• Source: List of prime numbers

      • This is a list of articles about prime numbers. A prime number (or prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself. By Euclid's theorem, there are an infinite number of prime numbers. Subsets of the prime numbers may be generated with various formulas for primes. The first 1000 primes are listed below, followed by lists of notable types of prime numbers in alphabetical order, giving their respective first terms. 1 is neither prime nor composite.


      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.

      (sequence A000040 in the OEIS).
      The Goldbach conjecture verification project reports that it has computed all primes smaller than 4×1018. That means 95,676,260,903,887,607 primes (nearly 1017), 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 2×1021) 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 the Riemann hypothesis is true.


      Lists of primes by type


      Below are listed the first prime numbers of many named forms and types. More details are in the article for the name. n is a natural number (including 0) in the definitions.


      = Balanced primes

      =
      Primes with equal-sized prime gaps after and before them, so that they are equal to the arithmetic mean of the nearest primes after and before.

      5, 53, 157, 173, 211, 257, 263, 373, 563, 593, 607, 653, 733, 947, 977, 1103, 1123, 1187, 1223, 1367, 1511, 1747, 1753, 1907, 2287, 2417, 2677, 2903, 2963, 3307, 3313, 3637, 3733, 4013, 4409, 4457, 4597, 4657, 4691, 4993, 5107, 5113, 5303, 5387, 5393 (OEIS: A006562).


      = Bell primes

      =
      Primes that are the number of partitions of a set with n members.
      2, 5, 877, 27644437, 35742549198872617291353508656626642567, 359334085968622831041960188598043661065388726959079837.
      The next term has 6,539 digits. (OEIS: A051131)


      = Chen primes

      =
      Where p is prime and p+2 is either a prime or semiprime.
      2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 47, 53, 59, 67, 71, 83, 89, 101, 107, 109, 113, 127, 131, 137, 139, 149, 157, 167, 179, 181, 191, 197, 199, 211, 227, 233, 239, 251, 257, 263, 269, 281, 293, 307, 311, 317, 337, 347, 353, 359, 379, 389, 401, 409 (OEIS: A109611)


      = Circular primes

      =
      A circular prime number is a number that remains prime on any cyclic rotation of its digits (in base 10).
      2, 3, 5, 7, 11, 13, 17, 31, 37, 71, 73, 79, 97, 113, 131, 197, 199, 311, 337, 373, 719, 733, 919, 971, 991, 1193, 1931, 3119, 3779, 7793, 7937, 9311, 9377, 11939, 19391, 19937, 37199, 39119, 71993, 91193, 93719, 93911, 99371, 193939, 199933, 319993, 331999, 391939, 393919, 919393, 933199, 939193, 939391, 993319, 999331 (OEIS: A068652)
      Some sources only list the smallest prime in each cycle, for example, listing 13, but omitting 31 (OEIS really calls this sequence circular primes, but not the above sequence):
      2, 3, 5, 7, 11, 13, 17, 37, 79, 113, 197, 199, 337, 1193, 3779, 11939, 19937, 193939, 199933, 1111111111111111111, 11111111111111111111111 (OEIS: A016114)
      All repunit primes are circular.


      = 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, 13, 17, 19, 23, ... (OEIS: A038134)
      All odd primes between 3 and 89, inclusive, are cluster primes. The first 10 primes that are not cluster primes are:
      2, 97, 127, 149, 191, 211, 223, 227, 229, 251.


      = Cousin primes

      =

      Where (p, p + 4) are both prime.
      (3, 7), (7, 11), (13, 17), (19, 23), (37, 41), (43, 47), (67, 71), (79, 83), (97, 101), (103, 107), (109, 113), (127, 131), (163, 167), (193, 197), (223, 227), (229, 233), (277, 281) (OEIS: A023200, OEIS: A046132)


