This is a list of articles aboutprime numbers. A prime number (orprime) is anatural number greater than 1 that has no positivedivisors other than 1 and itself. ByEuclid's theorem, there are an infinite number of prime numbers. Subsets of the prime numbers may be generated with variousformulas for primes.
The first 1,000 primes are listed below, followed by lists of notable types of prime numbers in alphabetical order, giving their respective first terms. The number 1 isneither prime nor composite.
The following table lists the first 1,000 primes, with 20 columns of consecutive primes in each of the 50 rows.[1]
TheGoldbach conjecture verification project reports that it has computed all primes smaller than 4×1018.[2] That means 95,676,260,903,887,607 primes[3] (nearly 1017), but they were not stored. There are known formulae to evaluate theprime-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 2×1021) smaller than 1023. A different computation found that there are 18,435,599,767,349,200,867,866 primes (roughly 2×1022) smaller than 1024, if theRiemann hypothesis is true.[4]
Below are listed the first prime numbers of many named forms and types. More details are in the article for the name.n is anatural number (including 0) in the definitions.
Balanced primes are primes with equal-sizedprime gaps before and after them, making them thearithmetic mean of their next larger and next smaller prime.
Bell primes are primes that are also the number ofpartitions of some finite set.
2,5,877, 27644437, 35742549198872617291353508656626642567, 359334085968622831041960188598043661065388726959079837.The next term has 6,539 digits. (OEIS: A051131)
Chen primes are primesp such thatp+2 is either a prime orsemiprime.
2,3,5,7,11,13,17,19,23,29,31,37,41,47,53,59,67,71,83,89,101,107,109,113,127,131,137,139,149,157,167,179,181,191,197,199,211,227,233,239,251,257,263,269,281,293,307,311,317,337,347,353,359,379,389,401,409 (OEIS: A109611)
A circular prime is a number that remains prime on any cyclic rotation of its base 10 digits.
2,3,5,7,11,13,17,31,37,71,73,79,97,113,131,197,199,311,337,373,719,733,919,971,991,1193,1931,3119,3779,7793,7937,9311,9377,11939,19391,19937,37199,39119,71993,91193,93719,93911,99371,193939,199933,319993,331999,391939,393919,919393,933199,939193,939391,993319,999331 (OEIS: A068652)
Some sources only include the smallest prime in each cycle. For example, listing 13, but omitting 31.
2,3,5,7,11,13,17,37,79,113,197,199,337,1193,3779,11939,19937,193939,199933, 1111111111111111111, 11111111111111111111111 (OEIS: A016114)
A cluster prime is a primep such that every evennatural numberk ≤p − 3 is the difference of two primes not exceedingp.
3,5,7,11,13,17,19,23, ... (OEIS: A038134)
All primes between 3 and 89, inclusive, are cluster primes. The first 10 primes that arenot cluster primes are:
2,97,127,149,191,211,223,227,229,251.
Cousin primes are pairs of primes that differ by four.
(3,7), (7,11), (13,17), (19,23), (37,41), (43,47), (67,71), (79,83), (97,101), (103,107), (109,113), (127,131), (163,167), (193,197), (223,227), (229,233), (277,281) (OEIS: A023200,OEIS: A046132)
Cuban primes are primes of the form where is a natural number.
7,19,37,61,127,271,331,397,547,631,919,1657,1801,1951,2269,2437,2791,3169,3571,4219,4447,5167,5419,6211,7057,7351,8269,9241,10267,11719,12097,13267,13669,16651,19441,19927,22447,23497,24571,25117,26227,27361,33391,35317 (OEIS: A002407)
The term is also used to refer to primes of the form where is a natural number.
13,109,193,433,769,1201,1453,2029,3469,3889,4801,10093,12289,13873,18253,20173,21169,22189,28813,37633,43201,47629,60493,63949,65713,69313,73009,76801,84673,106033,108301,112909,115249 (OEIS: A002648)
Cullen primes are primesp of the formp=k2k + 1, for some natural numberk.
