7000 (number)#7001 to 7099|7043]] || [[7000 (number)#7001 to 7099|7057]] || [[7000 (number)#7001 to 7099|7069]] || [[7000 (number)#7001 to 7099|7079]] || [[7000 (number)#7100 to 7199|7103]] || [[7000 (number)#7100 to 7199|7109]] || [[7000 (number)#7100 to 7199|7121]] || [[7000 (number)#7100 to 7199|7127]] || [[7000 (number)#7100 to 7199|7129]] || [[7000 (number)#7100 to 7199|7151]] || [[7000 (number)#7100 to 7199|7159]] || [[7000 (number)#7100 to 7199|7177]] || [[7000 (number)#7100 to 7199|7187]] || [[7000 (number)#7100 to 7199|7193]] || [[7000 (number)#7200 to 7299|7207]]
|- style="text-align: center;" 
! 921–940
| [[7000 (number)#7200 to 7299|7211]] || [[7000 (number)#7200 to 7299|7213]] || [[7000 (number)#7200 to 7299|7219]] || [[7000 (number)#7200 to 7299|7229]] || [[7000 (number)#7200 to 7299|7237]] || [[7000 (number)#7200 to 7299|7243]] || [[7000 (number)#7200 to 7299|7247]] || [[7000 (number)#7200 to 7299|7253]] || [[7000 (number)#7200 to 7299|7283]] || [[7000 (number)#7200 to 7299|7297]] || [[7000 (number)#7300 to 7399|7307]] || [[7000 (number)#7300 to 7399|7309]] || [[7000 (number)#7300 to 7399|7321]] || [[7000 (number)#7300 to 7399|7331]] || [[7000 (number)#7300 to 7399|7333]] || [[7000 (number)#7300 to 7399|7349]] || [[7000 (number)#7300 to 7399|7351]] || [[7000 (number)#7300 to 7399|7369]] || [[7000 (number)#7300 to 7399|7393]] || [[7000 (number)#7400 to 7499|7411]]
|- style="text-align: center;" 
! 941–960
| [[7000 (number)#7400 to 7499|7417]] || [[7000 (number)#7400 to 7499|7433]] || [[7000 (number)#7400 to 7499|7451]] || [[7000 (number)#7400 to 7499|7457]] || [[7000 (number)#7400 to 7499|7459]] || [[7000 (number)#7400 to 7499|7477]] || [[7000 (number)#7400 to 7499|7481]] || [[7000 (number)#7400 to 7499|7487]] || [[7000 (number)#7400 to 7499|7489]] || [[7000 (number)#7400 to 7499|7499]] || [[7000 (number)#7500 to 7599|7507]] || [[7000 (number)#7500 to 7599|7517]] || [[7000 (number)#7500 to 7599|7523]] || [[7000 (number)#7500 to 7599|7529]] || [[7000 (number)#7500 to 7599|7537]] || [[7000 (number)#7500 to 7599|7541]] || [[7000 (number)#7500 to 7599|7547]] || [[7000 (number)#7500 to 7599|7549]] || [[7000 (number)#7500 to 7599|7559]] || [[7000 (number)#7500 to 7599|7561]]
|- style="text-align: center;" 
! 961–980
| [[7000 (number)#7500 to 7599|7573]] || [[7000 (number)#7500 to 7599|7577]] || [[7000 (number)#7500 to 7599|7583]] || [[7000 (number)#7500 to 7599|7589]] || [[7000 (number)#7500 to 7599|7591]] || [[7000 (number)#7600 to 7699|7603]] || [[7000 (number)#7600 to 7699|7607]] || [[7000 (number)#7600 to 7699|7621]] || [[7000 (number)#7600 to 7699|7639]] || [[7000 (number)#7600 to 7699|7643]] || [[7000 (number)#7600 to 7699|7649]] || [[7000 (number)#7600 to 7699|7669]] || [[7000 (number)#7600 to 7699|7673]] || [[7000 (number)#7600 to 7699|7681]] || [[7000 (number)#7600 to 7699|7687]] || [[7000 (number)#7600 to 7699|7691]] || [[7000 (number)#7600 to 7699|7699]] || [[7000 (number)#7700 to 7799|7703]] || [[7000 (number)#7700 to 7799|7717]] || [[7000 (number)#7700 to 7799|7723]]
|- style="text-align: center;" 
! 981–1000
| [[7000 (number)#7700 to 7799|7727]] || [[7000 (number)#7700 to 7799|7741]] || [[7000 (number)#7700 to 7799|7753]] || [[7000 (number)#7700 to 7799|7757]] || [[7000 (number)#7700 to 7799|7759]] || [[7000 (number)#7700 to 7799|7789]] || [[7000 (number)#7700 to 7799|7793]] || [[7000 (number)#7800 to 7899|7817]] || [[7000 (number)#7800 to 7899|7823]] || [[7000 (number)#7800 to 7899|7829]] || [[7000 (number)#7800 to 7899|7841]] || [[7000 (number)#7800 to 7899|7853]] || [[7000 (number)#7800 to 7899|7867]] || [[7000 (number)#7800 to 7899|7873]] || [[7000 (number)#7800 to 7899|7877]] || [[7000 (number)#7800 to 7899|7879]] || [[7000 (number)#7800 to 7899|7883]] || [[7000 (number)#7900 to 7999|7901]] || [[7000 (number)#7900 to 7999|7907]] || [[7000 (number)#7900 to 7999|7919]]
|}

{{OEIS|A000040}}.

The [[Goldbach's conjecture|Goldbach conjecture]] verification project reports that it has computed all primes smaller than 4×10{{sup|18}}.<ref>Tomás Oliveira e Silva, [http://www.ieeta.pt/~tos/goldbach.html Goldbach conjecture verification] {{webarchive|url=https://web.archive.org/web/20110524084452/http://www.ieeta.pt/~tos/goldbach.html |date=24 May 2011 }}. Retrieved 16 July 2013</ref> That means 95,676,260,903,887,607 primes<ref>{{OEIS|id=A080127}}</ref> (nearly 10{{sup|17}}), but they were not stored. There are known formulae to evaluate the [[prime-counting function]] (the number of primes smaller than a given value) faster than computing the primes. This has been used to compute that there are 1,925,320,391,606,803,968,923 primes (roughly 2{{e|21}}) smaller than 10{{sup|23}}. A different computation found that there are 18,435,599,767,349,200,867,866 primes (roughly 2{{e|22}}) smaller than 10{{sup|24}}, if the [[Riemann hypothesis]] is true.<ref name="Franke">{{cite web|title=Conditional Calculation of pi(10{{sup|24}})|url=http://primes.utm.edu/notes/pi(10%5E24).html|author=Jens Franke|date=29 July 2010|access-date=17 May 2011|url-status=live|archive-url=http://archive.wikiwix.com/cache/20140824032441/http://primes.utm.edu/notes/pi(10%5E24).html|archive-date=24 August 2014}}</ref>

== Lists of primes by type ==
Below are listed the first prime numbers of many named forms and types. More details are in the article for the name. ''n'' is a [[natural number]] (including 0) in the definitions.

=== [[Balanced prime]]s ===
Primes with equal-sized [[Prime gap|prime gaps]] after and before them, so that they are equal to the [[arithmetic mean]] of the nearest primes after and before.

* [[5 (number)|5]], [[53 (number)|53]], 157, [[173 (number)|173]], 211, [[257 (number)|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|A006562}}).

=== [[Bell prime]]s ===
Primes that are the number of [[Partition of a set|partitions of a set]] with ''n'' members.

[[2]], [[5]], [[877 (number)|877]], 27644437, 35742549198872617291353508656626642567, 359334085968622831041960188598043661065388726959079837.
The next term has 6,539 digits. ({{OEIS2C|A051131}})

=== [[Chen primes]] ===
Where ''p'' is prime and ''p''+2 is either a prime or [[semiprime]].

[[2]], [[3]], [[5]], [[7]], [[11 (number)|11]], [[13 (number)|13]], [[17 (number)|17]], [[19 (number)|19]], [[23 (number)|23]], [[29 (number)|29]], [[31 (number)|31]], [[37 (number)|37]], [[41 (number)|41]], [[47 (number)|47]], [[53 (number)|53]], [[59 (number)|59]], [[67 (number)|67]], [[71 (number)|71]], [[83 (number)|83]], [[89 (number)|89]], [[101 (number)|101]], [[107 (number)|107]], [[109 (number)|109]], [[113 (number)|113]], [[127 (number)|127]], [[131 (number)|131]], [[137 (number)|137]], [[139 (number)|139]], [[149 (number)|149]], [[157 (number)|157]], [[167 (number)|167]], [[179 (number)|179]], [[181 (number)|181]], [[191 (number)|191]], [[197 (number)|197]], [[199 (number)|199]], [[211 (number)|211]], [[227 (number)|227]], [[233 (number)|233]], [[239 (number)|239]], [[251 (number)|251]], [[257 (number)|257]], [[263 (number)|263]], [[269 (number)|269]], [[281 (number)|281]], [[293 (number)|293]], [[307 (number)|307]], [[311 (number)|311]], [[317 (number)|317]], [[337 (number)|337]], [[347 (number)|347]], [[353 (number)|353]], [[359 (number)|359]], [[379 (number)|379]], [[389 (number)|389]], [[401 (number)|401]], [[409 (number)|409]] ({{OEIS2C|A109611}})

=== [[Circular prime]]s ===
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 (number)|11]], [[13 (number)|13]], [[17 (number)|17]], [[31 (number)|31]], [[37 (number)|37]], [[71 (number)|71]], [[73 (number)|73]], [[79 (number)|79]], [[97 (number)|97]], [[113 (number)|113]], [[131 (number)|131]], [[197 (number)|197]], [[199 (number)|199]], [[311 (number)|311]], [[337 (number)|337]], [[373 (number)|373]], [[719 (number)|719]], [[733 (number)|733]], [[919 (number)|919]], [[971 (number)|971]], [[991 (number)|991]], [[1193 (number)|1193]], [[1931 (number)|1931]], [[3119 (number)|3119]], [[3779 (number)|3779]], [[7000 (number)#7700 to 7799|7793]], [[7000 (number)#7900 to 7999|7937]], [[9000 (number)#9300 to 9399|9311]], [[9000 (number)#9300 to 9399|9377]], [[10000 (number)|11939]], [[10000 (number)|19391]], [[10000 (number)|19937]], [[30000 (number)|37199]], [[30000 (number)|39119]], [[70000 (number)|71993]], [[90000 (number)|91193]], [[90000 (number)|93719]], [[90000 (number)|93911]], [[90000 (number)|99371]], [[100000 (number)|193939]], [[100000 (number)|199933]], [[100000 (number)|319993]], [[100000 (number)|331999]], [[100000 (number)|391939]], [[100000 (number)|393919]], [[100000 (number)|919393]], [[100000 (number)|933199]], [[100000 (number)|939193]], [[100000 (number)|939391]], [[100000 (number)|993319]], [[100000 (number)|999331]] ({{OEIS2C|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 (number)|11]], [[13 (number)|13]], [[17 (number)|17]], [[37 (number)|37]], [[79 (number)|79]], [[113 (number)|113]], [[197 (number)|197]], [[199 (number)|199]], [[337 (number)|337]], [[1193 (number)|1193]], [[3779 (number)|3779]], [[10000 (number)|11939]], [[10000 (number)|19937]], [[100000 (number)|193939]], [[100000 (number)|199933]], 1111111111111111111, 11111111111111111111111 ({{OEIS2C|A016114}})

All [[repunit]] primes are circular.

=== [[Cluster prime|Cluster primes]] ===
A cluster prime is a prime ''p'' such that every even [[natural number]] ''k'' ≤ ''p'' − 3 is the difference of two primes not exceeding ''p''.

[[3]], [[5]], [[7]], [[11 (number)|11]], [[13 (number)|13]], [[17 (number)|17]], [[19 (number)|19]], [[23 (number)|23]], ... ({{OEIS2C|A038134}})

All odd primes between 3 and 89, inclusive, are cluster primes. The first 10 primes that are ''not'' cluster primes are:

[[2]], [[97 (number)|97]], [[127 (number)|127]], [[149 (number)|149]], [[191 (number)|191]], [[211 (number)|211]], [[223 (number)|223]], [[227 (number)|227]], [[229 (number)|229]], [[251 (number)|251]].

=== [[Cousin primes]] ===
{{see also|#Twin primes|#Prime triplets|#Prime quadruplets}}

Where (''p'', ''p'' + 4) are both prime.