      = Cuban primes

      =
      Of the form








      x

      3


      −

      y

      3




      x
      −
      y






      {\displaystyle {\tfrac {x^{3}-y^{3}}{x-y}}}

      where x = y + 1.
      7, 19, 37, 61, 127, 271, 331, 397, 547, 631, 919, 1657, 1801, 1951, 2269, 2437, 2791, 3169, 3571, 4219, 4447, 5167, 5419, 6211, 7057, 7351, 8269, 9241, 10267, 11719, 12097, 13267, 13669, 16651, 19441, 19927, 22447, 23497, 24571, 25117, 26227, 27361, 33391, 35317 (OEIS: A002407)
      Of the form








      x

      3


      −

      y

      3




      x
      −
      y






      {\displaystyle {\tfrac {x^{3}-y^{3}}{x-y}}}

      where x = y + 2.
      13, 109, 193, 433, 769, 1201, 1453, 2029, 3469, 3889, 4801, 10093, 12289, 13873, 18253, 20173, 21169, 22189, 28813, 37633, 43201, 47629, 60493, 63949, 65713, 69313, 73009, 76801, 84673, 106033, 108301, 112909, 115249 (OEIS: A002648)


      = Cullen primes

      =
      Of the form n×2n + 1.
      3, 393050634124102232869567034555427371542904833 (OEIS: A050920)


      = Delicate primes

      =
      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 (OEIS: A050249)


      = Dihedral primes

      =
      Primes that remain prime when read upside down or mirrored in a seven-segment display.
      2, 5, 11, 101, 181, 1181, 1811, 18181, 108881, 110881, 118081, 120121,
      121021, 121151, 150151, 151051, 151121, 180181, 180811, 181081 (OEIS: A134996)


      = Eisenstein primes without imaginary part

      =
      Eisenstein integers that are irreducible and real numbers (primes of the form 3n − 1).
      2, 5, 11, 17, 23, 29, 41, 47, 53, 59, 71, 83, 89, 101, 107, 113, 131, 137, 149, 167, 173, 179, 191, 197, 227, 233, 239, 251, 257, 263, 269, 281, 293, 311, 317, 347, 353, 359, 383, 389, 401 (OEIS: 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, 17, 31, 37, 71, 73, 79, 97, 107, 113, 149, 157, 167, 179, 199, 311, 337, 347, 359, 389, 701, 709, 733, 739, 743, 751, 761, 769, 907, 937, 941, 953, 967, 971, 983, 991 (OEIS: A006567)


      = Euclid primes

      =
      Of the form pn# + 1 (a subset of primorial primes).
      3, 7, 31, 211, 2311, 200560490131 (OEIS: A018239)


      = Euler irregular primes

      =
      A prime



      p


      {\displaystyle p}

      that divides Euler number




      E

      2
      n




      {\displaystyle E_{2n}}

      for some



      0
      ≤
      2
      n
      ≤
      p
      −
      3


      {\displaystyle 0\leq 2n\leq p-3}

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


      Euler (p, p − 3) irregular primes


      Primes



      p


      {\displaystyle p}

      such that



      (
      p
      ,
      p
      −
      3
      )


      {\displaystyle (p,p-3)}

      is an Euler irregular pair.
      149, 241, 2946901 (OEIS: A198245)


      = Factorial primes

      =
      Of the form n! − 1 or n! + 1.
      2, 3, 5, 7, 23, 719, 5039, 39916801, 479001599, 87178291199, 10888869450418352160768000001, 265252859812191058636308479999999, 263130836933693530167218012159999999, 8683317618811886495518194401279999999 (OEIS: A088054)


      = Fermat primes

      =
      Of the form 22n + 1.
      3, 5, 17, 257, 65537 (OEIS: A019434)
      As of June 2024 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.