3, 393050634124102232869567034555427371542904833 (OEIS: A050920)
Delicate primes are those primes that always become acomposite number when any of their base 10 digit is changed.
294001, 505447, 584141, 604171, 971767, 1062599, 1282529, 1524181, 2017963, 2474431, 2690201, 3085553, 3326489, 4393139 (OEIS: A050249)
Dihedral primes are primes that satisfy 180° rotational symmetry and mirror symmetry on aseven-segment display.
2,5,11,101,181,1181,1811,18181,108881,110881,118081,120121,121021,121151,150151,151051,151121,180181,180811,181081 (OEIS: A134996)
Real Eisenstein primes are realEisenstein integers that areirreducible. Equivalently, they are primes of the form 3k − 1, for a positive integerk.
2,5,11,17,23,29,41,47,53,59,71,83,89,101,107,113,131,137,149,167,173,179,191,197,227,233,239,251,257,263,269,281,293,311,317,347,353,359,383,389,401 (OEIS: A003627)
Emirps are those primes that become a different prime after their base 10 digits have been reversed. The name "emirp" is the reverse of the word "prime".
13,17,31,37,71,73,79,97,107,113,149,157,167,179,199,311,337,347,359,389,701,709,733,739,743,751,761,769,907,937,941,953,967,971,983,991 (OEIS: A006567)
Euclid primes are primesp such thatp−1 is aprimorial.
3,7,31,211,2311, 200560490131 (OEIS: A018239[5])
Euler irregular primes are primes that divide anEuler number for some
19,31,43,47,61,67,71,79,101,137,139,149,193,223,241,251,263,277,307,311,349,353,359,373,379,419,433,461,463,491,509,541,563,571,577,587 (OEIS: A120337)
Euler (p,p - 3) irregular primes are primesp that divide the (p + 3)rdEuler number.
149,241,2946901 (OEIS: A198245)
Factorial primes are primes whose distance to the nextfactorial number is one.
2,3,5,7,23,719,5039,39916801,479001599, 87178291199, 10888869450418352160768000001, 265252859812191058636308479999999, 263130836933693530167218012159999999, 8683317618811886495518194401279999999 (OEIS: A088054)
Fermat primes are primesp of the formp = 22k + 1, for anon-negative integerk. As of June 2024[update] only five Fermat primes have been discovered.
3,5,17,257,65537 (OEIS: A019434)
Generalized Fermat primes are primesp of the formp = a2k + 1, for anon-negative integerk and even natural numbera.
| Generalized Fermat primes with basea | |
|---|---|
| 2 | 3,5,17,257,65537, ... (OEIS: A019434) |
| 4 | 5,17,257,65537, ... |
| 6 | 7,37,1297, ... |
| 8 | (none exist) |
| 10 | 11,101, ... |
| 12 | 13, ... |
| 14 | 197, ... |
| 16 | 17,257,65537, ... |
| 18 | 19, ... |
| 20 | 401, 160001, ... |
| 22 | 23, ... |
| 24 | 577, 331777, ... |
Fibonacci primes are primes that appear in theFibonacci sequence.
2,3,5,13,89,233,1597, 28657, 514229, 433494437, 2971215073, 99194853094755497, 1066340417491710595814572169, 19134702400093278081449423917 (OEIS: A005478)
Fortunate primes are primes that are also Fortunate numbers. There are no known composite Fortunate numbers.[citation needed]
3,5,7,13,17,19,23,37,47,59,61,67,71,79,89,101,103,107,109,127,151,157,163,167,191,197,199,223,229,233,239,271,277,283,293,307,311,313,331,353,373,379,383,397 (OEIS: A046066)
Gaussian primes are primesp of the formp = 4k + 3, for anon-negative integerk.
3,7,11,19,23,31,43,47,59,67,71,79,83,103,107,127,131,139,151,163,167,179,191,199,211,223,227,239,251,263,271,283,307,311,331,347,359,367,379,383,419,431,439,443,463,467,479,487,491,499,503 (OEIS: A002145)
Good primes are primesp satisfyingab < p2, for all primesa andb such thata,b < p
5,11,17,29,37,41,53,59,67,71,97,101,127,149,179,191,223,227,251,257,269,307 (OEIS: A028388)
Happy primes are primes that are also happy numbers.