([[3]], [[7]]), ([[7]], [[11 (number)|11]]), ([[13 (number)|13]], [[17 (number)|17]]), ([[19 (number)|19]], [[23 (number)|23]]), ([[37 (number)|37]], [[41 (number)|41]]), ([[43 (number)|43]], [[47 (number)|47]]), ([[67 (number)|67]], [[71 (number)|71]]), ([[79 (number)|79]], [[83 (number)|83]]), ([[97 (number)|97]], [[101 (number)|101]]), ([[103 (number)|103]], [[107 (number)|107]]), ([[109 (number)|109]], [[113 (number)|113]]), ([[127 (number)|127]], [[131 (number)|131]]), ([[163 (number)|163]], [[167 (number)|167]]), ([[193 (number)|193]], [[197 (number)|197]]), ([[223 (number)|223]], [[227 (number)|227]]), ([[229 (number)|229]], [[233 (number)|233]]), ([[277 (number)|277]], [[281 (number)|281]]) ({{OEIS2C|A023200}}, {{OEIS2C|A046132}})

=== [[Cuban prime]]s ===
Of the form <math>\tfrac{x^3-y^3}{x-y}</math> where ''x''&nbsp;=&nbsp;''y'' + 1.

[[7]], [[19 (number)|19]], [[37 (number)|37]], [[61 (number)|61]], [[127 (number)|127]], [[271 (number)|271]], [[331 (number)|331]], [[397 (number)|397]], [[547 (number)|547]], [[631 (number)|631]], [[919 (number)|919]], [[1657 (number)|1657]], [[1801 (number)|1801]], [[1951 (number)|1951]], [[2269 (number)|2269]], [[2437 (number)|2437]], [[2791 (number)|2791]], [[3169 (number)|3169]], [[3571 (number)|3571]], [[4000 (number)#4200 to 4299|4219]], [[4000 (number)#4400 to 4499|4447]], [[5000 (number)#5100 to 5199|5167]], [[5000 (number)#5400 to 5499|5419]], [[6000 (number)#6200 to 6299|6211]], [[7000 (number)#7001 to 7099|7057]], [[7000 (number)#7300 to 7399|7351]], [[8000 (number)#8200 to 8299|8269]], [[9000 (number)#9200 to 9299|9241]], [[10000 (number)|10267]], [[10000 (number)|11719]], [[10000 (number)|12097]], [[10000 (number)|13267]], [[10000 (number)|13669]], [[10000 (number)|16651]], [[10000 (number)|19441]], [[10000 (number)|19927]], [[20000 (number)|22447]], [[20000 (number)|23497]], [[20000 (number)|24571]], [[20000 (number)|25117]], [[20000 (number)|26227]], [[20000 (number)|27361]], [[30000 (number)|33391]], [[30000 (number)|35317]] ({{OEIS2C|A002407}})

Of the form <math>\tfrac{x^3-y^3}{x-y}</math> where ''x''&nbsp;=&nbsp;''y'' + 2.

[[13 (number)|13]], [[109 (number)|109]], [[193 (number)|193]], [[433 (number)|433]], [[769 (number)|769]], [[1201 (number)|1201]], [[1453 (number)|1453]], [[2029 (number)|2029]], [[3469 (number)|3469]], [[3889 (number)|3889]], [[4000 (number)#4800 to 4899|4801]], [[10000 (number)|10093]], [[10000 (number)|12289]], [[10000 (number)|13873]], [[10000 (number)|18253]], [[20000 (number)|20173]], [[20000 (number)|21169]], [[20000 (number)|22189]], [[20000 (number)|28813]], [[30000 (number)|37633]], [[40000 (number)|43201]], [[40000 (number)|47629]], [[60000 (number)|60493]], [[60000 (number)|63949]], [[60000 (number)|65713]], [[60000 (number)|69313]], [[70000 (number)|73009]], [[70000 (number)|76801]], [[80000 (number)|84673]], [[100000 (number)|106033]], [[100000 (number)|108301]], [[100000 (number)|112909]], [[100000 (number)|115249]] ({{OEIS2C|A002648}})

=== [[Cullen number|Cullen primes]] ===
Of the form ''n''×2{{sup|''n''}} + 1.

[[3]], 393050634124102232869567034555427371542904833 ({{OEIS2C|A050920}})

=== [[Delicate prime]]s ===
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 ({{OEIS2C|A050249}})

=== [[Dihedral prime]]s ===
Primes that remain prime when read upside down or mirrored in a [[seven-segment display]].

[[2]], [[5]], [[11 (number)|11]], [[101 (number)|101]], [[181 (number)|181]], [[1181 (number)|1181]], [[1000 (number)#1800 to 1899|1811]], [[10000 (number)|18181]], [[100000 (number)|108881]], [[100000 (number)|110881]], [[100000 (number)|118081]], [[100000 (number)|120121]],
[[100000 (number)|121021]], [[100000 (number)|121151]], [[100000 (number)|150151]], [[100000 (number)|151051]], [[100000 (number)|151121]], [[100000 (number)|180181]], [[100000 (number)|180811]], [[100000 (number)|181081]] ({{OEIS2C|A134996}})

=== [[Eisenstein primes]] without imaginary part ===
[[Eisenstein integer]]s that are [[Irreducible element|irreducible]] and real numbers (primes of the form 3''n''&nbsp;−&nbsp;1).

[[2]], [[5]], [[11 (number)|11]], [[17 (number)|17]], [[23 (number)|23]], [[29 (number)|29]], [[41 (number)|41]], [[47 (number)|47]], [[53 (number)|53]], [[59 (number)|59]], [[71 (number)|71]], [[83 (number)|83]], [[89 (number)|89]], [[101 (number)|101]], [[107 (number)|107]], [[113 (number)|113]], [[131 (number)|131]], [[137 (number)|137]], [[149 (number)|149]], [[167 (number)|167]], [[173 (number)|173]], [[179 (number)|179]], [[191 (number)|191]], [[197 (number)|197]], [[227 (number)|227]], [[233 (number)|233]], [[239 (number)|239]], [[251 (number)|251]], [[257 (number)|257]], [[263 (number)|263]], [[269 (number)|269]], [[281 (number)|281]], [[293 (number)|293]], [[311 (number)|311]], [[317 (number)|317]], [[347 (number)|347]], [[353 (number)|353]], [[359 (number)|359]], [[383 (number)|383]], [[389 (number)|389]], [[401 (number)|401]] ({{OEIS2C|A003627}})

=== [[Emirps]] ===
Primes that become a different prime when their decimal digits are reversed. The name "emirp" is the reverse of the word "prime".

[[13 (number)|13]], [[17 (number)|17]], [[31 (number)|31]], [[37 (number)|37]], [[71 (number)|71]], [[73 (number)|73]], [[79 (number)|79]], [[97 (number)|97]], [[107 (number)|107]], [[113 (number)|113]], [[149 (number)|149]], [[157 (number)|157]], [[167 (number)|167]], [[179 (number)|179]], [[199 (number)|199]], [[311 (number)|311]], [[337 (number)|337]], [[347 (number)|347]], [[359 (number)|359]], [[389 (number)|389]], [[701 (number)|701]], [[709 (number)|709]], [[733 (number)|733]], [[739 (number)|739]], [[743 (number)|743]], [[751 (number)|751]], [[761 (number)|761]], [[769 (number)|769]], [[907 (number)|907]], [[937 (number)|937]], [[941 (number)|941]], [[953 (number)|953]], [[967 (number)|967]], [[971 (number)|971]], [[983 (number)|983]], [[991 (number)|991]] ({{OEIS2C|A006567}})

=== [[Euclid number|Euclid primes]] ===
Of the form ''p''{{sub|''n''}}#&nbsp;+&nbsp;1 (a subset of [[primorial prime]]s).

[[3]], [[7]], [[31 (number)|31]], [[211 (number)|211]], [[2311 (number)|2311]], 200560490131 ({{OEIS2C|A018239}}<ref name="A018239">{{OEIS2C|A018239}} includes 2 = [[empty product]] of first 0 primes plus 1, but 2 is excluded in this list.</ref>)

=== [[Regular prime#Euler irregular primes|Euler irregular primes]] ===

A prime <math>p</math> that divides [[Euler number]] <math>E_{2n}</math> for some <math>0\leq 2n\leq p-3</math>.

[[19 (number)|19]], [[31 (number)|31]], [[43 (number)|43]], [[47 (number)|47]], [[61 (number)|61]], [[67 (number)|67]], [[71 (number)|71]], [[79 (number)|79]], [[101 (number)|101]], [[137 (number)|137]], [[139 (number)|139]], [[149 (number)|149]], [[193 (number)|193]], [[223 (number)|223]], [[241 (number)|241]], [[251 (number)|251]], [[263 (number)|263]], [[277 (number)|277]], [[307 (number)|307]], [[311 (number)|311]], [[349 (number)|349]], [[353 (number)|353]], [[359 (number)|359]], [[373 (number)|373]], [[379 (number)|379]], [[419 (number)|419]], [[433 (number)|433]], [[461 (number)|461]], [[463 (number)|463]], [[491 (number)|491]], [[509 (number)|509]], [[541 (number)|541]], [[563 (number)|563]], [[571 (number)|571]], [[577 (number)|577]], [[587 (number)|587]] ({{OEIS2C|A120337}})

==== [[Regular prime#Irregular pairs|Euler (''p'', ''p'' − 3) irregular primes]] ====
Primes <math>p</math> such that <math>(p, p-3)</math> is an Euler irregular pair.

[[149 (number)|149]], [[241 (number)|241]], [[1000000 (number)|2946901]] ({{OEIS2C|A198245}})

=== [[Factorial prime]]s ===
Of the form ''n''[[factorial|!]] − 1 or ''n''[[factorial|!]] + 1.

[[2]], [[3]], [[5]], [[7]], [[23 (number)|23]], [[719 (number)|719]], [[5000 (number)#5001 to 5099|5039]], [[10000000 (number)|39916801]], [[100000000 (number)|479001599]], 87178291199, 10888869450418352160768000001, 265252859812191058636308479999999, 263130836933693530167218012159999999, 8683317618811886495518194401279999999 ({{OEIS2C|A088054}})

=== [[Fermat prime]]s ===
Of the form 2{{sup|2{{sup|''n''}}}}&nbsp;+&nbsp;1.

[[3]], [[5]], [[17 (number)|17]], [[257 (number)|257]], [[65537 (number)|65537]] ({{OEIS2C|A019434}})

{{As of|2024|6}} these are the only known Fermat primes, and conjecturally the only Fermat primes. The probability of the existence of another Fermat prime is less than one in a billion.<ref>{{Cite arXiv|last1=Boklan|first1=Kent D.|last2=Conway|first2=John H.|date=2016|title=Expect at most one billionth of a new Fermat Prime!|eprint=1605.01371|class=math.NT}}</ref>

==== Generalized [[Fermat primes]] ====
Of the form ''a''{{sup|2{{sup|''n''}}}}&nbsp;+&nbsp;1 for fixed integer ''a''.

''a'' = 2: [[3]], [[5]], [[17 (number)|17]], [[257 (number)|257]], [[65537 (number)|65537]] ({{OEIS2C|A019434}})

''a'' = 4: [[5]], [[17 (number)|17]], [[257 (number)|257]], [[65537 (number)|65537]]

''a'' = 6: [[7]], [[37 (number)|37]], [[1297 (number)|1297]]

''a'' = 8: (does not exist)

''a'' = 10: [[11 (number)|11]], [[101 (number)|101]]

''a'' = 12: [[13 (number)|13]]

''a'' = 14: [[197 (number)|197]]

''a'' = 16: [[17 (number)|17]], [[257 (number)|257]], [[65537 (number)|65537]]

''a'' = 18: [[19 (number)|19]]

''a'' = 20: [[401 (number)|401]], 160001

''a'' = 22: [[23 (number)|23]]

''a'' = 24: [[577 (number)|577]], 331777

=== [[Fibonacci prime]]s ===
Primes in the [[Fibonacci sequence]] ''F''{{sub|0}}&nbsp;=&nbsp;0, ''F''{{sub|1}}&nbsp;=&nbsp;1,
''F''{{sub|''n''}} = ''F''{{sub|''n''−1}} + ''F''{{sub|''n''−2}}.

[[2]], [[3]], [[5]], [[13 (number)|13]], [[89 (number)|89]], [[233 (number)|233]], [[1597 (number)|1597]], 28657, 514229, 433494437, 2971215073, 99194853094755497, 1066340417491710595814572169, 19134702400093278081449423917 ({{OEIS2C|A005478}})

=== [[Fortunate number|Fortunate primes]] ===
[[Fortunate number]]s that are prime (it has been conjectured they all are).