      Generalized Fermat primes


      Of the form a2n + 1 for fixed integer a.
      a = 2: 3, 5, 17, 257, 65537 (OEIS: A019434)
      a = 4: 5, 17, 257, 65537
      a = 6: 7, 37, 1297
      a = 8: (does not exist)
      a = 10: 11, 101
      a = 12: 13
      a = 14: 197
      a = 16: 17, 257, 65537
      a = 18: 19
      a = 20: 401, 160001
      a = 22: 23
      a = 24: 577, 331777


      = Fibonacci primes

      =
      Primes in the Fibonacci sequence F0 = 0, F1 = 1,
      Fn = Fn−1 + Fn−2.
      2, 3, 5, 13, 89, 233, 1597, 28657, 514229, 433494437, 2971215073, 99194853094755497, 1066340417491710595814572169, 19134702400093278081449423917 (OEIS: A005478)


      = Fortunate primes

      =
      Fortunate numbers that are prime (it has been conjectured they all are).
      3, 5, 7, 13, 17, 19, 23, 37, 47, 59, 61, 67, 71, 79, 89, 101, 103, 107, 109, 127, 151, 157, 163, 167, 191, 197, 199, 223, 229, 233, 239, 271, 277, 283, 293, 307, 311, 313, 331, 353, 373, 379, 383, 397 (OEIS: A046066)


      = Gaussian primes

      =
      Prime elements of the Gaussian integers; equivalently, primes of the form 4n + 3.
      3, 7, 11, 19, 23, 31, 43, 47, 59, 67, 71, 79, 83, 103, 107, 127, 131, 139, 151, 163, 167, 179, 191, 199, 211, 223, 227, 239, 251, 263, 271, 283, 307, 311, 331, 347, 359, 367, 379, 383, 419, 431, 439, 443, 463, 467, 479, 487, 491, 499, 503 (OEIS: A002145)


      = Good primes

      =
      Primes pn for which pn2 > pn−i pn+i for all 1 ≤ i ≤ n−1, where pn is the nth prime.
      5, 11, 17, 29, 37, 41, 53, 59, 67, 71, 97, 101, 127, 149, 179, 191, 223, 227, 251, 257, 269, 307 (OEIS: A028388)


      = Happy primes

      =
      Happy numbers that are prime.
      7, 13, 19, 23, 31, 79, 97, 103, 109, 139, 167, 193, 239, 263, 293, 313, 331, 367, 379, 383, 397, 409, 487, 563, 617, 653, 673, 683, 709, 739, 761, 863, 881, 907, 937, 1009, 1033, 1039, 1093 (OEIS: A035497)


      = Harmonic primes

      =
      Primes p for which there are no solutions to Hk ≡ 0 (mod p) and Hk ≡ −ωp (mod p) for 1 ≤ k ≤ p−2, where Hk denotes the k-th harmonic number and ωp denotes the Wolstenholme quotient.
      5, 13, 17, 23, 41, 67, 73, 79, 107, 113, 139, 149, 157, 179, 191, 193, 223, 239, 241, 251, 263, 277, 281, 293, 307, 311, 317, 331, 337, 349 (OEIS: A092101)


      = Higgs primes for squares

      =
      Primes p for which p − 1 divides the square of the product of all earlier terms.
      2, 3, 5, 7, 11, 13, 19, 23, 29, 31, 37, 43, 47, 53, 59, 61, 67, 71, 79, 101, 107, 127, 131, 139, 149, 151, 157, 173, 181, 191, 197, 199, 211, 223, 229, 263, 269, 277, 283, 311, 317, 331, 347, 349 (OEIS: A007459)


      = Highly cototient primes

      =
      Primes that are a cototient more often than any integer below it except 1.
      2, 23, 47, 59, 83, 89, 113, 167, 269, 389, 419, 509, 659, 839, 1049, 1259, 1889 (OEIS: A105440)


      = Home primes

      =
      For n ≥ 2, write the prime factorization of 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 (OEIS: A037274)


      = Irregular primes

      =
      Odd primes p that divide the class number of the p-th cyclotomic field.
      37, 59, 67, 101, 103, 131, 149, 157, 233, 257, 263, 271, 283, 293, 307, 311, 347, 353, 379, 389, 401, 409, 421, 433, 461, 463, 467, 491, 523, 541, 547, 557, 577, 587, 593, 607, 613 (OEIS: A000928)


      (p, p − 3) irregular primes


      (See Wolstenholme prime)


      (p, p − 5) irregular primes


      Primes p such that (p, p−5) is an irregular pair.
      37


      (p, p − 9) irregular primes


      Primes p such that (p, p − 9) is an irregular pair.
      67, 877 (OEIS: A212557)