7,13,19,23,31,79,97,103,109,139,167,193,239,263,293,313,331,367,379,383,397,409,487,563,617,653,673,683,709,739,761,863,881,907,937,1009,1033,1039,1093 (OEIS: A035497)
Harmonic primes are primesp for which there are no solutions toHk ≡ 0 (mod p) andHk ≡ −ωp (mod p), for 1 ≤ k ≤ p−2, whereHk denotes thek-thharmonic number andωp denotes theWolstenholme quotient.[6]
5,13,17,23,41,67,73,79,107,113,139,149,157,179,191,193,223,239,241,251,263,277,281,293,307,311,317,331,337,349 (OEIS: A092101)
Higgs primes are primesp for whichp − 1 divides the square of the product of all smaller Higgs primes.
2,3,5,7,11,13,19,23,29,31,37,43,47,53,59,61,67,71,79,101,107,127,131,139,149,151,157,173,181,191,197,199,211,223,229,263,269,277,283,311,317,331,347,349 (OEIS: A007459)
Highly cototient primes are primes that are acototient more often than any integer below it except 1.
2,23,47,59,83,89,113,167,269,389,419,509,659,839,1049,1259,1889 (OEIS: A105440)
Forn ≥ 2, write the prime factorization ofn in base 10 and concatenate the factors; iterate until a prime is reached.
For anon-negative integer, its home prime is obtained by concatenating its prime factors in increasing order repeatedly, until a prime is achieved.
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 (OEIS: A037274)
Irregular primes are odd primesp that divide theclass number of thep-thcyclotomic field.
37,59,67,101,103,131,149,157,233,257,263,271,283,293,307,311,347,353,379,389,401,409,421,433,461,463,467,491,523,541,547,557,577,587,593,607,613 (OEIS: A000928)
The (p,p - 3) irregular primes are primesp such that (p,p − 3) is an irregular pair.
The (p,p - 5) irregular primes are primesp such that (p,p − 5) is an irregular pair.[7]
The (p,p - 9) irregular primes are primesp such that (p,p − 9) is an irregular pair.[7]
Isolated primes are primesp such that bothp − 2 andp + 2 are both composite.
2,23,37,47,53,67,79,83,89,97,113,127,131,157,163,167,173,211,223,233,251,257,263,277,293,307,317,331,337,353,359,367,373,379,383,389,397,401,409,439,443,449,457,467,479,487,491,499,503,509,541,547,557,563,577,587,593,607,613,631,647,653,673,677,683,691,701,709,719,727,733,739,743,751,757,761,769,773,787,797,839,853,863,877,887,907,911,919,929,937,941,947,953,967,971,977,983,991,997 (OEIS: A007510)
Leyland primes are primesp of the formp = ab + ba, wherea andb are integers larger than one.
17,593, 32993, 2097593, 8589935681, 59604644783353249, 523347633027360537213687137, 43143988327398957279342419750374600193 (OEIS: A094133)
Long primes, or full reptend primes, are odd primesp for which is acyclic number. Bases other than 10 are also used.
7,17,19,23,29,47,59,61,97,109,113,131,149,167,179,181,193,223,229,233,257,263,269,313,337,367,379,383,389,419,433,461,487,491,499,503,509,541,571,577,593 (OEIS: A001913)
Lucas primes are primes that appear in the Lucas sequence.
2,[8]3,7,11,29,47,199,521,2207,3571, 9349, 3010349, 54018521, 370248451, 6643838879, 119218851371, 5600748293801, 688846502588399, 32361122672259149 (OEIS: A005479)
Lucky primes are primes that are also lucky numbers.
3,7,13,31,37,43,67,73,79,127,151,163,193,211,223,241,283,307,331,349,367,409,421,433,463,487,541,577,601,613,619,631,643,673,727,739,769,787,823,883,937,991,997 (OEIS: A031157)
Mersenne primes are primesp of the formp = 2k − 1, for somenon-negative integerk.