[[3]], [[5]], [[7]], [[13 (number)|13]], [[17 (number)|17]], [[19 (number)|19]], [[23 (number)|23]], [[37 (number)|37]], [[47 (number)|47]], [[59 (number)|59]], [[61 (number)|61]], [[67 (number)|67]], [[71 (number)|71]], [[79 (number)|79]], [[89 (number)|89]], [[101 (number)|101]], [[103 (number)|103]], [[107 (number)|107]], [[109 (number)|109]], [[127 (number)|127]], [[151 (number)|151]], [[157 (number)|157]], [[163 (number)|163]], [[167 (number)|167]], [[191 (number)|191]], [[197 (number)|197]], [[199 (number)|199]], [[223 (number)|223]], [[229 (number)|229]], [[233 (number)|233]], [[239 (number)|239]], [[271 (number)|271]], [[277 (number)|277]], [[283 (number)|283]], [[293 (number)|293]], [[307 (number)|307]], [[311 (number)|311]], [[313 (number)|313]], [[331 (number)|331]], [[353 (number)|353]], [[373 (number)|373]], [[379 (number)|379]], [[383 (number)|383]], [[397 (number)|397]] ({{OEIS2C|A046066}})

=== [[Gaussian prime]]s ===
[[Prime element]]s of the Gaussian integers; equivalently, primes of the form 4''n''&nbsp;+&nbsp;3.

[[3]], [[7]], [[11 (number)|11]], [[19 (number)|19]], [[23 (number)|23]], [[31 (number)|31]], [[43 (number)|43]], [[47 (number)|47]], [[59 (number)|59]], [[67 (number)|67]], [[71 (number)|71]], [[79 (number)|79]], [[83 (number)|83]], [[103 (number)|103]], [[107 (number)|107]], [[127 (number)|127]], [[131 (number)|131]], [[139 (number)|139]], [[151 (number)|151]], [[163 (number)|163]], [[167 (number)|167]], [[179 (number)|179]], [[191 (number)|191]], [[199 (number)|199]], [[211 (number)|211]], [[223 (number)|223]], [[227 (number)|227]], [[239 (number)|239]], [[251 (number)|251]], [[263 (number)|263]], [[271 (number)|271]], [[283 (number)|283]], [[307 (number)|307]], [[311 (number)|311]], [[331 (number)|331]], [[347 (number)|347]], [[359 (number)|359]], [[367 (number)|367]], [[379 (number)|379]], [[383 (number)|383]], [[419 (number)|419]], [[431 (number)|431]], [[439 (number)|439]], [[443 (number)|443]], [[463 (number)|463]], [[467 (number)|467]], [[479 (number)|479]], [[487 (number)|487]], [[491 (number)|491]], [[499 (number)|499]], [[503 (number)|503]] ({{OEIS2C|A002145}})

=== [[Good prime]]s ===
Primes ''p''{{sub|''n''}} for which ''p''{{sub|''n''}}{{sup|2}}&nbsp;>&nbsp;''p''{{sub|''n''−''i''}}&nbsp;''p''{{sub|''n''+''i''}} for all 1&nbsp;≤&nbsp;''i''&nbsp;≤&nbsp;''n''−1, where ''p''{{sub|''n''}} is the ''n''th prime.

[[5]], [[11 (number)|11]], [[17 (number)|17]], [[29 (number)|29]], [[37 (number)|37]], [[41 (number)|41]], [[53 (number)|53]], [[59 (number)|59]], [[67 (number)|67]], [[71 (number)|71]], [[97 (number)|97]], [[101 (number)|101]], [[127 (number)|127]], [[149 (number)|149]], [[179 (number)|179]], [[191 (number)|191]], [[223 (number)|223]], [[227 (number)|227]], [[251 (number)|251]], [[257 (number)|257]], [[269 (number)|269]], [[307 (number)|307]] ({{OEIS2C|A028388}})

=== [[Happy prime]]s ===
Happy numbers that are prime.

[[7]], [[13 (number)|13]], [[19 (number)|19]], [[23 (number)|23]], [[31 (number)|31]], [[79 (number)|79]], [[97 (number)|97]], [[103 (number)|103]], [[109 (number)|109]], [[139 (number)|139]], [[167 (number)|167]], [[193 (number)|193]], [[239 (number)|239]], [[263 (number)|263]], [[293 (number)|293]], [[313 (number)|313]], [[331 (number)|331]], [[367 (number)|367]], [[379 (number)|379]], [[383 (number)|383]], [[397 (number)|397]], [[409 (number)|409]], [[487 (number)|487]], [[563 (number)|563]], [[617 (number)|617]], [[653 (number)|653]], [[673 (number)|673]], [[683 (number)|683]], [[709 (number)|709]], [[739 (number)|739]], [[761 (number)|761]], [[863 (number)|863]], [[881 (number)|881]], [[907 (number)|907]], [[937 (number)|937]], [[1009 (number)|1009]], [[1033 (number)|1033]], [[1039 (number)|1039]], [[1093 (number)|1093]] ({{OEIS2C|A035497}})

=== Harmonic primes ===
Primes ''p'' for which there are no solutions to ''H''{{sub|''k''}}&nbsp;≡&nbsp;0&nbsp;(mod&nbsp;''p'') and ''H''{{sub|''k''}}&nbsp;≡&nbsp;−''ω''{{sub|''p''}}&nbsp;(mod&nbsp;''p'') for 1&nbsp;≤&nbsp;''k''&nbsp;≤&nbsp;''p''−2, where ''H''{{sub|''k''}} denotes the ''k''-th [[harmonic number]] and ''ω''{{sub|''p''}} denotes the [[Wolstenholme quotient]].<ref>{{Cite journal | last1 = Boyd | first1 = D. W. | title = A ''p''-adic Study of the Partial Sums of the Harmonic Series | url = http://projecteuclid.org/euclid.em/1048515811 | doi = 10.1080/10586458.1994.10504298 | journal = [[Experimental Mathematics (journal)|Experimental Mathematics]] | volume = 3 | issue = 4 | pages = 287–302 | year = 1994 | id = [[CiteSeerX]]: {{URL|1=citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.56.7026|2=10.1.1.56.7026}} | zbl = 0838.11015 | url-status = live | archive-url = https://web.archive.org/web/20160127080246/http://projecteuclid.org/euclid.em/1048515811 | archive-date = 27 January 2016}}</ref>

[[5]], [[13 (number)|13]], [[17 (number)|17]], [[23 (number)|23]], [[41 (number)|41]], [[67 (number)|67]], [[73 (number)|73]], [[79 (number)|79]], [[107 (number)|107]], [[113 (number)|113]], [[139 (number)|139]], [[149 (number)|149]], [[157 (number)|157]], [[179 (number)|179]], [[191 (number)|191]], [[193 (number)|193]], [[223 (number)|223]], [[239 (number)|239]], [[241 (number)|241]], [[251 (number)|251]], [[263 (number)|263]], [[277 (number)|277]], [[281 (number)|281]], [[293 (number)|293]], [[307 (number)|307]], [[311 (number)|311]], [[317 (number)|317]], [[331 (number)|331]], [[337 (number)|337]], [[349 (number)|349]] ({{OEIS2C|A092101}})

=== [[Higgs prime]]s for squares ===
Primes ''p'' for which ''p'' − 1 divides the square of the product of all earlier terms.

[[2]], [[3]], [[5]], [[7]], [[11 (number)|11]], [[13 (number)|13]], [[19 (number)|19]], [[23 (number)|23]], [[29 (number)|29]], [[31 (number)|31]], [[37 (number)|37]], [[43 (number)|43]], [[47 (number)|47]], [[53 (number)|53]], [[59 (number)|59]], [[61 (number)|61]], [[67 (number)|67]], [[71 (number)|71]], [[79 (number)|79]], [[101 (number)|101]], [[107 (number)|107]], [[127 (number)|127]], [[131 (number)|131]], [[139 (number)|139]], [[149 (number)|149]], [[151 (number)|151]], [[157 (number)|157]], [[173 (number)|173]], [[181 (number)|181]], [[191 (number)|191]], [[197 (number)|197]], [[199 (number)|199]], [[211 (number)|211]], [[223 (number)|223]], [[229 (number)|229]], [[263 (number)|263]], [[269 (number)|269]], [[277 (number)|277]], [[283 (number)|283]], [[311 (number)|311]], [[317 (number)|317]], [[331 (number)|331]], [[347 (number)|347]], [[349 (number)|349]] ({{OEIS2C|A007459}})

=== [[Highly cototient number| Highly cototient primes]] ===
Primes that are a [[cototient]] more often than any integer below it except 1.

[[2]], [[23 (number)|23]], [[47 (number)|47]], [[59 (number)|59]], [[83 (number)|83]], [[89 (number)|89]], [[113 (number)|113]], [[167 (number)|167]], [[269 (number)|269]], [[389 (number)|389]], [[419 (number)|419]], [[509 (number)|509]], [[659 (number)|659]], [[839 (number)|839]], [[1049 (number)|1049]], [[1259 (number)|1259]], [[1889 (number)|1889]] ({{OEIS2C|A105440}})

=== [[Home prime]]s ===
For {{math|''n'' ≥ 2}}, write the prime factorization of {{mvar|n}} in base 10 and concatenate the factors; iterate until a prime is reached.

2, 3, 211, 5, 23, 7, 3331113965338635107, 311, 773, 11, 223, 13, 13367, 1129, 31636373, 17, 233, 19, 3318308475676071413, 37, 211, 23, 331319, 773, 3251, 13367, 227, 29, 547, 31, 241271, 311, 31397, 1129, 71129, 37, 373, 313, 3314192745739, 41, 379, 43, 22815088913, 3411949, 223, 47, 6161791591356884791277 ({{OEIS2C|A037274}})

=== [[Irregular prime]]s ===
Odd primes ''p'' that divide the [[Class number (number theory)|class number]] of the ''p''-th [[cyclotomic field]].

[[37 (number)|37]], [[59 (number)|59]], [[67 (number)|67]], [[101 (number)|101]], [[103 (number)|103]], [[131 (number)|131]], [[149 (number)|149]], [[157 (number)|157]], [[233 (number)|233]], [[257 (number)|257]], [[263 (number)|263]], [[271 (number)|271]], [[283 (number)|283]], [[293 (number)|293]], [[307 (number)|307]], [[311 (number)|311]], [[347 (number)|347]], [[353 (number)|353]], [[379 (number)|379]], [[389 (number)|389]], [[401 (number)|401]], [[409 (number)|409]], [[421 (number)|421]], [[433 (number)|433]], [[461 (number)|461]], [[463 (number)|463]], [[467 (number)|467]], [[491 (number)|491]], [[523 (number)|523]], [[541 (number)|541]], [[547 (number)|547]], [[557 (number)|557]], [[577 (number)|577]], [[587 (number)|587]], [[593 (number)|593]], [[607 (number)|607]], [[613 (number)|613]] ({{OEIS2C|A000928}})

==== [[Regular prime#Irregular pairs|(''p'', ''p'' − 3) irregular primes]] ====

(See [[Wolstenholme prime]])

==== [[Regular prime#Irregular pairs|(''p'', ''p'' − 5) irregular primes]] ====
Primes ''p'' such that (''p'', ''p''−5) is an irregular pair.<ref name="Johnson">{{cite journal
 |last=Johnson 
 |first=W. 
 |title=Irregular Primes and Cyclotomic Invariants 
 |journal=[[Mathematics of Computation]] 
 |volume=29 
 |issue=129 
 |pages=113–120 
 |publisher=[[American Mathematical Society|AMS]] 
 |year=1975 
 |doi=10.2307/2005468 
 |jstor=2005468 
 |doi-access=free 
 }}</ref>

[[37 (number)|37]]

==== [[Regular prime#Irregular pairs|(''p'', ''p'' − 9) irregular primes]] ====
Primes ''p'' such that (''p'', ''p'' − 9) is an irregular pair.<ref name="Johnson" />

[[67 (number)|67]], [[877 (number)|877]] ({{OEIS2C|A212557}})

=== [[Isolated prime]]s ===
Primes ''p'' such that neither ''p'' − 2 nor ''p'' + 2 is prime.