      = Isolated primes

      =
      Primes p such that neither p − 2 nor p + 2 is prime.
      2, 23, 37, 47, 53, 67, 79, 83, 89, 97, 113, 127, 131, 157, 163, 167, 173, 211, 223, 233, 251, 257, 263, 277, 293, 307, 317, 331, 337, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 439, 443, 449, 457, 467, 479, 487, 491, 499, 503, 509, 541, 547, 557, 563, 577, 587, 593, 607, 613, 631, 647, 653, 673, 677, 683, 691, 701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797, 839, 853, 863, 877, 887, 907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997 (OEIS: A007510)


      = Leyland primes

      =
      Of the form xy + yx, with 1 < x < y.
      17, 593, 32993, 2097593, 8589935681, 59604644783353249, 523347633027360537213687137, 43143988327398957279342419750374600193 (OEIS: A094133)


      = Long primes

      =
      Primes p for which, in a given base b,







      b

      p
      −
      1


      −
      1

      p




      {\displaystyle {\frac {b^{p-1}-1}{p}}}

      gives a cyclic number. They are also called full reptend primes. Primes p for base 10:
      7, 17, 19, 23, 29, 47, 59, 61, 97, 109, 113, 131, 149, 167, 179, 181, 193, 223, 229, 233, 257, 263, 269, 313, 337, 367, 379, 383, 389, 419, 433, 461, 487, 491, 499, 503, 509, 541, 571, 577, 593 (OEIS: A001913)


      = Lucas primes

      =
      Primes in the Lucas number sequence L0 = 2, L1 = 1,
      Ln = Ln−1 + Ln−2.
      2, 3, 7, 11, 29, 47, 199, 521, 2207, 3571, 9349, 3010349, 54018521, 370248451, 6643838879, 119218851371, 5600748293801, 688846502588399, 32361122672259149 (OEIS: A005479)


      = Lucky primes

      =
      Lucky numbers that are prime.
      3, 7, 13, 31, 37, 43, 67, 73, 79, 127, 151, 163, 193, 211, 223, 241, 283, 307, 331, 349, 367, 409, 421, 433, 463, 487, 541, 577, 601, 613, 619, 631, 643, 673, 727, 739, 769, 787, 823, 883, 937, 991, 997 (OEIS: A031157)


      = Mersenne primes

      =
      Of the form 2n − 1.
      3, 7, 31, 127, 8191, 131071, 524287, 2147483647, 2305843009213693951, 618970019642690137449562111, 162259276829213363391578010288127, 170141183460469231731687303715884105727 (OEIS: 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 2136,279,841-1, which is the 52nd Mersenne prime.


      Mersenne divisors


      Primes p that divide 2n − 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 (OEIS: A122094)
      All Mersenne primes are, by definition, members of this sequence.


      Mersenne prime exponents


      Primes p 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 (OEIS: A000043)
      As of October 2024, four more are known to be in the sequence, but it is not known whether they are the next:
      74207281, 77232917, 82589933, 136279841


      Double Mersenne primes


      A subset of Mersenne primes of the form 22p−1 − 1 for prime p.
      7, 127, 2147483647, 170141183460469231731687303715884105727 (primes in OEIS: A077586)


      Generalized repunit primes


      Of the form (an − 1) / (a − 1) for fixed integer a.
      For a = 2, these are the Mersenne primes, while for a = 10 they are the repunit primes. For other small a, they are given below:
      a = 3: 13, 1093, 797161, 3754733257489862401973357979128773, 6957596529882152968992225251835887181478451547013 (OEIS: A076481)
      a = 4: 5 (the only prime for a = 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 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 bn − (b − 1)n, including the Mersenne primes and the cuban primes as special cases
      Williams primes, of the form (b − 1)·bn − 1


      = Mills primes

      =
      Of the form ⌊θ3n⌋, where θ is Mills' constant. This form is prime for all positive integers n.
      2, 11, 1361, 2521008887, 16022236204009818131831320183 (OEIS: A051254)