3,7,31,127,8191,131071,524287,2147483647, 2305843009213693951, 618970019642690137449562111, 162259276829213363391578010288127, 170141183460469231731687303715884105727 (OEIS: A000668)
As of 2024[update], there are 52 known Mersenne primes.[citation needed] The 13th, 14th, and 52nd have respectively 157, 183, and 41,024,320 digits.[citation needed] The largest known prime 2136,279,841−1 is the 52nd Mersenne prime.[citation needed]
Mersenne divisors are primes that divide 2k − 1, for some primek. Every Mersenne primep is also a Mersenne divisor, withk =p.
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 (OEIS: A122094)
Primesp such that 2p − 1 is prime.
2,3,5,7,13,17,19,31,61,89,107,127,521,607,1279,2203,2281,3217, 4253, 4423,9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049,216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011,24036583, 25964951, 30402457, 32582657, 37156667, 42643801,43112609, 57885161, 74207281, 77232917 (OEIS: A000043)
As of September 2025[update], two more are known to be in the sequence, but it is not known whether they are the next:
82589933, 136279841
A subset of Mersenne primes of the form 22p−1 − 1 for primep.
7,127,2147483647, 170141183460469231731687303715884105727 (primes inOEIS: A077586)
Of the form (an − 1) / (a − 1) for fixed integera.
Fora = 2, these are the Mersenne primes, while fora = 10 they are therepunit primes. For other smalla, they are given below:
a = 3:13,1093, 797161, 3754733257489862401973357979128773, 6957596529882152968992225251835887181478451547013 (OEIS: A076481)
a = 4:5 (the only prime fora = 4)
a = 5:31, 19531, 12207031, 305175781, 177635683940025046467781066894531, 14693679385278593849609206715278070972733319459651094018859396328480215743184089660644531 (OEIS: A086122)
a = 6:7,43, 55987, 7369130657357778596659, 3546245297457217493590449191748546458005595187661976371 (OEIS: A165210)
a = 7: 2801, 16148168401, 85053461164796801949539541639542805770666392330682673302530819774105141531698707146930307290253537320447270457
a = 8:73 (the only prime fora = 8)
a = 9: none exist
Many generalizations of Mersenne primes have been defined. This include the following:
Of the form ⌊θ3n⌋, where θ is Mills' constant. This form is prime for all positive integersn.
2,11,1361, 2521008887, 16022236204009818131831320183 (OEIS: A051254)
Primes for which there is no shortersub-sequence of the decimal digits that form a prime. There are exactly 26 minimal primes:
2,3,5,7,11,19,41,61,89,409,449,499,881,991, 6469, 6949,9001, 9049, 9649, 9949, 60649, 666649, 946669, 60000049, 66000049, 66600049 (OEIS: A071062)
Newman–Shanks–Williams numbers that are prime.
7,41,239, 9369319, 63018038201, 489133282872437279, 19175002942688032928599 (OEIS: A088165)
Primesp for which the least positiveprimitive root is not a primitive root ofp2. Three such primes are known; it is not known whether there are more.[12]
2, 40487, 6692367337 (OEIS: A055578)
Primes that remain the same when their decimal digits are read backwards.
2,3,5,7,11,101,131,151,181,191,313,353,373,383,727,757,787,797,919,929, 10301, 10501, 10601, 11311, 11411, 12421, 12721, 12821, 13331, 13831, 13931, 14341, 14741 (OEIS: A002385)
Primes of the form with.[13] This means all digits except the middle digit are equal.
101,131,151,181,191,313,353,373,383,727,757,787,797,919,929, 11311, 11411, 33533, 77377, 77477, 77977, 1114111, 1117111, 3331333, 3337333, 7772777, 7774777, 7778777, 111181111, 111191111, 777767777, 77777677777, 99999199999 (OEIS: A077798)
Partition function values that are prime.
2,3,5,7,11,101, 17977, 10619863, 6620830889, 80630964769, 228204732751, 1171432692373, 1398341745571, 10963707205259, 15285151248481, 10657331232548839, 790738119649411319, 18987964267331664557 (OEIS: A049575)
Primes in the Pell number sequenceP0 = 0,P1 = 1,Pn = 2Pn−1 + Pn−2.
2,5,29, 5741, 33461, 44560482149, 1746860020068409, 68480406462161287469, 13558774610046711780701, 4125636888562548868221559797461449 (OEIS: A086383)
Any permutation of the decimal digits is a prime.
2,3,5,7,11,13,17,31,37,71,73,79,97,113,131,199,311,337,373,733,919,991, 1111111111111111111, 11111111111111111111111 (OEIS: A003459)
Primes in the Perrin number sequenceP(0) = 3,P(1) = 0,P(2) = 2,P(n) = P(n−2) + P(n−3).
2,3,5,7,17,29,277,367,853, 14197, 43721, 1442968193, 792606555396977, 187278659180417234321, 66241160488780141071579864797 (OEIS: A074788)
Of the form 2u3v + 1 for someintegersu,v ≥ 0.
These are alsoclass 1- primes.
2,3,5,7,13,17,19,37,73,97,109,163,193,257,433,487,577,769,1153,1297,1459,2593,2917,3457,3889, 10369, 12289, 17497, 18433, 39367, 52489,65537, 139969, 147457 (OEIS: A005109)
Primesp for which there existn > 0 such thatp dividesn! + 1 andn does not dividep − 1.
23,29,59,61,67,71,79,83,109,137,139,149,193,227,233,239,251,257,269,271,277,293,307,311,317,359,379,383,389,397,401,419,431,449,461,463,467,479,499 (OEIS: A063980)
2,17,257,1297,65537, 160001, 331777, 614657, 1336337, 4477457, 5308417, 8503057, 9834497, 29986577, 40960001, 45212177, 59969537, 65610001, 126247697, 193877777, 303595777, 384160001, 406586897, 562448657, 655360001 (OEIS: A037896)
Primes for which there are more prime permutations of some or all the decimal digits than for any smaller number.
2,13,37,107,113,137,1013,1237,1367, 10079 (OEIS: A119535)
Of the formpn# ± 1.
3,5,7,29,31,211,2309,2311, 30029, 200560490131, 304250263527209, 23768741896345550770650537601358309 (union ofOEIS: A057705 andOEIS: A018239[5])
Of the formk×2n + 1, with oddk andk < 2n.
3,5,13,17,41,97,113,193,241,257,353,449,577,641,673,769,929,1153,1217,1409,1601,2113,2689,2753,3137,3329,3457, 4481, 4993, 6529, 7297, 7681, 7937, 9473, 9601, 9857 (OEIS: A080076)
Of the form 4n + 1.
5,13,17,29,37,41,53,61,73,89,97,101,109,113,137,149,157,173,181,193,197,229,233,241,257,269,277,281,293,313,317,337,349,353,373,389,397,401,409,421,433,449 (OEIS: A002144)
Where (p,p+2,p+6,p+8) are all prime.
(5,7,11,13), (11, 13,17,19), (101,103,107,109), (191,193,197,199), (821,823,827,829), (1481,1483,1487,1489), (1871,1873,1877,1879), (2081,2083,2087,2089), (3251,3253,3257,3259), (3461,3463,3467,3469), (5651, 5653, 5657, 5659), (9431, 9433, 9437, 9439) (OEIS: A007530,OEIS: A136720,OEIS: A136721,OEIS: A090258)
Of the formx4 + y4, wherex,y > 0.
2,17,97,257,337,641,881 (OEIS: A002645)
IntegersRn that are the smallest to give at leastn primes fromx/2 tox for allx ≥ Rn (all such integers are primes).
2,11,17,29,41,47,59,67,71,97,101,107,127,149,151,167,179,181,227,229,233,239,241,263,269,281,307,311,347,349,367,373,401,409,419,431,433,439,461,487,491 (OEIS: A104272)
Primesp that do not divide theclass number of thep-thcyclotomic field.
3,5,7,11,13,17,19,23,29,31,41,43,47,53,61,71,73,79,83,89,97,107,109,113,127,137,139,151,163,167,173,179,181,191,193,197,199,211,223,227,229,239,241,251,269,277,281 (OEIS: A007703)
Primes containing only the decimal digit 1.
11, 1111111111111111111 (19 digits), 11111111111111111111111 (23 digits) (OEIS: A004022)
The next have 317, 1031, 49081, 86453, 109297, and 270343 digits, respectively (OEIS: A004023).
Of the forman +d for fixed integersa andd. Also called primes congruent todmoduloa.
The primes of the form 2n+1 are the odd primes, including all primes other than 2. Some sequences have alternate names: 4n+1 are Pythagorean primes, 4n+3 are the integer Gaussian primes, and 6n+5 are the Eisenstein primes (with 2 omitted). The classes 10n+d (d = 1, 3, 7, 9) are primes ending in the decimal digitd.
Ifa andd are relatively prime, the arithmetic progression contains infinitely many primes.
2n+1:3,5,7,11,13,17,19,23,29,31,37,41,43,47,53 (OEIS: A065091)
4n+1: 5, 13, 17, 29, 37, 41, 53,61,73,89,97,101,109,113,137 (OEIS: A002144)
4n+3: 3, 7, 11, 19, 23, 31, 43, 47,59,67,71,79,83,103,107 (OEIS: A002145)
6n+1: 7, 13, 19, 31, 37, 43, 61, 67, 73, 79, 97, 103, 109,127,139 (OEIS: A002476)
6n+5: 5, 11, 17, 23, 29, 41, 47, 53, 59, 71, 83, 89, 101, 107, 113 (OEIS: A007528)
8n+1: 17, 41, 73, 89, 97, 113, 137,193,233,241,257,281,313,337,353 (OEIS: A007519)
8n+3: 3, 11, 19, 43, 59, 67, 83, 107,131, 139,163,179,211,227,251 (OEIS: A007520)
8n+5: 5, 13, 29, 37, 53, 61, 101, 109,149,157,173,181,197,229,269 (OEIS: A007521)
8n+7: 7, 23, 31, 47, 71, 79, 103, 127,151,167,191,199,223,239,263 (OEIS: A007522)
10n+1: 11, 31, 41, 61, 71, 101, 131, 151, 181, 191, 211, 241, 251,271, 281 (OEIS: A030430)
10n+3: 3, 13, 23, 43, 53, 73, 83, 103, 113, 163, 173, 193, 223, 233, 263 (OEIS: A030431)
10n+7: 7, 17, 37, 47, 67, 97, 107, 127, 137, 157, 167, 197, 227, 257,277 (OEIS: A030432)
10n+9: 19, 29, 59, 79, 89, 109, 139, 149, 179, 199, 229, 239, 269,349,359 (OEIS: A030433)
12n+1: 13, 37, 61, 73, 97, 109, 157, 181, 193, 229, 241, 277, 313, 337, 349 (OEIS: A068228)
12n+5: 5, 17, 29, 41, 53, 89, 101, 113, 137, 149, 173, 197, 233, 257, 269 (OEIS: A040117)
12n+7: 7, 19, 31, 43, 67, 79, 103, 127, 139, 151, 163, 199, 211, 223, 271 (OEIS: A068229)
12n+11: 11, 23, 47, 59, 71, 83, 107, 131, 167, 179, 191, 227, 239, 251, 263 (OEIS: A068231)
Wherep and (p−1) / 2 are both prime.
5,7,11,23,47,59,83,107,167,179,227,263,347,359,383,467,479,503,563,587,719,839,863,887,983,1019,1187,1283,1307,1319,1367,1439,1487,1523,1619,1823,1907 (OEIS: A005385)
Primes that cannot be generated by any integer added to the sum of its decimal digits.
3,5,7,31,53,97,211,233,277,367,389,457,479,547,569,613,659,727,839,883,929,1021,1087,1109,1223,1289,1447,1559,1627,1693,1783,1873 (OEIS: A006378)
Where (p,p + 6) are both prime.
(5,11), (7,13), (11,17), (13,19), (17,23), (23,29), (31,37), (37,43), (41,47), (47,53), (53,59), (61,67), (67,73), (73,79), (83,89), (97,103), (101,107), (103,109), (107,113), (131,137), (151,157), (157,163), (167,173), (173,179), (191,197), (193,199) (OEIS: A023201,OEIS: A046117)
Primes that are the concatenation of the firstn primes written in decimal.
The fourth Smarandache-Wellin prime is the 355-digit concatenation of the first 128 primes that end with 719.
Of the form 2k − c1·2k−1 − c2·2k−2 − ... − ck.
Wherep and 2p + 1 are both prime. A Sophie Germain prime has a correspondingsafe prime.
2,3,5,11,23,29,41,53,83,89,113,131,173,179,191,233,239,251,281,293,359,419,431,443,491,509,593,641,653,659,683,719,743,761,809,911,953 (OEIS: A005384)
Primes that are not the sum of a smaller prime and twice the square of a nonzero integer.
2,3,17,137,227,977,1187,1493 (OEIS: A042978)
As of 2011[update], these are the only known Stern primes, and possibly the only existing.
Primes with prime-numbered indexes in the sequence of prime numbers (the 2nd, 3rd, 5th, ... prime).
3,5,11,17,31,41,59,67,83,109,127,157,179,191,211,241,277,283,331,353,367,401,431,461,509,547,563,587,599,617,709,739,773,797,859,877,919,967,991 (OEIS: A006450)
There are exactly fifteen supersingular primes:
2,3,5,7,11,13,17,19,23,29,31,41,47,59,71 (OEIS: A002267)
Of the form 3×2n − 1.
2,5,11,23,47,191,383, 6143, 786431, 51539607551, 824633720831, 26388279066623, 108086391056891903, 55340232221128654847, 226673591177742970257407 (OEIS: A007505)
The primes of the form 3×2n + 1 are related.
7,13,97,193,769, 12289, 786433, 3221225473, 206158430209, 6597069766657 (OEIS: A039687)
Where (p,p+2,p+6) or (p,p+4,p+6) are all prime.
(5,7,11), (7, 11,13), (11, 13,17), (13, 17,19), (17, 19,23), (37,41,43), (41, 43,47), (67,71,73), (97,101,103), (101, 103,107), (103, 107,109), (107, 109,113), (191,193,197), (193, 197,199), (223,227,229), (227, 229,233), (277,281,283), (307,311,313), (311, 313,317), (347,349,353) (OEIS: A007529,OEIS: A098414,OEIS: A098415)
Primes that remain prime when the leading decimal digit is successively removed.
2,3,5,7,13,17,23,37,43,47,53,67,73,83,97,113,137,167,173,197,223,283,313,317,337,347,353,367,373,383,397,443,467,523,547,613,617,643,647,653,673,683 (OEIS: A024785)
Primes that remain prime when the least significant decimal digit is successively removed.
2,3,5,7,23,29,31,37,53,59,71,73,79,233,239,293,311,313,317,373,379,593,599,719,733,739,797,2333,2339,2393,2399,2939,3119,3137,3733,3739,3793,3797 (OEIS: A024770)
Primes that are both left-truncatable and right-truncatable. There are exactly fifteen two-sided primes:
2,3,5,7,23,37,53,73,313,317,373,797,3137,3797, 739397 (OEIS: A020994)
Where (p,p+2) are both prime.
(3,5), (5,7), (11,13), (17,19), (29,31), (41,43), (59,61), (71,73), (101,103), (107,109), (137,139), (149,151), (179,181), (191,193), (197,199), (227,229), (239,241), (269,271), (281,283), (311,313), (347,349), (419,421), (431,433), (461,463) (OEIS: A001359,OEIS: A006512)
The list of primesp for which theperiod length of the decimal expansion of 1/p is unique (no other prime gives the same period).
3,11,37,101, 9091, 9901, 333667, 909091, 99990001, 999999000001, 9999999900000001, 909090909090909091, 1111111111111111111, 11111111111111111111111, 900900900900990990990991 (OEIS: A040017)
Of the form (2n + 1) / 3.
3,11,43,683,2731, 43691, 174763, 2796203, 715827883, 2932031007403, 768614336404564651, 201487636602438195784363, 845100400152152934331135470251, 56713727820156410577229101238628035243 (OEIS: A000979)
Values ofn:
3,5,7, 11,13,17,19,23,31, 43,61,79,101,127,167,191,199,313,347,701,1709,2617,3539,5807,10501, 10691, 11279, 12391, 14479, 42737, 83339, 95369, 117239, 127031, 138937, 141079, 267017, 269987, 374321 (OEIS: A000978)
A primep > 5, ifp2 divides theFibonacci number, where theLegendre symbol is defined as
As of 2022[update], no Wall-Sun-Sun primes have been found below (about).[17]
 | This sectionappears to contradict the articleWieferich prime on The definition of a Weiferich prime. Please see thetalk page for more information.(October 2025) |
Primesp such thatap − 1 ≡ 1 (modp2) for fixed integera > 1.
2p − 1 ≡ 1 (modp2):1093,3511 (OEIS: A001220)
3p − 1 ≡ 1 (modp2):11, 1006003 (OEIS: A014127)[18][19][20]
4p − 1 ≡ 1 (modp2):1093,3511
5p − 1 ≡ 1 (modp2):2, 20771, 40487, 53471161, 1645333507, 6692367337, 188748146801 (OEIS: A123692)
6p − 1 ≡ 1 (modp2): 66161, 534851, 3152573 (OEIS: A212583)
7p − 1 ≡ 1 (modp2):5, 491531 (OEIS: A123693)
8p − 1 ≡ 1 (modp2):3,1093,3511
9p − 1 ≡ 1 (modp2):2,11, 1006003
10p − 1 ≡ 1 (modp2):3,487, 56598313 (OEIS: A045616)
11p − 1 ≡ 1 (modp2):71[21]
12p − 1 ≡ 1 (modp2):2693, 123653 (OEIS: A111027)
13p − 1 ≡ 1 (modp2):2,863, 1747591 (OEIS: A128667)[21]
14p − 1 ≡ 1 (modp2):29,353, 7596952219 (OEIS: A234810)
15p − 1 ≡ 1 (modp2): 29131, 119327070011 (OEIS: A242741)
16p − 1 ≡ 1 (modp2):1093,3511
17p − 1 ≡ 1 (modp2):2,3, 46021, 48947 (OEIS: A128668)[21]
18p − 1 ≡ 1 (modp2):5,7,37,331, 33923, 1284043 (OEIS: A244260)
19p − 1 ≡ 1 (modp2):3,7,13,43,137, 63061489 (OEIS: A090968)[21]
20p − 1 ≡ 1 (modp2):281, 46457, 9377747, 122959073 (OEIS: A242982)
21p − 1 ≡ 1 (modp2):2
22p − 1 ≡ 1 (modp2):13,673, 1595813, 492366587, 9809862296159 (OEIS: A298951)
23p − 1 ≡ 1 (modp2):13, 2481757, 13703077, 15546404183, 2549536629329 (OEIS: A128669)
24p − 1 ≡ 1 (modp2):5, 25633
25p − 1 ≡ 1 (modp2):2, 20771, 40487, 53471161, 1645333507, 6692367337, 188748146801
As of 2018[update], these are all known Wieferich primes witha ≤ 25.
Primesp for whichp2 divides (p−1)! + 1.
As of 2018[update], these are the only known Wilson primes.
Primesp for which thebinomial coefficient
16843, 2124679 (OEIS: A088164)
As of 2018[update], these are the only known Wolstenholme primes.
Of the formn×2n − 1.
7,23,383, 32212254719, 2833419889721787128217599, 195845982777569926302400511, 4776913109852041418248056622882488319 (OEIS: A050918)