[[2]], [[23 (number)|23]], [[37 (number)|37]], [[47 (number)|47]], [[53 (number)|53]], [[67 (number)|67]], [[79 (number)|79]], [[83 (number)|83]], [[89 (number)|89]], [[97 (number)|97]], [[113 (number)|113]], [[127 (number)|127]], [[131 (number)|131]], [[157 (number)|157]], [[163 (number)|163]], [[167 (number)|167]], [[173 (number)|173]], [[211 (number)|211]], [[223 (number)|223]], [[233 (number)|233]], [[251 (number)|251]], [[257 (number)|257]], [[263 (number)|263]], [[277 (number)|277]], [[293 (number)|293]], [[307 (number)|307]], [[317 (number)|317]], [[331 (number)|331]], [[337 (number)|337]], [[353 (number)|353]], [[359 (number)|359]], [[367 (number)|367]], [[373 (number)|373]], [[379 (number)|379]], [[383 (number)|383]], [[389 (number)|389]], [[397 (number)|397]], [[401 (number)|401]], [[409 (number)|409]], [[439 (number)|439]], [[443 (number)|443]], [[449 (number)|449]], [[457 (number)|457]], [[467 (number)|467]], [[479 (number)|479]], [[487 (number)|487]], [[491 (number)|491]], [[499 (number)|499]], [[503 (number)|503]], [[509 (number)|509]], [[541 (number)|541]], [[547 (number)|547]], [[557 (number)|557]], [[563 (number)|563]], [[577 (number)|577]], [[587 (number)|587]], [[593 (number)|593]], [[607 (number)|607]], [[613 (number)|613]], [[631 (number)|631]], [[647 (number)|647]], [[653 (number)|653]], [[673 (number)|673]], [[677 (number)|677]], [[683 (number)|683]], [[691 (number)|691]], [[701 (number)|701]], [[709 (number)|709]], [[719 (number)|719]], [[727 (number)|727]], [[733 (number)|733]], [[739 (number)|739]], [[743 (number)|743]], [[751 (number)|751]], [[757 (number)|757]], [[761 (number)|761]], [[769 (number)|769]], [[773 (number)|773]], [[787 (number)|787]], [[797 (number)|797]], [[839 (number)|839]], [[853 (number)|853]], [[863 (number)|863]], [[877 (number)|877]], [[887 (number)|887]], [[907 (number)|907]], [[911 (number)|911]], [[919 (number)|919]], [[929 (number)|929]], [[937 (number)|937]], [[941 (number)|941]], [[947 (number)|947]], [[953 (number)|953]], [[967 (number)|967]], [[971 (number)|971]], [[977 (number)|977]], [[983 (number)|983]], [[991 (number)|991]], [[997 (number)|997]] ({{OEIS2C|A007510}})

=== [[Leyland number|Leyland primes]] ===
Of the form ''x''{{sup|''y''}}&nbsp;+&nbsp;''y''{{sup|''x''}}, with 1&nbsp;<&nbsp;''x''&nbsp;<&nbsp;''y''.

[[17 (number)|17]], [[593 (number)|593]], 32993, 2097593, 8589935681, 59604644783353249, 523347633027360537213687137, 43143988327398957279342419750374600193 ({{OEIS2C|A094133}})

=== [[Full reptend prime|Long primes]] ===
Primes ''p'' for which, in a given base ''b'', <math>\frac{b^{p-1}-1}{p}</math> gives a [[cyclic number]]. They are also called full reptend primes. Primes ''p'' for base 10:

[[7]], [[17 (number)|17]], [[19 (number)|19]], [[23 (number)|23]], [[29 (number)|29]], [[47 (number)|47]], [[59 (number)|59]], [[61 (number)|61]], [[97 (number)|97]], [[109 (number)|109]], [[113 (number)|113]], [[131 (number)|131]], [[149 (number)|149]], [[167 (number)|167]], [[179 (number)|179]], [[181 (number)|181]], [[193 (number)|193]], [[223 (number)|223]], [[229 (number)|229]], [[233 (number)|233]], [[257 (number)|257]], [[263 (number)|263]], [[269 (number)|269]], [[313 (number)|313]], [[337 (number)|337]], [[367 (number)|367]], [[379 (number)|379]], [[383 (number)|383]], [[389 (number)|389]], [[419 (number)|419]], [[433 (number)|433]], [[461 (number)|461]], [[487 (number)|487]], [[491 (number)|491]], [[499 (number)|499]], [[503 (number)|503]], [[509 (number)|509]], [[541 (number)|541]], [[571 (number)|571]], [[577 (number)|577]], [[593 (number)|593]] ({{OEIS2C|A001913}})

=== [[Lucas prime]]s ===
Primes in the Lucas number sequence ''L''{{sub|0}}&nbsp;=&nbsp;2, ''L''{{sub|1}}&nbsp;=&nbsp;1,
''L''{{sub|''n''}}&nbsp;=&nbsp;''L''{{sub|''n''−1}}&nbsp;+&nbsp;''L''{{sub|''n''−2}}.

[[2]],<ref>It varies whether ''L''{{sub|0}}&nbsp;=&nbsp;2 is included in the Lucas numbers.</ref> [[3]], [[7]], [[11 (number)|11]], [[29 (number)|29]], [[47 (number)|47]], [[199 (number)|199]], [[521 (number)|521]], [[2207 (number)|2207]], [[3571 (number)|3571]], 9349, 3010349, 54018521, 370248451, 6643838879, 119218851371, 5600748293801, 688846502588399, 32361122672259149 ({{OEIS2C|A005479}})

=== [[Lucky number|Lucky primes]] ===
Lucky numbers that are prime.

[[3]], [[7]], [[13 (number)|13]], [[31 (number)|31]], [[37 (number)|37]], [[43 (number)|43]], [[67 (number)|67]], [[73 (number)|73]], [[79 (number)|79]], [[127 (number)|127]], [[151 (number)|151]], [[163 (number)|163]], [[193 (number)|193]], [[211 (number)|211]], [[223 (number)|223]], [[241 (number)|241]], [[283 (number)|283]], [[307 (number)|307]], [[331 (number)|331]], [[349 (number)|349]], [[367 (number)|367]], [[409 (number)|409]], [[421 (number)|421]], [[433 (number)|433]], [[463 (number)|463]], [[487 (number)|487]], [[541 (number)|541]], [[577 (number)|577]], [[601 (number)|601]], [[613 (number)|613]], [[619 (number)|619]], [[631 (number)|631]], [[643 (number)|643]], [[673 (number)|673]], [[727 (number)|727]], [[739 (number)|739]], [[769 (number)|769]], [[787 (number)|787]], [[823 (number)|823]], [[883 (number)|883]], [[937 (number)|937]], [[991 (number)|991]], [[997 (number)|997]] ({{OEIS2C|A031157}})

=== [[Mersenne prime]]s ===
Of the form 2{{sup|''n''}} − 1.

[[3]], [[7]], [[31 (number)|31]], [[127 (number)|127]], [[8191 (number)|8191]], [[131071 (number)|131071]], [[524287 (number)|524287]], [[2147483647]], 2305843009213693951, 618970019642690137449562111, 162259276829213363391578010288127, 170141183460469231731687303715884105727 ({{OEIS2C|A000668}})

{{As of|2024}}, there are 52 known Mersenne primes. The 13th, 14th, and 52nd have respectively 157, 183, and 41,024,320 digits. This includes the largest known prime 2<sup>136,279,841</sup>-1, which is the 52nd Mersenne prime.

==== Mersenne divisors ====
Primes ''p'' that divide 2{{sup|''n''}} − 1, for some prime number n.

3, 7, 23, 31, 47, 89, 127, 167, 223, 233, 263, 359, 383, 431, 439, 479, 503, 719, 839, 863, 887, 983, 1103, 1319, 1367, 1399, 1433, 1439, 1487, 1823, 1913, 2039, 2063, 2089, 2207, 2351, 2383, 2447, 2687, 2767, 2879, 2903, 2999, 3023, 3119, 3167, 3343 ({{OEIS2C|A122094}})

All Mersenne primes are, by definition, members of this sequence.

==== Mersenne prime exponents ====
Primes ''p'' such that 2{{sup|''p''}} − 1 is prime.

<!--1st-10th exponent-->
[[2]], [[3]], [[5]], [[7]], [[13 (number)|13]], [[17 (number)|17]], [[19 (number)|19]], [[31 (number)|31]], [[61 (number)|61]], [[89 (number)|89]],
<!--11th-20th exponent-->
[[107 (number)|107]], [[127 (number)|127]], [[521 (number)|521]], [[607 (number)|607]], [[1279 (number)|1279]], [[2203 (number)|2203]], [[2281 (number)|2281]], [[3217 (number)|3217]], 4253, 4423,
<!--21st-30th exponent-->
9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049,
<!--31st-40th exponent-->
216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011,
<!--41st-45th exponent-->
24036583, 25964951, 30402457, 32582657, 37156667, 42643801, [[43,112,609 (number)|43112609]], 57885161 ({{OEIS2C|A000043}})

{{As of|2024|10}}, four more are known to be in the sequence, but it is not known whether they are the next:<br />
<!--48th-51st known exponent-->74207281, 77232917, 82589933, 136279841

==== [[Double Mersenne prime]]s ====
A subset of Mersenne primes of the form 2{{sup|2{{sup|''p''}}−1}}&nbsp;−&nbsp;1 for prime ''p''.

[[7]], [[127 (number)|127]], [[2147483647]], 170141183460469231731687303715884105727 (primes in {{OEIS2C|A077586}})

==== Generalized [[Repunit#Repunit primes|repunit primes]] ====
Of the form (''a''{{sup|''n''}}&nbsp;−&nbsp;1)&nbsp;/&nbsp;(''a''&nbsp;−&nbsp;1) for fixed integer ''a''.

For ''a'' = 2, these are the Mersenne primes, while for ''a'' = 10 they are the [[#Repunit primes|repunit primes]].  For other small ''a'', they are given below:

''a'' = 3: [[13 (number)|13]], [[1093 (number)|1093]], 797161, 3754733257489862401973357979128773, 6957596529882152968992225251835887181478451547013 ({{OEIS2C|A076481}})

''a'' = 4: [[5]] (the only prime for ''a'' = 4)

''a'' = 5: [[31 (number)|31]], 19531, 12207031, 305175781, 177635683940025046467781066894531, 14693679385278593849609206715278070972733319459651094018859396328480215743184089660644531 ({{OEIS2C|A086122}})

''a'' = 6: [[7]], [[43 (number)|43]], 55987, 7369130657357778596659, 3546245297457217493590449191748546458005595187661976371 ({{OEIS2C|A165210}})

''a'' = 7: 2801, 16148168401, 85053461164796801949539541639542805770666392330682673302530819774105141531698707146930307290253537320447270457

''a'' = 8: [[73 (number)|73]] (the only prime for ''a'' = 8)

''a'' = 9: none exist

==== Other generalizations and variations ====
Many generalizations of Mersenne primes have been defined.  This include the following:
* Primes of the form {{math|''b<sup>n</sup>'' − (''b'' − 1)<sup>''n''</sup>}},<ref>{{Cite OEIS |A121091|Smallest nexus prime of the form n^p - (n-1)^p, where p is an odd prime}}</ref><ref>{{Cite OEIS |A121616|Primes of form (n+1)^5 - n^5}}</ref><ref>{{Cite OEIS |A121618|Nexus primes of order 7 or primes of form n^7 - (n-1)^7}}</ref> including the Mersenne primes and the [[cuban prime]]s as special cases
* [[Williams prime]]s, of the form {{math|(''b'' − 1)·''b<sup>n</sup>'' − 1}}

=== [[Mills' constant|Mills primes]] ===
Of the form ⌊θ{{sup|3{{sup|''n''}}}}⌋, where θ is Mills' constant. This form is prime for all positive integers ''n''.

[[2]], [[11 (number)|11]], [[1361 (number)|1361]], 2521008887, 16022236204009818131831320183 ({{OEIS2C|A051254}})

=== [[Minimal prime (number theory)|Minimal primes]] ===
Primes for which there is no shorter [[subsequence|sub-sequence]] of the decimal digits that form a prime. There are exactly 26 minimal primes:

[[2]], [[3]], [[5]], [[7]], [[11 (number)|11]], [[19 (number)|19]], [[41 (number)|41]], [[61 (number)|61]], [[89 (number)|89]], [[409 (number)|409]], [[449 (number)|449]], [[499 (number)|499]], [[881 (number)|881]], [[991 (number)|991]], 6469, 6949, [[9001 (number)|9001]], 9049, 9649, 9949, 60649, 666649, 946669, 60000049, 66000049, 66600049 ({{OEIS2C|A071062}})

=== [[Newman–Shanks–Williams prime]]s ===
Newman–Shanks–Williams numbers that are prime.