      = Minimal primes

      =
      Primes for which there is no shorter sub-sequence of the decimal digits that form a prime. There are exactly 26 minimal primes:
      2, 3, 5, 7, 11, 19, 41, 61, 89, 409, 449, 499, 881, 991, 6469, 6949, 9001, 9049, 9649, 9949, 60649, 666649, 946669, 60000049, 66000049, 66600049 (OEIS: A071062)


      = Newman–Shanks–Williams primes

      =
      Newman–Shanks–Williams numbers that are prime.
      7, 41, 239, 9369319, 63018038201, 489133282872437279, 19175002942688032928599 (OEIS: A088165)


      = Non-generous primes

      =
      Primes p for which the least positive primitive root is not a primitive root of p2. Three such primes are known; it is not known whether there are more.
      2, 40487, 6692367337 (OEIS: A055578)


      = Palindromic primes

      =
      Primes that remain the same when their decimal digits are read backwards.
      2, 3, 5, 7, 11, 101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, 797, 919, 929, 10301, 10501, 10601, 11311, 11411, 12421, 12721, 12821, 13331, 13831, 13931, 14341, 14741 (OEIS: A002385)


      = Palindromic wing primes

      =
      Primes of the form






      a


      (



      10

      m


      −
      1


      )



      9


      ±
      b
      ×

      10



      m
      −
      1

      2





      {\displaystyle {\frac {a{\big (}10^{m}-1{\big )}}{9}}\pm b\times 10^{\frac {m-1}{2}}}

      with



      0
      ≤
      a
      ±
      b
      <
      10


      {\displaystyle 0\leq a\pm b<10}

      . This means all digits except the middle digit are equal.
      101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, 797, 919, 929, 11311, 11411, 33533, 77377, 77477, 77977, 1114111, 1117111, 3331333, 3337333, 7772777, 7774777, 7778777, 111181111, 111191111, 777767777, 77777677777, 99999199999 (OEIS: A077798)


      = Partition primes

      =
      Partition function values that are prime.
      2, 3, 5, 7, 11, 101, 17977, 10619863, 6620830889, 80630964769, 228204732751, 1171432692373, 1398341745571, 10963707205259, 15285151248481, 10657331232548839, 790738119649411319, 18987964267331664557 (OEIS: A049575)


      = Pell primes

      =
      Primes in the Pell number sequence P0 = 0, P1 = 1,
      Pn = 2Pn−1 + Pn−2.
      2, 5, 29, 5741, 33461, 44560482149, 1746860020068409, 68480406462161287469, 13558774610046711780701, 4125636888562548868221559797461449 (OEIS: A086383)


      = Permutable primes

      =
      Any permutation of the decimal digits is a prime.
      2, 3, 5, 7, 11, 13, 17, 31, 37, 71, 73, 79, 97, 113, 131, 199, 311, 337, 373, 733, 919, 991, 1111111111111111111, 11111111111111111111111 (OEIS: A003459)


      = Perrin primes

      =
      Primes in the Perrin number sequence P(0) = 3, P(1) = 0, P(2) = 2,
      P(n) = P(n−2) + P(n−3).
      2, 3, 5, 7, 17, 29, 277, 367, 853, 14197, 43721, 1442968193, 792606555396977, 187278659180417234321, 66241160488780141071579864797 (OEIS: A074788)


      = Pierpont primes

      =
      Of the form 2u3v + 1 for some integers u,v ≥ 0.
      These are also class 1- primes.
      2, 3, 5, 7, 13, 17, 19, 37, 73, 97, 109, 163, 193, 257, 433, 487, 577, 769, 1153, 1297, 1459, 2593, 2917, 3457, 3889, 10369, 12289, 17497, 18433, 39367, 52489, 65537, 139969, 147457 (OEIS: A005109)


      = Pillai primes

      =
      Primes p for which there exist n > 0 such that p divides n! + 1 and n does not divide p − 1.
      23, 29, 59, 61, 67, 71, 79, 83, 109, 137, 139, 149, 193, 227, 233, 239, 251, 257, 269, 271, 277, 293, 307, 311, 317, 359, 379, 383, 389, 397, 401, 419, 431, 449, 461, 463, 467, 479, 499 (OEIS: A063980)


      = Primes of the form n4 + 1

      =
      Of the form n4 + 1.
      2, 17, 257, 1297, 65537, 160001, 331777, 614657, 1336337, 4477457, 5308417, 8503057, 9834497, 29986577, 40960001, 45212177, 59969537, 65610001, 126247697, 193877777, 303595777, 384160001, 406586897, 562448657, 655360001 (OEIS: A037896)


      = Primeval primes

      =
      Primes for which there are more prime permutations of some or all the decimal digits than for any smaller number.
      2, 13, 37, 107, 113, 137, 1013, 1237, 1367, 10079 (OEIS: A119535)


      = Primorial primes

      =
      Of the form pn# ± 1.
      3, 5, 7, 29, 31, 211, 2309, 2311, 30029, 200560490131, 304250263527209, 23768741896345550770650537601358309 (union of OEIS: A057705 and OEIS: A018239)


      = Proth primes

      =
      Of the form k×2n + 1, with odd k and k < 2n.
      3, 5, 13, 17, 41, 97, 113, 193, 241, 257, 353, 449, 577, 641, 673, 769, 929, 1153, 1217, 1409, 1601, 2113, 2689, 2753, 3137, 3329, 3457, 4481, 4993, 6529, 7297, 7681, 7937, 9473, 9601, 9857 (OEIS: A080076)


      = Pythagorean primes

      =
      Of the form 4n + 1.
      5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101, 109, 113, 137, 149, 157, 173, 181, 193, 197, 229, 233, 241, 257, 269, 277, 281, 293, 313, 317, 337, 349, 353, 373, 389, 397, 401, 409, 421, 433, 449 (OEIS: A002144)


      = Prime quadruplets

      =

      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)


      = Quartan primes

      =
      Of the form x4 + y4, where x,y > 0.
      2, 17, 97, 257, 337, 641, 881 (OEIS: A002645)


      = Ramanujan primes

      =
      Integers Rn that are the smallest to give at least n primes from x/2 to x for all x ≥ Rn (all such integers are primes).
      2, 11, 17, 29, 41, 47, 59, 67, 71, 97, 101, 107, 127, 149, 151, 167, 179, 181, 227, 229, 233, 239, 241, 263, 269, 281, 307, 311, 347, 349, 367, 373, 401, 409, 419, 431, 433, 439, 461, 487, 491 (OEIS: A104272)


      = Regular primes

      =
      Primes p that do not divide the class number of the p-th cyclotomic field.
      3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 43, 47, 53, 61, 71, 73, 79, 83, 89, 97, 107, 109, 113, 127, 137, 139, 151, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 239, 241, 251, 269, 277, 281 (OEIS: A007703)


      = Repunit primes

      =
      Primes containing only the decimal digit 1.
      11, 1111111111111111111 (19 digits), 11111111111111111111111 (23 digits) (OEIS: A004022)
      The next have 317, 1031, 49081, 86453, 109297, 270343 digits (OEIS: A004023)


      = Residue classes of primes

      =
      Of the form an + d for fixed integers a and d. Also called primes congruent to d modulo a.
      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 digit d.
      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)


      = Safe primes

      =
      Where p and (p−1) / 2 are both prime.
      5, 7, 11, 23, 47, 59, 83, 107, 167, 179, 227, 263, 347, 359, 383, 467, 479, 503, 563, 587, 719, 839, 863, 887, 983, 1019, 1187, 1283, 1307, 1319, 1367, 1439, 1487, 1523, 1619, 1823, 1907 (OEIS: A005385)


      = Self primes in base 10

      =
      Primes that cannot be generated by any integer added to the sum of its decimal digits.
      3, 5, 7, 31, 53, 97, 211, 233, 277, 367, 389, 457, 479, 547, 569, 613, 659, 727, 839, 883, 929, 1021, 1087, 1109, 1223, 1289, 1447, 1559, 1627, 1693, 1783, 1873 (OEIS: A006378)