[[7]], [[41 (number)|41]], [[239 (number)|239]], 9369319, 63018038201, 489133282872437279, 19175002942688032928599 ({{OEIS2C|A088165}})

=== Non-generous primes ===
Primes ''p'' for which the least positive [[primitive root modulo n|primitive root]] is not a primitive root of ''p''<sup>2</sup>.  Three such primes are known; it is not known whether there are more.<ref>{{cite journal
|last = Paszkiewicz 
| first = Andrzej
| title = A new prime <math>p</math> for which the least primitive root <math>(\textrm{mod } p)</math> and the least primitive root <math>(\textrm{mod } p^2)</math> are not equal
| journal = Math. Comp.
| volume = 78
| year = 2009
| issue = 266
| pages = 1193&ndash;1195
| url = https://www.ams.org/journals/mcom/2009-78-266/S0025-5718-08-02090-5/S0025-5718-08-02090-5.pdf
| doi = 10.1090/S0025-5718-08-02090-5 
| publisher = American Mathematical Society
| bibcode = 2009MaCom..78.1193P
| doi-access = free
}}</ref>

[[2]], 40487, 6692367337 ({{OEIS2C|A055578}})

=== [[Palindromic primes]] ===
Primes that remain the same when their decimal digits are read backwards.

[[2]], [[3]], [[5]], [[7]], [[11 (number)|11]], [[101 (number)|101]], [[131 (number)|131]], [[151 (number)|151]], [[181 (number)|181]], [[191 (number)|191]], [[313 (number)|313]], [[353 (number)|353]], [[373 (number)|373]], [[383 (number)|383]], [[727 (number)|727]], [[757 (number)|757]], [[787 (number)|787]], [[797 (number)|797]], [[919 (number)|919]], [[929 (number)|929]], 10301, 10501, 10601, 11311, 11411, 12421, 12721, 12821, 13331, 13831, 13931, 14341, 14741 ({{OEIS2C|A002385}})

=== Palindromic wing primes ===
Primes of the form <math>\frac{a \big( 10^m-1 \big)}{9} \pm b \times 10^{\frac{ m-1 }{2}}</math> with <math>0 \le a \pm b < 10</math>.<ref>{{Cite journal | last1 = Caldwell | first1 = C. | author1-link = Chris Caldwell (mathematician)| last2 = Dubner | first2 = H. | author2-link = Harvey Dubner | title = The near repdigit primes <math>A_{n-k-1}B_1A_k</math>, especially <math>9_{n-k-1}8_19_k</math> | journal = [[Journal of Recreational Mathematics]] | volume = 28 | issue = 1 | pages = 1–9 | year = 1996–97}}</ref> This means all digits except the middle digit are equal.

[[101 (number)|101]], [[131 (number)|131]], [[151 (number)|151]], [[181 (number)|181]], [[191 (number)|191]], [[313 (number)|313]], [[353 (number)|353]], [[373 (number)|373]], [[383 (number)|383]], [[727 (number)|727]], [[757 (number)|757]], [[787 (number)|787]], [[797 (number)|797]], [[919 (number)|919]], [[929 (number)|929]], 11311, 11411, 33533, 77377, 77477, 77977, 1114111, 1117111, 3331333, 3337333, 7772777, 7774777, 7778777, 111181111, 111191111, 777767777, 77777677777, 99999199999 ({{OEIS2C|A077798}})

=== [[Partition function (number theory)|Partition primes]] ===
Partition function values that are prime.

[[2]], [[3]], [[5]], [[7]], [[11 (number)|11]], [[101 (number)|101]], 17977, 10619863, 6620830889, 80630964769, 228204732751, 1171432692373, 1398341745571, 10963707205259, 15285151248481, 10657331232548839, 790738119649411319, 18987964267331664557 ({{OEIS2C|A049575}})

=== [[Pell number|Pell primes]] ===
Primes in the Pell number sequence ''P''{{sub|0}}&nbsp;=&nbsp;0, ''P''{{sub|''1''}}&nbsp;=&nbsp;1,
''P''{{sub|''n''}}&nbsp;=&nbsp;2''P''{{sub|''n''−1}}&nbsp;+&nbsp;''P''{{sub|''n''−2}}.

[[2]], [[5]], [[29 (number)|29]], 5741, 33461, 44560482149, 1746860020068409, 68480406462161287469, 13558774610046711780701, 4125636888562548868221559797461449 ({{OEIS2C|A086383}})

=== [[Permutable prime]]s ===
Any permutation of the decimal digits is a prime.

[[2]], [[3]], [[5]], [[7]], [[11 (number)|11]], [[13 (number)|13]], [[17 (number)|17]], [[31 (number)|31]], [[37 (number)|37]], [[71 (number)|71]], [[73 (number)|73]], [[79 (number)|79]], [[97 (number)|97]], [[113 (number)|113]], [[131 (number)|131]], [[199 (number)|199]], [[311 (number)|311]], [[337 (number)|337]], [[373 (number)|373]], [[733 (number)|733]], [[919 (number)|919]], [[991 (number)|991]], 1111111111111111111, 11111111111111111111111 ({{OEIS2C|A003459}})

=== [[Perrin number|Perrin primes]] ===
Primes in the Perrin number sequence ''P''(0)&nbsp;=&nbsp;3, ''P''(1)&nbsp;=&nbsp;0, ''P''(2)&nbsp;=&nbsp;2,
''P''(''n'')&nbsp;=&nbsp;''P''(''n''−2)&nbsp;+&nbsp;''P''(''n''−3).

[[2]], [[3]], [[5]], [[7]], [[17 (number)|17]], [[29 (number)|29]], [[277 (number)|277]], [[367 (number)|367]], [[853 (number)|853]], 14197, 43721, 1442968193, 792606555396977, 187278659180417234321, 66241160488780141071579864797 ({{OEIS2C|A074788}})

=== [[Pierpont prime]]s ===
Of the form 2{{sup|''u''}}3{{sup|''v''}}&nbsp;+&nbsp;1 for some [[integer]]s ''u'',''v''&nbsp;≥&nbsp;0.

These are also [[Prime number#Classification of prime numbers|class 1- primes]].

[[2]], [[3]], [[5]], [[7]], [[13 (number)|13]], [[17 (number)|17]], [[19 (number)|19]], [[37 (number)|37]], [[73 (number)|73]], [[97 (number)|97]], [[109 (number)|109]], [[163 (number)|163]], [[193 (number)|193]], [[257 (number)|257]], [[433 (number)|433]], [[487 (number)|487]], [[577 (number)|577]], [[769 (number)|769]], [[1153 (number)|1153]], [[1297 (number)|1297]], [[1459 (number)|1459]], [[2593 (number)|2593]], [[2917 (number)|2917]], [[3457 (number)|3457]], [[3889 (number)|3889]], 10369, 12289, 17497, 18433, 39367, 52489, [[65537 (number)|65537]], 139969, 147457 ({{OEIS2C|A005109}})

=== [[Pillai prime]]s ===
Primes ''p'' for which there exist ''n''&nbsp;>&nbsp;0 such that ''p'' divides ''n''! + 1 and ''n'' does not divide ''p'' − 1.

[[23 (number)|23]], [[29 (number)|29]], [[59 (number)|59]], [[61 (number)|61]], [[67 (number)|67]], [[71 (number)|71]], [[79 (number)|79]], [[83 (number)|83]], [[109 (number)|109]], [[137 (number)|137]], [[139 (number)|139]], [[149 (number)|149]], [[193 (number)|193]], [[227 (number)|227]], [[233 (number)|233]], [[239 (number)|239]], [[251 (number)|251]], [[257 (number)|257]], [[269 (number)|269]], [[271 (number)|271]], [[277 (number)|277]], [[293 (number)|293]], [[307 (number)|307]], [[311 (number)|311]], [[317 (number)|317]], [[359 (number)|359]], [[379 (number)|379]], [[383 (number)|383]], [[389 (number)|389]], [[397 (number)|397]], [[401 (number)|401]], [[419 (number)|419]], [[431 (number)|431]], [[449 (number)|449]], [[461 (number)|461]], [[463 (number)|463]], [[467 (number)|467]], [[479 (number)|479]], [[499 (number)|499]] ({{OEIS2C|A063980}})

=== Primes of the form ''n''<sup>4</sup> + 1 ===
Of the form ''n''<sup>4</sup> + 1.<ref>{{cite journal | last = Lal | first = M. | title = Primes of the Form n<sup>4</sup> + 1 | journal = Mathematics of Computation | volume = 21 | pages = 245–247 | publisher = [[American Mathematical Society|AMS]] | date = 1967 | url = https://www.ams.org/journals/mcom/1967-21-098/S0025-5718-1967-0222007-9/S0025-5718-1967-0222007-9.pdf | issn = 1088-6842 | doi = 10.1090/S0025-5718-1967-0222007-9 | url-status = live | archive-url = https://web.archive.org/web/20150113214845/http://www.ams.org/journals/mcom/1967-21-098/S0025-5718-1967-0222007-9/S0025-5718-1967-0222007-9.pdf | archive-date = 13 January 2015| doi-access = free }}</ref><ref>{{cite journal | last = Bohman | first = J. | title = New primes of the form ''n''<sup>4</sup> + 1 | journal = BIT Numerical Mathematics | volume = 13 | issue = 3 | pages = 370–372 | publisher = Springer | date = 1973 | issn = 1572-9125 | doi = 10.1007/BF01951947| s2cid = 123070671 }}</ref>

[[2]], [[17 (number)|17]], [[257 (number)|257]], [[1297 (number)|1297]], [[65537 (number)|65537]], 160001, 331777, 614657, 1336337, 4477457, 5308417, 8503057, 9834497, 29986577, 40960001, 45212177, 59969537, 65610001, 126247697, 193877777, 303595777, 384160001, 406586897, 562448657, 655360001 ({{OEIS2C|A037896}})

=== [[Primeval prime]]s ===
Primes for which there are more prime permutations of some or all the decimal digits than for any smaller number.

[[2]], [[13 (number)|13]], [[37 (number)|37]], [[107 (number)|107]], [[113 (number)|113]], [[137 (number)|137]], [[1013 (number)|1013]], [[1237 (number)|1237]], [[1367 (number)|1367]], 10079 ({{OEIS2C|A119535}})

=== [[Primorial prime]]s ===
Of the form ''p''{{sub|''n''}}#&nbsp;± 1.

[[3]], [[5]], [[7]], [[29 (number)|29]], [[31 (number)|31]], [[211 (number)|211]], [[2309 (number)|2309]], [[2311 (number)|2311]], 30029, 200560490131, 304250263527209, 23768741896345550770650537601358309 (union of {{OEIS2C|A057705}} and {{OEIS2C|A018239}}<ref name="A018239"/>)

=== [[Proth number|Proth primes]] ===
Of the form ''k''×2{{sup|''n''}}&nbsp;+&nbsp;1, with odd ''k'' and ''k''&nbsp;<&nbsp;2{{sup|''n''}}.

[[3]], [[5]], [[13 (number)|13]], [[17 (number)|17]], [[41 (number)|41]], [[97 (number)|97]], [[113 (number)|113]], [[193 (number)|193]], [[241 (number)|241]], [[257 (number)|257]], [[353 (number)|353]], [[449 (number)|449]], [[577 (number)|577]], [[641 (number)|641]], [[673 (number)|673]], [[769 (number)|769]], [[929 (number)|929]], [[1153 (number)|1153]], [[1217 (number)|1217]], [[1409 (number)|1409]], [[1601 (number)|1601]], [[2113 (number)|2113]], [[2689 (number)|2689]], [[2753 (number)|2753]], [[3137 (number)|3137]], [[3329 (number)|3329]], [[3457 (number)|3457]], 4481, 4993, 6529, 7297, 7681, 7937, 9473, 9601, 9857 ({{OEIS2C|A080076}})

=== [[Pythagorean prime]]s ===
Of the form 4''n'' + 1.

[[5]], [[13 (number)|13]], [[17 (number)|17]], [[29 (number)|29]],  [[37 (number)|37]], [[41 (number)|41]], [[53 (number)|53]], [[61 (number)|61]], [[73 (number)|73]], [[89 (number)|89]], [[97 (number)|97]], [[101 (number)|101]], [[109 (number)|109]], [[113 (number)|113]], [[137 (number)|137]], [[149 (number)|149]], [[157 (number)|157]], [[173 (number)|173]], [[181 (number)|181]], [[193 (number)|193]], [[197 (number)|197]], [[229 (number)|229]], [[233 (number)|233]], [[241 (number)|241]], [[257 (number)|257]], [[269 (number)|269]], [[277 (number)|277]], [[281 (number)|281]], [[293 (number)|293]], [[313 (number)|313]], [[317 (number)|317]], [[337 (number)|337]], [[349 (number)|349]], [[353 (number)|353]], [[373 (number)|373]], [[389 (number)|389]], [[397 (number)|397]], [[401 (number)|401]], [[409 (number)|409]], [[421 (number)|421]], [[433 (number)|433]], [[449 (number)|449]] ({{OEIS2C|A002144}})

=== [[Prime quadruplet|Prime quadruplets]] ===
{{see also|#Cousin primes|#Twin primes|#Prime triplets}}

Where (''p'', ''p''+2, ''p''+6, ''p''+8) are all prime.

([[5]], [[7]], [[11 (number)|11]], [[13 (number)|13]]), (11, 13, [[17 (number)|17]], [[19 (number)|19]]), ([[101 (number)|101]], [[103 (number)|103]], [[107 (number)|107]], [[109 (number)|109]]), ([[191 (number)|191]], [[193 (number)|193]], [[197 (number)|197]], [[199 (number)|199]]), ([[821 (number)|821]], [[823 (number)|823]], [[827 (number)|827]], [[829 (number)|829]]), ([[1481 (number)|1481]], [[1483 (number)|1483]], [[1487 (number)|1487]], [[1489 (number)|1489]]), ([[1871 (number)|1871]], [[1873 (number)|1873]], [[1877 (number)|1877]], [[1879 (number)|1879]]), ([[2081 (number)|2081]], [[2083 (number)|2083]], [[2087 (number)|2087]], [[2089 (number)|2089]]), ([[3251 (number)|3251]], [[3253 (number)|3253]], [[3257 (number)|3257]], [[3259 (number)|3259]]), ([[3461 (number)|3461]], [[3463 (number)|3463]], [[3467 (number)|3467]], [[3469 (number)|3469]]), (5651, 5653, 5657, 5659), (9431, 9433, 9437, 9439) ({{OEIS2C|A007530}}, {{OEIS2C|A136720}}, {{OEIS2C|A136721}}, {{OEIS2C|A090258}})

=== [[Quartan prime]]s ===
Of the form ''x''{{sup|4}}&nbsp;+&nbsp;''y''{{sup|4}}, where ''x'',''y''&nbsp;>&nbsp;0.

[[2]], [[17 (number)|17]], [[97 (number)|97]], [[257 (number)|257]], [[337 (number)|337]], [[641 (number)|641]], [[881 (number)|881]] ({{OEIS2C|A002645}})

=== [[Ramanujan prime]]s ===
Integers ''R''{{sub|''n''}} that are the smallest to give at least ''n'' primes from ''x''/2 to ''x'' for all ''x''&nbsp;≥&nbsp;''R''{{sub|''n''}} (all such integers are primes).

[[2]], [[11 (number)|11]], [[17 (number)|17]], [[29 (number)|29]], [[41 (number)|41]], [[47 (number)|47]], [[59 (number)|59]], [[67 (number)|67]], [[71 (number)|71]], [[97 (number)|97]], [[101 (number)|101]], [[107 (number)|107]], [[127 (number)|127]], [[149 (number)|149]], [[151 (number)|151]], [[167 (number)|167]], [[179 (number)|179]], [[181 (number)|181]], [[227 (number)|227]], [[229 (number)|229]], [[233 (number)|233]], [[239 (number)|239]], [[241 (number)|241]], [[263 (number)|263]], [[269 (number)|269]], [[281 (number)|281]], [[307 (number)|307]], [[311 (number)|311]], [[347 (number)|347]], [[349 (number)|349]], [[367 (number)|367]], [[373 (number)|373]], [[401 (number)|401]], [[409 (number)|409]], [[419 (number)|419]], [[431 (number)|431]], [[433 (number)|433]], [[439 (number)|439]], [[461 (number)|461]], [[487 (number)|487]], [[491 (number)|491]] ({{OEIS2C|A104272}})

=== [[Regular prime]]s ===
Primes ''p'' that do not divide the [[Class number (number theory)|class number]] of the ''p''-th [[cyclotomic field]].

[[3]], [[5]], [[7]], [[11 (number)|11]], [[13 (number)|13]], [[17 (number)|17]], [[19 (number)|19]], [[23 (number)|23]], [[29 (number)|29]], [[31 (number)|31]], [[41 (number)|41]], [[43 (number)|43]], [[47 (number)|47]], [[53 (number)|53]], [[61 (number)|61]], [[71 (number)|71]], [[73 (number)|73]], [[79 (number)|79]], [[83 (number)|83]], [[89 (number)|89]], [[97 (number)|97]], [[107 (number)|107]], [[109 (number)|109]], [[113 (number)|113]], [[127 (number)|127]], [[137 (number)|137]], [[139 (number)|139]], [[151 (number)|151]], [[163 (number)|163]], [[167 (number)|167]], [[173 (number)|173]], [[179 (number)|179]], [[181 (number)|181]], [[191 (number)|191]], [[193 (number)|193]], [[197 (number)|197]], [[199 (number)|199]], [[211 (number)|211]], [[223 (number)|223]], [[227 (number)|227]], [[229 (number)|229]], [[239 (number)|239]], [[241 (number)|241]], [[251 (number)|251]], [[269 (number)|269]], [[277 (number)|277]], [[281 (number)|281]] ({{OEIS2C|A007703}})

=== [[Repunit#Decimal repunit primes|Repunit primes]] ===
Primes containing only the decimal digit 1.

[[11 (number)|11]], 1111111111111111111 (19 digits), 11111111111111111111111  (23 digits) ({{OEIS2C|A004022}})

The next have 317, 1031, 49081, 86453, 109297, 270343 digits ({{OEIS2C|A004023}})

=== [[Dirichlet's theorem on arithmetic progressions|Residue classes of primes]] ===
Of the form ''an'' + ''d'' for fixed integers ''a'' and ''d''. Also called primes congruent to ''d'' [[Modular arithmetic|modulo]] ''a''.

The primes of the form 2''n''+1 are the odd primes, including all primes other than 2.  Some sequences have alternate names: 4''n''+1 are Pythagorean primes, 4''n''+3 are the integer Gaussian primes, and 6''n''+5 are the Eisenstein primes (with 2 omitted).  The classes 10''n''+''d'' (''d'' = 1, 3, 7, 9) are primes ending in the decimal digit ''d''.

2''n''+1: [[3]], [[5]], [[7]], [[11 (number)|11]], [[13 (number)|13]], [[17 (number)|17]], [[19 (number)|19]], [[23 (number)|23]], [[29 (number)|29]], [[31 (number)|31]], [[37 (number)|37]], [[41 (number)|41]], [[43 (number)|43]], [[47 (number)|47]], [[53 (number)|53]] ({{OEIS2C|A065091}})<br>
4''n''+1: 5, 13, 17, 29, 37, 41, 53, [[61 (number)|61]], [[73 (number)|73]], [[89 (number)|89]], [[97 (number)|97]], [[101 (number)|101]], [[109 (number)|109]], [[113 (number)|113]], [[137 (number)|137]] ({{OEIS2C|A002144}})<br>
4''n''+3: 3, 7, 11, 19, 23, 31, 43, 47, [[59 (number)|59]], [[67 (number)|67]], [[71 (number)|71]], [[79 (number)|79]], [[83 (number)|83]], [[103 (number)|103]], [[107 (number)|107]] ({{OEIS2C|A002145}})<br>
6''n''+1: 7, 13, 19, 31, 37, 43, 61, 67, 73, 79, 97, 103, 109, [[127 (number)|127]], [[139 (number)|139]] ({{OEIS2C|A002476}})<br>
6''n''+5: 5, 11, 17, 23, 29, 41, 47, 53, 59, 71, 83, 89, 101, 107, 113 ({{OEIS2C|A007528}})<br>
8''n''+1: 17, 41, 73, 89, 97, 113, 137, [[193 (number)|193]], [[233 (number)|233]], [[241 (number)|241]], [[257 (number)|257]], [[281 (number)|281]], [[313 (number)|313]], [[337 (number)|337]], [[353 (number)|353]] ({{OEIS2C|A007519}})<br>
8''n''+3: 3, 11, 19, 43, 59, 67, 83, 107, [[131 (number)|131]], 139, [[163 (number)|163]], [[179 (number)|179]], [[211 (number)|211]], [[227 (number)|227]], [[251 (number)|251]] ({{OEIS2C|A007520}})<br>
8''n''+5: 5, 13, 29, 37, 53, 61, 101, 109, [[149 (number)|149]], [[157 (number)|157]], [[173 (number)|173]], [[181 (number)|181]], [[197 (number)|197]], [[229 (number)|229]], [[269 (number)|269]] ({{OEIS2C|A007521}})<br>
8''n''+7: 7, 23, 31, 47, 71, 79, 103, 127, [[151 (number)|151]], [[167 (number)|167]], [[191 (number)|191]], [[199 (number)|199]], [[223 (number)|223]], [[239 (number)|239]], [[263 (number)|263]] ({{OEIS2C|A007522}})<br>
10''n''+1: 11, 31, 41, 61, 71, 101, 131, 151, 181, 191, 211, 241, 251, [[271 (number)|271]], 281 ({{OEIS2C|A030430}})<br>
10''n''+3: 3, 13, 23, 43, 53, 73, 83, 103, 113, 163, 173, 193, 223, 233, 263 ({{OEIS2C|A030431}})<br>
10''n''+7: 7, 17, 37, 47, 67, 97, 107, 127, 137, 157, 167, 197, 227, 257, [[277 (number)|277]] ({{OEIS2C|A030432}})<br>
10''n''+9: 19, 29, 59, 79, 89, 109, 139, 149, 179, 199, 229, 239, 269, [[349 (number)|349]], [[359 (number)|359]] ({{OEIS2C|A030433}})<br>
12''n''+1: 13, 37, 61, 73, 97, 109, 157, 181, 193, 229, 241, 277, 313, 337, 349 ({{OEIS2C|A068228}})<br>
12''n''+5: 5, 17, 29, 41, 53, 89, 101, 113, 137, 149, 173, 197, 233, 257, 269 ({{OEIS2C|A040117}})<br>
12''n''+7: 7, 19, 31, 43, 67, 79, 103, 127, 139, 151, 163, 199, 211, 223, 271 ({{OEIS2C|A068229}})<br>
12''n''+11: 11, 23, 47, 59, 71, 83, 107, 131, 167, 179, 191, 227, 239, 251, 263 ({{OEIS2C|A068231}})

=== [[Safe prime]]s ===
Where ''p'' and (''p''−1)&nbsp;/&nbsp;2 are both prime.

[[5]], [[7]], [[11 (number)|11]], [[23 (number)|23]], [[47 (number)|47]], [[59 (number)|59]], [[83 (number)|83]], [[107 (number)|107]], [[167 (number)|167]], [[179 (number)|179]], [[227 (number)|227]], [[263 (number)|263]], [[347 (number)|347]], [[359 (number)|359]], [[383 (number)|383]], [[467 (number)|467]], [[479 (number)|479]], [[503 (number)|503]], [[563 (number)|563]], [[587 (number)|587]], [[719 (number)|719]], [[839 (number)|839]], [[863 (number)|863]], [[887 (number)|887]], [[983 (number)|983]], [[1019 (number)|1019]], [[1187 (number)|1187]], [[1283 (number)|1283]], [[1307 (number)|1307]], [[1319 (number)|1319]], [[1367 (number)|1367]], [[1439 (number)|1439]], [[1487 (number)|1487]], [[1523 (number)|1523]], [[1619 (number)|1619]], [[1823 (number)|1823]], [[1907 (number)|1907]] ({{OEIS2C|A005385}})

=== [[Self number|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 (number)|31]], [[53 (number)|53]], [[97 (number)|97]], [[211 (number)|211]], [[233 (number)|233]], [[277 (number)|277]], [[367 (number)|367]], [[389 (number)|389]], [[457 (number)|457]], [[479 (number)|479]], [[547 (number)|547]], [[569 (number)|569]], [[613 (number)|613]], [[659 (number)|659]], [[727 (number)|727]], [[839 (number)|839]], [[883 (number)|883]], [[929 (number)|929]], [[1021 (number)|1021]], [[1087 (number)|1087]], [[1109 (number)|1109]], [[1223 (number)|1223]], [[1289 (number)|1289]], [[1447 (number)|1447]], [[1559 (number)|1559]], [[1627 (number)|1627]], [[1693 (number)|1693]], [[1783 (number)|1783]], [[1873 (number)|1873]] ({{OEIS2C|A006378}})

=== [[Sexy prime]]s ===
Where (''p'', ''p'' + 6) are both prime.

([[5]], [[11 (number)|11]]), ([[7]], [[13 (number)|13]]), (11, [[17 (number)|17]]), (13, [[19 (number)|19]]), (17, [[23 (number)|23]]), (23, [[29 (number)|29]]), ([[31 (number)|31]], [[37 (number)|37]]), (37, [[43 (number)|43]]), ([[41 (number)|41]], [[47 (number)|47]]), (47, [[53 (number)|53]]), (53, [[59 (number)|59]]), ([[61 (number)|61]], [[67 (number)|67]]), (67, [[73 (number)|73]]), (73, [[79 (number)|79]]), ([[83 (number)|83]], [[89 (number)|89]]), ([[97 (number)|97]], [[103 (number)|103]]), ([[101 (number)|101]], [[107 (number)|107]]), (103, [[109 (number)|109]]), (107, [[113 (number)|113]]), ([[131 (number)|131]], [[137 (number)|137]]), ([[151 (number)|151]], [[157 (number)|157]]), (157, [[163 (number)|163]]), ([[167 (number)|167]], [[173 (number)|173]]), (173, [[179 (number)|179]]), ([[191 (number)|191]], [[197 (number)|197]]), ([[193 (number)|193]], [[199 (number)|199]]) ({{OEIS2C|A023201}}, {{OEIS2C|A046117}})

=== [[Smarandache–Wellin number|Smarandache–Wellin primes]] ===
Primes that are the concatenation of the first ''n'' primes written in decimal.

[[2]], [[23 (number)|23]], [[2357 (number)|2357]] ({{OEIS2C|A069151}})

The fourth Smarandache-Wellin prime is the 355-digit concatenation of the first 128 primes that end with 719.

=== [[Solinas prime]]s ===
Of the form 2{{sup|''a''}}&nbsp;±&nbsp;2{{sup|''b''}}&nbsp;±&nbsp;1, where 0&nbsp;<&nbsp;''b''&nbsp;<&nbsp;''a''.

[[3]], [[5]], [[7]], [[11 (number)|11]], [[13 (number)|13]] ({{OEIS2C|A165255}})

=== [[Sophie Germain prime]]s ===
Where ''p'' and 2''p'' + 1 are both prime. A Sophie Germain prime has a corresponding [[#Safe primes|safe prime]].

[[2]], [[3]], [[5]], [[11 (number)|11]], [[23 (number)|23]], [[29 (number)|29]], [[41 (number)|41]], [[53 (number)|53]], [[83 (number)|83]], [[89 (number)|89]], [[113 (number)|113]], [[131 (number)|131]], [[173 (number)|173]], [[179 (number)|179]], [[191 (number)|191]], [[233 (number)|233]], [[239 (number)|239]], [[251 (number)|251]], [[281 (number)|281]], [[293 (number)|293]], [[359 (number)|359]], [[419 (number)|419]], [[431 (number)|431]], [[443 (number)|443]], [[491 (number)|491]], [[509 (number)|509]], [[593 (number)|593]], [[641 (number)|641]], [[653 (number)|653]], [[659 (number)|659]], [[683 (number)|683]], [[719 (number)|719]], [[743 (number)|743]], [[761 (number)|761]], [[809 (number)|809]], [[911 (number)|911]], [[953 (number)|953]] ({{OEIS2C|A005384}})

=== [[Stern prime]]s ===
Primes that are not the sum of a smaller prime and twice the square of a nonzero integer.

[[2]], [[3]], [[17 (number)|17]], [[137 (number)|137]], [[227 (number)|227]], [[977 (number)|977]], [[1187 (number)|1187]], [[1493 (number)|1493]] ({{OEIS2C|A042978}})

{{As of|2011}}, these are the only known Stern primes, and possibly the only existing.

=== [[Super-prime]]s ===
Primes with prime-numbered indexes in the sequence of prime numbers (the 2nd, 3rd, 5th, ... prime).

[[3]], [[5]], [[11 (number)|11]], [[17 (number)|17]], [[31 (number)|31]], [[41 (number)|41]], [[59 (number)|59]], [[67 (number)|67]], [[83 (number)|83]], [[109 (number)|109]], [[127 (number)|127]], [[157 (number)|157]], [[179 (number)|179]], [[191 (number)|191]], [[211 (number)|211]], [[241 (number)|241]], [[277 (number)|277]], [[283 (number)|283]], [[331 (number)|331]], [[353 (number)|353]], [[367 (number)|367]], [[401 (number)|401]], [[431 (number)|431]], [[461 (number)|461]], [[509 (number)|509]], [[547 (number)|547]], [[563 (number)|563]], [[587 (number)|587]], [[599 (number)|599]], [[617 (number)|617]], [[709 (number)|709]], [[739 (number)|739]], [[773 (number)|773]], [[797 (number)|797]], [[859 (number)|859]], [[877 (number)|877]], [[919 (number)|919]], [[967 (number)|967]], [[991 (number)|991]] ({{OEIS2C|A006450}})

=== [[Supersingular prime (moonshine theory)|Supersingular primes]] ===
There are exactly fifteen supersingular primes:

[[2]], [[3]], [[5]], [[7]], [[11 (number)|11]], [[13 (number)|13]], [[17 (number)|17]], [[19 (number)|19]], [[23 (number)|23]], [[29 (number)|29]], [[31 (number)|31]], [[41 (number)|41]], [[47 (number)|47]], [[59 (number)|59]], [[71 (number)|71]] ({{OEIS2C|A002267}})

=== [[Thabit prime]]s ===
Of the form 3×2{{sup|''n''}} − 1.

[[2]], [[5]], [[11 (number)|11]], [[23 (number)|23]], [[47 (number)|47]], [[191 (number)|191]], [[383 (number)|383]], 6143, 786431, 51539607551, 824633720831, 26388279066623, 108086391056891903, 55340232221128654847, 226673591177742970257407 ({{OEIS2C|A007505}})

The primes of the form 3×2{{sup|''n''}} + 1 are related.

[[7]], [[13 (number)|13]], [[97 (number)|97]], [[193 (number)|193]], [[769 (number)|769]], 12289, 786433, 3221225473, 206158430209, 6597069766657 ({{OEIS2C|A039687}})

=== [[Prime triplet|Prime triplets]] ===
{{see also|#Cousin primes|#Twin primes|#Prime quadruplets}}

Where (''p'', ''p''+2, ''p''+6) or (''p'', ''p''+4, ''p''+6) are all prime.

([[5]], [[7]], [[11 (number)|11]]), (7, 11, [[13 (number)|13]]), (11, 13, [[17 (number)|17]]), (13, 17, [[19 (number)|19]]), (17, 19, [[23 (number)|23]]), ([[37 (number)|37]], [[41 (number)|41]], [[43 (number)|43]]), (41, 43, [[47 (number)|47]]), ([[67 (number)|67]], [[71 (number)|71]], [[73 (number)|73]]), ([[97 (number)|97]], [[101 (number)|101]], [[103 (number)|103]]), (101, 103, [[107 (number)|107]]), (103, 107, [[109 (number)|109]]), (107, 109, [[113 (number)|113]]), ([[191 (number)|191]], [[193 (number)|193]], [[197 (number)|197]]), (193, 197, [[199 (number)|199]]), ([[223 (number)|223]], [[227 (number)|227]], [[229 (number)|229]]), (227, 229, [[233 (number)|233]]), ([[277 (number)|277]], [[281 (number)|281]], [[283 (number)|283]]), ([[307 (number)|307]], [[311 (number)|311]], [[313 (number)|313]]), (311, 313, [[317 (number)|317]]), ([[347 (number)|347]], [[349 (number)|349]], [[353 (number)|353]]) ({{OEIS2C|A007529}}, {{OEIS2C|A098414}}, {{OEIS2C|A098415}})

=== [[Truncatable prime]] ===
==== Left-truncatable ====
Primes that remain prime when the leading decimal digit is successively removed.

[[2]], [[3]], [[5]], [[7]], [[13 (number)|13]], [[17 (number)|17]], [[23 (number)|23]], [[37 (number)|37]], [[43 (number)|43]], [[47 (number)|47]], [[53 (number)|53]], [[67 (number)|67]], [[73 (number)|73]], [[83 (number)|83]], [[97 (number)|97]], [[113 (number)|113]], [[137 (number)|137]], [[167 (number)|167]], [[173 (number)|173]], [[197 (number)|197]], [[223 (number)|223]], [[283 (number)|283]], [[313 (number)|313]], [[317 (number)|317]], [[337 (number)|337]], [[347 (number)|347]], [[353 (number)|353]], [[367 (number)|367]], [[373 (number)|373]], [[383 (number)|383]], [[397 (number)|397]], [[443 (number)|443]], [[467 (number)|467]], [[523 (number)|523]], [[547 (number)|547]], [[613 (number)|613]], [[617 (number)|617]], [[643 (number)|643]], [[647 (number)|647]], [[653 (number)|653]], [[673 (number)|673]], [[683 (number)|683]] ({{OEIS2C|A024785}})

==== Right-truncatable ====
Primes that remain prime when the least significant decimal digit is successively removed.

[[2]], [[3]], [[5]], [[7]], [[23 (number)|23]], [[29 (number)|29]], [[31 (number)|31]], [[37 (number)|37]], [[53 (number)|53]], [[59 (number)|59]], [[71 (number)|71]], [[73 (number)|73]], [[79 (number)|79]], [[233 (number)|233]], [[239 (number)|239]], [[293 (number)|293]], [[311 (number)|311]], [[313 (number)|313]], [[317 (number)|317]], [[373 (number)|373]], [[379 (number)|379]], [[593 (number)|593]], [[599 (number)|599]], [[719 (number)|719]], [[733 (number)|733]], [[739 (number)|739]], [[797 (number)|797]], [[2333 (number)|2333]], [[2339 (number)|2339]], [[2393 (number)|2393]], [[2399 (number)|2399]], [[2939 (number)|2939]], [[3119 (number)|3119]], [[3137 (number)|3137]], [[3733 (number)|3733]], [[3739 (number)|3739]], [[3793 (number)|3793]], [[3797 (number)|3797]] ({{OEIS2C|A024770}})

==== Two-sided ====
Primes that are both left-truncatable and right-truncatable. There are exactly fifteen two-sided primes:

[[2]], [[3]], [[5]], [[7]], [[23 (number)|23]], [[37 (number)|37]], [[53 (number)|53]], [[73 (number)|73]], [[313 (number)|313]], [[317 (number)|317]], [[373 (number)|373]], [[797 (number)|797]], [[3137 (number)|3137]], [[3797 (number)|3797]], 739397 ({{OEIS2C|A020994}})

=== [[Twin prime]]s ===
{{see also|#Cousin primes|#Prime triplets|#Prime quadruplets}}

Where (''p'', ''p''+2) are both prime.

([[3]], [[5]]), (5, [[7]]), ([[11 (number)|11]], [[13 (number)|13]]), ([[17 (number)|17]], [[19 (number)|19]]), ([[29 (number)|29]], [[31 (number)|31]]), ([[41 (number)|41]], [[43 (number)|43]]), ([[59 (number)|59]], [[61 (number)|61]]), ([[71 (number)|71]], [[73 (number)|73]]), ([[101 (number)|101]], [[103 (number)|103]]), ([[107 (number)|107]], [[109 (number)|109]]), ([[137 (number)|137]], [[139 (number)|139]]), ([[149 (number)|149]], [[151 (number)|151]]), ([[179 (number)|179]], [[181 (number)|181]]), ([[191 (number)|191]], [[193 (number)|193]]), ([[197 (number)|197]], [[199 (number)|199]]), ([[227 (number)|227]], [[229 (number)|229]]), ([[239 (number)|239]], [[241 (number)|241]]), ([[269 (number)|269]], [[271 (number)|271]]), ([[281 (number)|281]], [[283 (number)|283]]), ([[311 (number)|311]], [[313 (number)|313]]), ([[347 (number)|347]], [[349 (number)|349]]), ([[419 (number)|419]], [[421 (number)|421]]), ([[431 (number)|431]], [[433 (number)|433]]), ([[461 (number)|461]], [[463 (number)|463]]) ({{OEIS2C|A001359}}, {{OEIS2C|A006512}})

=== [[Unique prime]]s ===
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 (number)|11]], [[37 (number)|37]], [[101 (number)|101]], 9091, 9901, 333667, 909091, 99990001, 999999000001, 9999999900000001, 909090909090909091, 1111111111111111111, 11111111111111111111111, 900900900900990990990991 ({{OEIS2C|A040017}})

=== [[Wagstaff prime]]s ===
Of the form (2{{sup|''n''}} + 1) / 3.

[[3]], [[11 (number)|11]], [[43 (number)|43]], [[683 (number)|683]], [[2731 (number)|2731]], 43691, 174763, 2796203, 715827883, 2932031007403, 768614336404564651, 201487636602438195784363, 845100400152152934331135470251, 56713727820156410577229101238628035243 ({{OEIS2C|A000979}})

Values of ''n'':

3, [[5]], [[7]], 11, [[13 (number)|13]], [[17 (number)|17]], [[19 (number)|19]], [[23 (number)|23]], [[31 (number)|31]], 43, [[61 (number)|61]], [[79 (number)|79]], [[101 (number)|101]], [[127 (number)|127]], [[167 (number)|167]], [[191 (number)|191]], [[199 (number)|199]], [[313 (number)|313]], [[347 (number)|347]], [[701 (number)|701]], [[1709 (number)|1709]], [[2617 (number)|2617]], [[3539 (number)|3539]], [[5807 (number)|5807]], [[10501 (number)|10501]], 10691, 11279, 12391, 14479, 42737, 83339, 95369, 117239, 127031, 138937, 141079, 267017, 269987, 374321 ({{OEIS2C|A000978}})

=== [[Wall–Sun–Sun prime]]s ===
A prime ''p''&nbsp;>&nbsp;5, if ''p''{{sup|2}} divides the [[Fibonacci number]] <math>F_{p - \left(\frac{{p}}{{5}}\right)}</math>, where the [[Legendre symbol]] <math>\left(\frac{{p}}{{5}}\right)</math> is defined as
:<math>\left(\frac{p}{5}\right) = \begin{cases} 1 &\textrm{if}\;p \equiv \pm1 \pmod 5\\ -1 &\textrm{if}\;p \equiv \pm2 \pmod 5. \end{cases}</math>

{{As of|2018}}, no Wall-Sun-Sun primes are known.

=== [[Wieferich prime]]s ===
Primes ''p'' such that {{nowrap|''a''<sup>''p'' − 1</sup> &equiv; 1 (mod ''p''<sup>2</sup>)}} for fixed integer ''a'' > 1.

2<sup>''p'' − 1</sup> &equiv; 1 (mod ''p''<sup>2</sup>): [[1093 (number)|1093]], [[3511 (number)|3511]] ({{OEIS2C|A001220}})<br>
3<sup>''p'' − 1</sup> &equiv; 1 (mod ''p''<sup>2</sup>): [[11 (number)|11]], 1006003 ({{OEIS2C|A014127}})<ref>{{cite book | last = Ribenboim | first = P. | author-link = Paulo Ribenboim | title = The new book of prime number records | publisher = Springer-Verlag | location = New York | page = 347 | url = https://books.google.com/books?id=72eg8bFw40kC&q=ribenboim | isbn = 0-387-94457-5| date = 22 February 1996 }}</ref><ref>{{cite web | title = Mirimanoff's Congruence: Other Congruences | url = http://www.museumstuff.com/learn/topics/Mirimanoff%27s_congruence::sub::Other_Congruences | access-date = 26 January 2011}}</ref><ref>{{cite journal | last1 = Gallot | first1 = Y. | last2 = Moree | first2 = P. | last3 = Zudilin | first3 = W. | title = The Erdös-Moser equation 1{{sup|''k''}} + 2{{sup|''k''}} +...+ (m−1){{sup|''k''}} = m{{sup|''k''}} revisited using continued fractions | journal = Mathematics of Computation | volume = 80 | pages = 1221–1237 | publisher = American Mathematical Society | year = 2011 | url = http://www.mpim-bonn.mpg.de/preprints/send?bid=4053 | doi = 10.1090/S0025-5718-2010-02439-1 | arxiv=0907.1356| s2cid = 16305654 }}</ref><br>
4<sup>''p'' − 1</sup> &equiv; 1 (mod ''p''<sup>2</sup>): [[1093 (number)|1093]], [[3511 (number)|3511]]<br>
5<sup>''p'' − 1</sup> &equiv; 1 (mod ''p''<sup>2</sup>): [[2]], 20771, 40487, 53471161, 1645333507, 6692367337, 188748146801 ({{OEIS2C|A123692}})<br>
6<sup>''p'' − 1</sup> &equiv; 1 (mod ''p''<sup>2</sup>): 66161, 534851, 3152573 ({{OEIS2C|A212583}})<br>
7<sup>''p'' − 1</sup> &equiv; 1 (mod ''p''<sup>2</sup>): [[5]], 491531 ({{OEIS2C|A123693}})<br>
8<sup>''p'' − 1</sup> &equiv; 1 (mod ''p''<sup>2</sup>): [[3]], [[1093 (number)|1093]], [[3511 (number)|3511]]<br>
9<sup>''p'' �� 1</sup> &equiv; 1 (mod ''p''<sup>2</sup>): [[2]], [[11 (number)|11]], 1006003<br>
10<sup>''p'' − 1</sup> &equiv; 1 (mod ''p''<sup>2</sup>): [[3]], [[487 (number)|487]], 56598313 ({{OEIS2C|A045616}})<br>
11<sup>''p'' − 1</sup> &equiv; 1 (mod ''p''<sup>2</sup>): [[71 (number)|71]]<ref name="RibenboimWelt">{{Cite book | last = Ribenboim | first = P. | author-link = Paulo Ribenboim | title = Die Welt der Primzahlen | publisher = Springer | year = 2006 | location = Berlin | page = 240 | url = https://www.gbv.de/dms/bs/toc/495799599.pdf | isbn = 3-540-34283-4 }}</ref><br>
12<sup>''p'' − 1</sup> &equiv; 1 (mod ''p''<sup>2</sup>): [[2693 (number)|2693]], 123653 ({{OEIS2C|A111027}})<br>
13<sup>''p'' − 1</sup> &equiv; 1 (mod ''p''<sup>2</sup>): [[2]], [[863 (number)|863]], 1747591 ({{OEIS2C|A128667}})<ref name="RibenboimWelt" /><br>
14<sup>''p'' − 1</sup> &equiv; 1 (mod ''p''<sup>2</sup>): [[29 (number)|29]], [[353 (number)|353]], 7596952219 ({{OEIS2C|A234810}})<br>
15<sup>''p'' − 1</sup> &equiv; 1 (mod ''p''<sup>2</sup>): 29131, 119327070011 ({{OEIS2C|A242741}})<br>
16<sup>''p'' − 1</sup> &equiv; 1 (mod ''p''<sup>2</sup>): [[1093 (number)|1093]], [[3511 (number)|3511]]<br>
17<sup>''p'' − 1</sup> &equiv; 1 (mod ''p''<sup>2</sup>): [[2]], [[3]], 46021, 48947 ({{OEIS2C|A128668}})<ref name="RibenboimWelt" /><br>
18<sup>''p'' − 1</sup> &equiv; 1 (mod ''p''<sup>2</sup>): [[5]], [[7]], [[37 (number)|37]], [[331 (number)|331]], 33923, 1284043 ({{OEIS2C|A244260}})<br>
19<sup>''p'' − 1</sup> &equiv; 1 (mod ''p''<sup>2</sup>): [[3]], [[7]], [[13 (number)|13]], [[43 (number)|43]], [[137 (number)|137]], 63061489 ({{OEIS2C|A090968}})<ref name="RibenboimWelt" /><br>
20<sup>''p'' − 1</sup> &equiv; 1 (mod ''p''<sup>2</sup>): [[281 (number)|281]], 46457, 9377747, 122959073 ({{OEIS2C|A242982}})<br>
21<sup>''p'' − 1</sup> &equiv; 1 (mod ''p''<sup>2</sup>): [[2]]<br>
22<sup>''p'' − 1</sup> &equiv; 1 (mod ''p''<sup>2</sup>): [[13 (number)|13]], [[673 (number)|673]], 1595813, 492366587, 9809862296159 ({{OEIS2C|A298951}})<br>
23<sup>''p'' − 1</sup> &equiv; 1 (mod ''p''<sup>2</sup>): [[13 (number)|13]], 2481757, 13703077, 15546404183, 2549536629329 ({{OEIS2C|A128669}})<br>
24<sup>''p'' − 1</sup> &equiv; 1 (mod ''p''<sup>2</sup>): [[5]], 25633<br>
25<sup>''p'' − 1</sup> &equiv; 1 (mod ''p''<sup>2</sup>): [[2]], 20771, 40487, 53471161, 1645333507, 6692367337, 188748146801

{{As of|2018}}, these are all known Wieferich primes with ''a'' ≤ 25.

=== [[Wilson prime]]s ===
Primes ''p'' for which ''p''{{sup|2}} divides (''p''−1)! + 1.

[[5]], [[13 (number)|13]], [[563 (number)|563]] ({{OEIS2C|A007540}})

{{As of|2018}}, these are the only known Wilson primes.

=== [[Wolstenholme prime]]s ===
Primes ''p'' for which the [[binomial coefficient]] <math>{{2p-1}\choose{p-1}} \equiv 1 \pmod{p^4}.</math>

[[16843 (number)|16843]], 2124679 ({{OEIS2C|A088164}})

{{As of|2018}}, these are the only known Wolstenholme primes.

=== [[Woodall number|Woodall primes]] ===
Of the form ''n''×2{{sup|''n''}} − 1.

[[7]], [[23 (number)|23]], [[383 (number)|383]], 32212254719, 2833419889721787128217599, 195845982777569926302400511, 4776913109852041418248056622882488319 ({{OEIS2C|A050918}})

== See also ==
{{Portal|Mathematics}}
<!-- Please keep entries in alphabetical order & add a short description [[WP:SEEALSO]] -->
{{div col|colwidth=20em|small=yes}}
*{{Annotated link|Illegal prime}}
*{{Annotated link|Largest known prime number}}
*{{Annotated link|List of largest known primes and probable primes}}
*{{Annotated link|List of numbers}}
*{{Annotated link|Prime gap}}
*{{Annotated link|Prime number theorem}}
*{{Annotated link|Probable prime}}
*{{Annotated link|Pseudoprime}}
*{{Annotated link|Strong prime}}
*{{Annotated link|Table of prime factors}}
*{{Annotated link|Wieferich pair}}
{{div col end}}
<!-- please keep entries in alphabetical order -->

== References ==
{{reflist}}

== External links ==
* [https://prime-numbers.de] All prime numbers from 31 to 6,469,693,189 for free download.
* [http://primes.utm.edu/lists/ Lists of Primes] at the Prime Pages.
* [http://primes.utm.edu/nthprime/ The Nth Prime Page] Nth prime through n=10^12, pi(x) through x=3*10^13, Random primes in same range.
* [http://www.rsok.com/~jrm/printprimes.html Interface to a list of the first 98 million primes] (primes less than 2,000,000,000)
* {{MathWorld|title=Prime Number Sequences|urlname=topics/PrimeNumberSequences|ref=none}}
* [http://oeis.org/wiki/Index_to_OEIS:_Section_Pri Selected prime related sequences] in [[On-Line Encyclopedia of Integer Sequences|OEIS]].
* Fischer, R. [http://www.fermatquotient.com/FermatQuotienten/FermQ_Sort.txt Thema: Fermatquotient B^(P−1) == 1 (mod P^2)] {{in lang|de}} (Lists Wieferich primes in all bases up to 1052)
* {{Cite web |last=Padilla |first=Tony |date=7 February 2013 |title=New Largest Known Prime Number |url=https://www.youtube.com/watch?v=QSEKzFGpCQs |url-status=live |archive-url=https://ghostarchive.org/varchive/youtube/20211102/QSEKzFGpCQs |archive-date=2021-11-02 |website=Numberphile |publisher=[[Brady Haran]] |ref=none}}{{cbignore}}

{{Prime number classes}}

[[Category:Prime numbers|*]]
[[Category:Classes of prime numbers|*]]
[[Category:Mathematics-related lists|Prime numbers]]
[[Category:Number-related lists|Prime]]
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