      = Sexy primes

      =
      Where (p, p + 6) are both prime.
      (5, 11), (7, 13), (11, 17), (13, 19), (17, 23), (23, 29), (31, 37), (37, 43), (41, 47), (47, 53), (53, 59), (61, 67), (67, 73), (73, 79), (83, 89), (97, 103), (101, 107), (103, 109), (107, 113), (131, 137), (151, 157), (157, 163), (167, 173), (173, 179), (191, 197), (193, 199) (OEIS: A023201, OEIS: A046117)


      = Smarandache–Wellin primes

      =
      Primes that are the concatenation of the first n primes written in decimal.
      2, 23, 2357 (OEIS: A069151)
      The fourth Smarandache-Wellin prime is the 355-digit concatenation of the first 128 primes that end with 719.


      = Solinas primes

      =
      Of the form 2a ± 2b ± 1, where 0 < b < a.
      3, 5, 7, 11, 13 (OEIS: A165255)


      = Sophie Germain primes

      =
      Where p and 2p + 1 are both prime. A Sophie Germain prime has a corresponding safe 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)


      = Stern primes

      =
      Primes that are not the sum of a smaller prime and twice the square of a nonzero integer.
      2, 3, 17, 137, 227, 977, 1187, 1493 (OEIS: A042978)
      As of 2011, these are the only known Stern primes, and possibly the only existing.


      = Super-primes

      =
      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)


      = Supersingular primes

      =
      There are exactly fifteen supersingular primes:
      2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59, 71 (OEIS: A002267)


      = Thabit primes

      =
      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)


      = Prime triplets

      =

      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)


      = Truncatable prime

      =


      Left-truncatable


      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)


      Right-truncatable


      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)


      Two-sided


      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)


      = Twin primes

      =

      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)


      = Unique primes

      =
      The list of primes p for which the period length of the decimal expansion of 1/p is unique (no other prime gives the same period).
      3, 11, 37, 101, 9091, 9901, 333667, 909091, 99990001, 999999000001, 9999999900000001, 909090909090909091, 1111111111111111111, 11111111111111111111111, 900900900900990990990991 (OEIS: A040017)


      = Wagstaff primes

      =
      Of the form (2n + 1) / 3.
      3, 11, 43, 683, 2731, 43691, 174763, 2796203, 715827883, 2932031007403, 768614336404564651, 201487636602438195784363, 845100400152152934331135470251, 56713727820156410577229101238628035243 (OEIS: A000979)
      Values of n:
      3, 5, 7, 11, 13, 17, 19, 23, 31, 43, 61, 79, 101, 127, 167, 191, 199, 313, 347, 701, 1709, 2617, 3539, 5807, 10501, 10691, 11279, 12391, 14479, 42737, 83339, 95369, 117239, 127031, 138937, 141079, 267017, 269987, 374321 (OEIS: A000978)


      = Wall–Sun–Sun primes

      =
      A prime p > 5, if p2 divides the Fibonacci number




      F

      p
      −

      (


      p
      5


      )





      {\displaystyle F_{p-\left({\frac {p}{5}}\right)}}

      , where the Legendre symbol




      (


      p
      5


      )



      {\displaystyle \left({\frac {p}{5}}\right)}

      is defined as





      (


      p
      5


      )

      =


      {



      1




      if



      p
      ≡
      ±
      1


      (
      mod

      5
      )





      −
      1




      if



      p
      ≡
      ±
      2


      (
      mod

      5
      )

      .








      {\displaystyle \left({\frac {p}{5}}\right)={\begin{cases}1&{\textrm {if}}\;p\equiv \pm 1{\pmod {5}}\\-1&{\textrm {if}}\;p\equiv \pm 2{\pmod {5}}.\end{cases}}}


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


      = Wieferich primes

      =
      Primes p such that ap − 1 ≡ 1 (mod p2) for fixed integer a > 1.
      2p − 1 ≡ 1 (mod p2): 1093, 3511 (OEIS: A001220)
      3p − 1 ≡ 1 (mod p2): 11, 1006003 (OEIS: A014127)
      4p − 1 ≡ 1 (mod p2): 1093, 3511
      5p − 1 ≡ 1 (mod p2): 2, 20771, 40487, 53471161, 1645333507, 6692367337, 188748146801 (OEIS: A123692)
      6p − 1 ≡ 1 (mod p2): 66161, 534851, 3152573 (OEIS: A212583)
      7p − 1 ≡ 1 (mod p2): 5, 491531 (OEIS: A123693)
      8p − 1 ≡ 1 (mod p2): 3, 1093, 3511
      9p − 1 ≡ 1 (mod p2): 2, 11, 1006003
      10p − 1 ≡ 1 (mod p2): 3, 487, 56598313 (OEIS: A045616)
      11p − 1 ≡ 1 (mod p2): 71
      12p − 1 ≡ 1 (mod p2): 2693, 123653 (OEIS: A111027)
      13p − 1 ≡ 1 (mod p2): 2, 863, 1747591 (OEIS: A128667)
      14p − 1 ≡ 1 (mod p2): 29, 353, 7596952219 (OEIS: A234810)
      15p − 1 ≡ 1 (mod p2): 29131, 119327070011 (OEIS: A242741)
      16p − 1 ≡ 1 (mod p2): 1093, 3511
      17p − 1 ≡ 1 (mod p2): 2, 3, 46021, 48947 (OEIS: A128668)
      18p − 1 ≡ 1 (mod p2): 5, 7, 37, 331, 33923, 1284043 (OEIS: A244260)
      19p − 1 ≡ 1 (mod p2): 3, 7, 13, 43, 137, 63061489 (OEIS: A090968)
      20p − 1 ≡ 1 (mod p2): 281, 46457, 9377747, 122959073 (OEIS: A242982)
      21p − 1 ≡ 1 (mod p2): 2
      22p − 1 ≡ 1 (mod p2): 13, 673, 1595813, 492366587, 9809862296159 (OEIS: A298951)
      23p − 1 ≡ 1 (mod p2): 13, 2481757, 13703077, 15546404183, 2549536629329 (OEIS: A128669)
      24p − 1 ≡ 1 (mod p2): 5, 25633
      25p − 1 ≡ 1 (mod p2): 2, 20771, 40487, 53471161, 1645333507, 6692367337, 188748146801
      As of 2018, these are all known Wieferich primes with a ≤ 25.


      = Wilson primes

      =
      Primes p for which p2 divides (p−1)! + 1.
      5, 13, 563 (OEIS: A007540)
      As of 2018, these are the only known Wilson primes.


      = Wolstenholme primes

      =
      Primes p for which the binomial coefficient






      (



      2
      p
      −
      1


      p
      −
      1



      )



      ≡
      1


      (
      mod


      p

      4


      )

      .


      {\displaystyle {{2p-1} \choose {p-1}}\equiv 1{\pmod {p^{4}}}.}


      16843, 2124679 (OEIS: A088164)
      As of 2018, these are the only known Wolstenholme primes.


      = Woodall primes

      =
      Of the form n×2n − 1.
      7, 23, 383, 32212254719, 2833419889721787128217599, 195845982777569926302400511, 4776913109852041418248056622882488319 (OEIS: A050918)


      See also




      References




      External links


      [1] All prime numbers from 31 to 6,469,693,189 for free download.
      Lists of Primes at the Prime Pages.
      The Nth Prime Page Nth prime through n=10^12, pi(x) through x=3*10^13, Random primes in same range.
      Interface to a list of the first 98 million primes (primes less than 2,000,000,000)
      Weisstein, Eric W. "Prime Number Sequences". MathWorld.
      Selected prime related sequences in OEIS.
      Fischer, R. Thema: Fermatquotient B^(P−1) == 1 (mod P^2) (in German) (Lists Wieferich primes in all bases up to 1052)
      Padilla, Tony (7 February 2013). "New Largest Known Prime Number". Numberphile. Brady Haran. Archived from the original on 2 November 2021.

    Kata Kunci Pencarian: