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 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 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 is neither prime nor composite.

The first 1,000 prime numbers

The following table lists the first 1,000 primes, with 20 columns of consecutive primes in each of the 50 rows.

{| class="wikitable" style="font-size:90%"

!

! 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 || 13 ||17 || 19 || 23 || 29 || 31 || 37|| 41 || 43 || 47 || 53 || 59 || 61 || 67 || 71

|- style="text-align: center;"

! 21–40

| 73 || 79 || 83 || 89 || 97 || 101|| 103 || 107 || 109 || 113|| 127 || 131 || 137 || 139 || 149 || 151 || 157 || 163 || 167 || 173

|- style="text-align: center;"

! 41–60

| 179 || 181 || 191 || 193 || 197 || 199 || 211 || 223 || 227 || 229|| 233 || 239 || 241 || 251 || 257 || 263 || 269 || 271 || 277 || 281

|- style="text-align: center;"

! 61–80

| 283 || 293 || 307 || 311 || 313 || 317 || 331 || 337 || 347 || 349|| 353 || 359 || 367 || 373 || 379 || 383 || 389 || 397 || 401 || 409

|- style="text-align: center;"

! 81–100

| 419 || 421 || 431 || 433 || 439 || 443 || 449 || 457 || 461 || 463|| 467 || 479 || 487 || 491 || 499 || 503 || 509 || 521 || 523 || 541

|- style="text-align: center;"

! 101–120

| 547 || 557 || 563 || 569 || 571 || 577 || 587 || 593 || 599 || 601|| 607 || 613 || 617 || 619 || 631 || 641 || 643 || 647

|| 653 || 659

|- style="text-align: center;"

! 121–140

| 661 || 673 || 677 || 683 || 691 || 701 || 709 || 719 || 727 || 733|| 739 || 743 || 751 || 757 || 761 || 769 || 773 || 787 || 797 || 809

|- style="text-align: center;"

! 141–160

| 811 || 821 || 823 || 827 || 829 || 839 || 853 || 857 || 859 || 863|| 877 || 881 || 883 || 887 || 907 || 911 || 919 || 929 || 937 || 941

|- style="text-align: center;"

! 161–180

| 947 || 953 || 967 || 971 || 977 || 983 || 991 || 997 || 1009 || 1013|| 1019 || 1021 || 1031 || 1033 || 1039 || 1049 || 1051 || 1061 || 1063 || 1069

|- style="text-align: center;"

! 181–200

| 1087 || 1091 || 1093 || 1097 || 1103 || 1109 || 1117 || 1123 || 1129 || 1151|| 1153 || 1163 || 1171 || 1181 || 1187 || 1193 || 1201 || 1213 || 1217 || 1223

|- style="text-align: center;"

! 201–220

| 1229 || 1231 || 1237 || 1249 || 1259 || 1277 || 1279 || 1283 || 1289 || 1291|| 1297 || 1301 || 1303 || 1307 || 1319 || 1321 || 1327 || 1361 || 1367 || 1373

|- style="text-align: center;"

! 221–240

| 1381 || 1399 || 1409 || 1423 || 1427 || 1429 || 1433 || 1439 || 1447 || 1451|| 1453 || 1459 || 1471 || 1481 || 1483 || 1487 || 1489 || 1493 || 1499 || 1511

|- style="text-align: center;"

! 241–260

| 1523 || 1531 || 1543 || 1549 || 1553 || 1559 || 1567 || 1571 || 1579 || 1583|| 1597 || 1601 || 1607 || 1609 || 1613 || 1619 || 1621 || 1627 || 1637 || 1657

|- style="text-align: center;"

! 261–280

| 1663 || 1667 || 1669 || 1693 || 1697 || 1699 || 1709 || 1721 || 1723 || 1733|| 1741 || 1747 || 1753 || 1759 || 1777 || 1783 || 1787 || 1789 || 1801 || 1811

|- style="text-align: center;"

! 281–300

| 1823 || 1831 || 1847 || 1861 || 1867 || 1871 || 1873 || 1877 || 1879 || 1889|| 1901 || 1907 || 1913 || 1931 || 1933 || 1949 || 1951 || 1973 || 1979 || 1987

|- style="text-align: center;"

! 301–320

| 1993 || 1997 || 1999 || 2003 || 2011 || 2017 || 2027 || 2029 || 2039 || 2053|| 2063 || 2069 || 2081 || 2083 || 2087 || 2089 || 2099 || 2111 || 2113 || 2129

|- style="text-align: center;"

! 321–340

| 2131 || 2137 || 2141 || 2143 || 2153 || 2161 || 2179 || 2203 || 2207 || 2213|| 2221 || 2237 || 2239 || 2243 || 2251 || 2267 || 2269 || 2273 || 2281 || 2287

|- style="text-align: center;"

! 341–360

| 2293 || 2297 || 2309 || 2311 || 2333 || 2339 || 2341 || 2347 || 2351 || 2357|| 2371 || 2377 || 2381 || 2383 || 2389 || 2393 || 2399 || 2411 || 2417 || 2423

|- style="text-align: center;"

! 361–380

| 2437 || 2441 || 2447 || 2459 || 2467 || 2473 || 2477 || 2503 || 2521 || 2531|| 2539 || 2543 || 2549 || 2551 || 2557 || 2579 || 2591 || 2593 || 2609 || 2617

|- style="text-align: center;"

! 381–400

| 2621 || 2633 || 2647 || 2657 || 2659 || 2663 || 2671 || 2677 || 2683 || 2687|| 2689 || 2693 || 2699 || 2707 || 2711 || 2713 || 2719 || 2729 || 2731 || 2741

|- style="text-align: center;"

! 401–420

| 2749 || 2753 || 2767 || 2777 || 2789 || 2791 || 2797 || 2801 || 2803 || 2819|| 2833 || 2837 || 2843 || 2851 || 2857 || 2861 || 2879 || 2887 || 2897 || 2903

|- style="text-align: center;"

! 421–440

| 2909 || 2917 || 2927 || 2939 || 2953 || 2957 || 2963 || 2969 || 2971 || 2999|| 3001 || 3011 || 3019 || 3023 || 3037 || 3041 || 3049 || 3061 || 3067 || 3079

|- style="text-align: center;"

! 441–460

| 3083 || 3089 || 3109 || 3119 || 3121 || 3137 || 3163 || 3167 || 3169 || 3181|| 3187 || 3191 || 3203 || 3209 || 3217 || 3221 || 3229 || 3251 || 3253 || 3257

|- style="text-align: center;"

! 461–480

| 3259 || 3271 || 3299 || 3301 || 3307 || 3313 || 3319 || 3323 || 3329 || 3331|| 3343 || 3347 || 3359 || 3361 || 3371 || 3373 || 3389 || 3391 || 3407 || 3413

|- style="text-align: center;"

! 481–500

| 3433 || 3449 || 3457|| 3461 || 3463 || 3467 || 3469 || 3491 || 3499 || 3511|| 3517 || 3527 || 3529 || 3533 || 3539 || 3541 || 3547 || 3557 || 3559 || 3571

|- style="text-align: center;"

! 501–520

| 3581 || 3583 || 3593 || 3607 || 3613 || 3617 || 3623 || 3631 || 3637 || 3643 || 3659 || 3671 || 3673 || 3677 || 3691 || 3697 || 3701 || 3709 || 3719 || 3727

|- style="text-align: center;"

! 521–540

| 3733 || 3739 || 3761 || 3767 || 3769 || 3779 || 3793 || 3797 || 3803 || 3821 || 3823 || 3833 || 3847 || 3851 || 3853 || 3863 || 3877 || 3881 || 3889 || 3907

|- style="text-align: center;"

! 541–560

| 3911 || 3917 || 3919 || 3923 || 3929 || 3931 || 3943 || 3947 || 3967 || 3989 || 4001 || 4003 || 4007 || 4013 || 4019 || 4021 || 4027 || 4049 || 4051 || 4057

|- style="text-align: center;"

! 561–580

| 4073 || 4079 || 4091 || 4093 || 4099 || 4111 || 4127 || 4129 || 4133 || 4139 || 4153 || 4157 || 4159 || 4177 || 4201 || 4211 || 4217 || 4219 || 4229 || 4231

|- style="text-align: center;"

! 581–600

| 4241 || 4243 || 4253 || 4259 || 4261 || 4271 || 4273 || 4283 || 4289 || 4297 || 4327 || 4337 || 4339 || 4349 || 4357 || 4363 || 4373 || 4391 || 4397 || 4409

|- style="text-align: center;"

! 601–620

| 4421 || 4423 || 4441 || 4447 || 4451 || 4457 || 4463 || 4481 || 4483 || 4493 || 4507 || 4513 || 4517 || 4519 || 4523 || 4547 || 4549 || 4561 || 4567 || 4583

|- style="text-align: center;"

! 621–640

| 4591 || 4597 || 4603 || 4621 || 4637 || 4639 || 4643 || 4649 || 4651 || 4657 || 4663 || 4673 || 4679 || 4691 || 4703 || 4721 || 4723 || 4729 || 4733 || 4751

|- style="text-align: center;"

! 641–660

| 4759 || 4783 || 4787 || 4789 || 4793 || 4799 || 4801 || 4813 || 4817 || 4831 || 4861 || 4871 || 4877 || 4889 || 4903 || 4909 || 4919 || 4931 || 4933 || 4937

|- style="text-align: center;"

! 661–680

| 4943 || 4951 || 4957 || 4967 || 4969 || 4973 || 4987 || 4993 || 4999 || 5003 || 5009 || 5011 || 5021 || 5023 || 5039 || 5051 || 5059 || 5077 || 5081 || 5087

|- style="text-align: center;"

! 681–700

| 5099 || 5101 || 5107 || 5113 || 5119 || 5147 || 5153 || 5167 || 5171 || 5179 || 5189 || 5197 || 5209 || 5227 || 5231 || 5233 || 5237 || 5261 || 5273 || 5279

|- style="text-align: center;"

! 701–720

| 5281 || 5297 || 5303 || 5309 || 5323 || 5333 || 5347 || 5351 || 5381 || 5387 || 5393 || 5399 || 5407 || 5413 || 5417 || 5419 || 5431 || 5437 || 5441 || 5443

|- style="text-align: center;"

! 721–740

| 5449 || 5471 || 5477 || 5479 || 5483 || 5501 || 5503 || 5507 || 5519 || 5521 || 5527 || 5531 || 5557 || 5563 || 5569 || 5573 || 5581 || 5591 || 5623 || 5639

|- style="text-align: center;"

! 741–760

| 5641 || 5647 || 5651 || 5653 || 5657 || 5659 || 5669 || 5683 || 5689 || 5693 || 5701 || 5711 || 5717 || 5737 || 5741 || 5743 || 5749 || 5779 || 5783 || 5791

|- style="text-align: center;"

! 761–780

| 5801 || 5807 || 5813 || 5821 || 5827 || 5839 || 5843 || 5849 || 5851 || 5857 || 5861 || 5867 || 5869 || 5879 || 5881 || 5897 || 5903 || 5923 || 5927 || 5939

|- style="text-align: center;"

! 781–800

| 5953 || 5981 || 5987 || 6007 || 6011 || 6029 || 6037 || 6043 || 6047 || 6053 || 6067 || 6073 || 6079 || 6089 || 6091 || 6101 || 6113 || 6121 || 6131 || 6133

|- style="text-align: center;"

! 801–820

| 6143 || 6151 || 6163 || 6173 || 6197 || 6199 || 6203 || 6211 || 6217 || 6221 || 6229 || 6247 || 6257 || 6263 || 6269 || 6271 || 6277 || 6287 || 6299 || 6301

|- style="text-align: center;"

! 821–840

| 6311 || 6317 || 6323 || 6329 || 6337 || 6343 || 6353 || 6359 || 6361 || 6367 || 6373 || 6379 || 6389 || 6397 || 6421 || 6427 || 6449 || 6451 || 6469 || 6473

|- style="text-align: center;"

! 841–860

| 6481 || 6491 || 6521 || 6529 || 6547 || 6551 || 6553 || 6563 || 6569 || 6571 || 6577 || 6581 || 6599 || 6607 || 6619 || 6637 || 6653 || 6659 || 6661 || 6673

|- style="text-align: center;"

! 861–880

| 6679 || 6689 || 6691 || 6701 || 6703 || 6709 || 6719 || 6733 || 6737 || 6761 || 6763 || 6779 || 6781 || 6791 || 6793 || 6803 || 6823 || 6827 || 6829 || 6833

|- style="text-align: center;"

! 881–900

| 6841 || 6857 || 6863 || 6869 || 6871 || 6883 || 6899 || 6907 || 6911 || 6917 || 6947 || 6949 || 6959 || 6961 || 6967 || 6971 || 6977 || 6983 || 6991 || 6997

|- style="text-align: center;"

! 901–920

| 7001 || 7013 || 7019 || 7027 || 7039 || 7043 || 7057 || 7069 || 7079 || 7103 || 7109 || 7121 || 7127 || 7129 || 7151 || 7159 || 7177 || 7187 || 7193 || 7207

|- style="text-align: center;"

! 921–940

| 7211 || 7213 || 7219 || 7229 || 7237 || 7243 || 7247 || 7253 || 7283 || 7297 || 7307 || 7309 || 7321 || 7331 || 7333 || 7349 || 7351 || 7369 || 7393 || 7411

|- style="text-align: center;"

! 941–960

| 7417 || 7433 || 7451 || 7457 || 7459 || 7477 || 7481 || 7487 || 7489 || 7499 || 7507 || 7517 || 7523 || 7529 || 7537 || 7541 || 7547 || 7549 || 7559 || 7561

|- style="text-align: center;"

! 961–980

| 7573 || 7577 || 7583 || 7589 || 7591 || 7603 || 7607 || 7621 || 7639 || 7643 || 7649 || 7669 || 7673 || 7681 || 7687 || 7691 || 7699 || 7703 || 7717 || 7723

|- style="text-align: center;"

!

| 7727 || 7741 || 7753 || 7757 || 7759 || 7789 || 7793 || 7817 || 7823 || 7829 || 7841 || 7853 || 7867 || 7873 || 7877 || 7879 || 7883 || 7901 || 7907 || 7919

|}

.

The Goldbach conjecture verification project reports that it has computed all primes smaller than 4×10. That means 95,676,260,903,887,607 primes (nearly 10), 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) smaller than 10. A different computation found that there are 18,435,599,767,349,200,867,866 primes (roughly 2) smaller than 10, 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

Balanced primes are primes with equal-sized prime gaps before and after them, making them the arithmetic mean of their next larger and next smaller prime.

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

Bell primes

Bell primes are primes that are also the number of partitions of some finite set.

2, 5, 877, 27644437, 35742549198872617291353508656626642567, 359334085968622831041960188598043661065388726959079837.

The next term has 6,539 digits. ()

Chen primes

Chen primes are primes p such that 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 ()

Circular primes

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 ()

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 ()

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, ... ()

All 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

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) (, )

Cuban primes

Cuban primes are primes <math>p</math> of the form <math>p = k^3 - (k - 1)^3,</math> where <math>k</math> 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 ()

The term is also used to refer to primes <math>p</math> of the form <math>p = (k^3 - (k - 2)^3)/2,</math> where <math>k</math> 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 ()

Cullen primes

Cullen primes are primes p of the form p=k2 + 1, for some natural number k.

3, 393050634124102232869567034555427371542904833 ()

Delicate primes

Delicate primes are those primes that always become a composite number when any of their base 10 digit is changed.

294001, 505447, 584141, 604171, 971767, 1062599, 1282529, 1524181, 2017963, 2474431, 2690201, 3085553, 3326489, 4393139 ()

Dihedral primes

Dihedral primes are primes that satisfy 180° rotational symmetry and mirror symmetry on a seven-segment display.

2, 5, 11, 101, 181, 1181, 1811, 18181, 108881, 110881, 118081, 120121,

121021, 121151, 150151, 151051, 151121, 180181, 180811, 181081 ()

Real Eisenstein primes

Real Eisenstein primes are real Eisenstein integers that are irreducible. Equivalently, they are primes of the form 3k&nbsp;−&nbsp;1, for a positive integer k.

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 ()

Emirps

Emirps are 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 ()

Euclid primes

Euclid primes are primes p such that p−1 is a primorial.

3, 7, 31, 211, 2311, 200560490131 ()

Euler irregular primes

Euler irregular primes are primes <math>p</math> that divide an Euler number <math>E_{2n},</math> for some <math>0\leq 2n\leq p-3.</math>

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 ()

Euler (p, p − 3) irregular primes

Euler (p, p - 3) irregular primes are primes p that divide the (p + 3)rd Euler number.

149, 241, 2946901 ()

Factorial primes

Factorial primes are of the form n! ± 1.

2, 3, 5, 7, 23, 719, 5039, 39916801, 479001599, 87178291199, 10888869450418352160768000001, 265252859812191058636308479999999, 263130836933693530167218012159999999, 8683317618811886495518194401279999999 ()

Fermat primes

Fermat primes are primes p of the form p&nbsp;=&nbsp;2&nbsp;+&nbsp;1, for a non-negative integer k. only five Fermat primes have been discovered.

3, 5, 17, 257, 65537 ()

Generalized Fermat primes

Generalized Fermat primes are primes p of the form p&nbsp;=&nbsp;a&nbsp;+&nbsp;1, for a non-negative integer k and even natural number a.

{|class="wikitable"

!<math>a</math>

!Generalized Fermat primes with base a

|-

|2

|3, 5, 17, 257, 65537, ... ()

|-

|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

Fibonacci primes are primes that appear in the Fibonacci sequence.

2, 3, 5, 13, 89, 233, 1597, 28657, 514229, 433494437, 2971215073, 99194853094755497, 1066340417491710595814572169, 19134702400093278081449423917 ()

Fortunate primes

Fortunate primes are primes that are also Fortunate numbers. There are no known composite Fortunate numbers.

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 ()

Gaussian primes

Gaussian primes are primes p of the form p&nbsp;=&nbsp;4k&nbsp;+&nbsp;3, for a non-negative integer k.

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 ()

Good primes

Good primes are primes p satisfying ab&nbsp;<&nbsp;p, for all primes a and b such that a,b&nbsp;<&nbsp;p

5, 11, 17, 29, 37, 41, 53, 59, 67, 71, 97, 101, 127, 149, 179, 191, 223, 227, 251, 257, 269, 307 ()

Happy primes

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 ()

Harmonic primes

Harmonic primes are primes p for which there are no solutions to H&nbsp;≡&nbsp;0&nbsp;(mod&nbsp;p) and H&nbsp;≡&nbsp;−ω&nbsp;(mod&nbsp;p), for 1&nbsp;≤&nbsp;k&nbsp;≤&nbsp;p−2, where H denotes the k-th harmonic number and ω 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 ()

Higgs primes

Higgs primes are primes p for which p − 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 ()

Highly cototient primes

Highly cototient primes are 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 ()

Home primes

For , write the prime factorization of in base 10 and concatenate the factors; iterate until a prime is reached.

For a non-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 ()

Irregular primes

Irregular primes are 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 ()

(p, p − 3) irregular primes

The (p, p - 3) irregular primes are primes p such that (p, p − 3) is an irregular pair.

16843, 2124679 ()

(p, p − 5) irregular primes

The (p, p - 5) irregular primes are primes p such that (p, p − 5) is an irregular pair.

37

(p, p − 9) irregular primes

The (p, p - 9) irregular primes are primes p such that (p, p − 9) is an irregular pair. 3, 7, 11, 29, 47, 199, 521, 2207, 3571, 9349, 3010349, 54018521, 370248451, 6643838879, 119218851371, 5600748293801, 688846502588399, 32361122672259149 ()

Lucky primes

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 ()

Mersenne primes

Mersenne primes are primes p of the form p = 2 − 1, for some non-negative integer k.

3, 7, 31, 127, 8191, 131071, 524287, 2147483647, 2305843009213693951, 618970019642690137449562111, 162259276829213363391578010288127, 170141183460469231731687303715884105727 ()

, there are 52 known Mersenne primes. The 13th, 14th, and 52nd have respectively 157, 183, and 41,024,320 digits.

Mersenne divisors

Mersenne divisors are primes that divide 2 − 1, for some prime k. Every Mersenne prime p is also a Mersenne divisor, with k = 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 ()

Mersenne prime exponents

Primes p such that 2 − 1 is prime.

<!--1st-10th exponent-->

2, 3, 5, 7, 13, 17, 19, 31, 61, 89,

<!--11th-20th exponent-->

107, 127, 521, 607, 1279, 2203, 2281, 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, 43112609, 57885161, 74207281, 77232917 ()

, two more are known to be in the sequence, but it is not known whether they are the next:<br />

<!--50th-51st known exponent-->82589933, 136279841

Double Mersenne primes

A subset of Mersenne primes of the form 2&nbsp;−&nbsp;1 for prime p.

7, 127, 2147483647, 170141183460469231731687303715884105727 (primes in )

Generalized repunit primes

Of the form (a&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. For other small a, they are given below:

a = 3: 13, 1093, 797161, 3754733257489862401973357979128773, 6957596529882152968992225251835887181478451547013 ()

a = 4: 5 (the only prime for a = 4)

a = 5: 31, 19531, 12207031, 305175781, 177635683940025046467781066894531, 14693679385278593849609206715278070972733319459651094018859396328480215743184089660644531 ()

a = 6: 7, 43, 55987, 7369130657357778596659, 3546245297457217493590449191748546458005595187661976371 ()

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 , including the Mersenne primes and the cuban primes as special cases
• Williams primes, of the form

Mills primes

Of the form ⌊θ⌋, where θ is Mills' constant. This form is prime for all positive integers n.

2, 11, 1361, 2521008887, 16022236204009818131831320183 ()

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 ()

Newman–Shanks–Williams primes

Newman–Shanks–Williams numbers that are prime.

7, 41, 239, 9369319, 63018038201, 489133282872437279, 19175002942688032928599 ()

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 ()

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 ()

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>. 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 ()

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 ()

Pell primes

Primes in the Pell number sequence P&nbsp;=&nbsp;0, P&nbsp;=&nbsp;1,

P&nbsp;=&nbsp;2P&nbsp;+&nbsp;P.

2, 5, 29, 5741, 33461, 44560482149, 1746860020068409, 68480406462161287469, 13558774610046711780701, 4125636888562548868221559797461449 ()

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 ()

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, 29, 277, 367, 853, 14197, 43721, 1442968193, 792606555396977, 187278659180417234321, 66241160488780141071579864797 ()

Pierpont primes

Of the form 23&nbsp;+&nbsp;1 for some integers u,v&nbsp;≥&nbsp;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 ()

Pillai primes

Primes p for which there exist n&nbsp;>&nbsp;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 ()

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 ()

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 ()

Primorial primes

Of the form p#&nbsp;± 1.

3, 5, 7, 29, 31, 211, 2309, 2311, 30029, 200560490131, 304250263527209, 23768741896345550770650537601358309 (union of and

• 2&nbsp;−&nbsp;59, the largest prime that fits into 64 bits of memory.

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 ()

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 ()

, 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 ()

Supersingular primes

There are exactly fifteen supersingular primes:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59, 71 ()

Thabit primes

Of the form 3×2 − 1.

2, 5, 11, 23, 47, 191, 383, 6143, 786431, 51539607551, 824633720831, 26388279066623, 108086391056891903, 55340232221128654847, 226673591177742970257407 ()

The primes of the form 3×2 + 1 are related.

7, 13, 97, 193, 769, 12289, 786433, 3221225473, 206158430209, 6597069766657 ()

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) (, , )

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 ()

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 ()

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 ()

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) (, )

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 ()

Wagstaff primes

Of the form (2 + 1) / 3.

3, 11, 43, 683, 2731, 43691, 174763, 2796203, 715827883, 2932031007403, 768614336404564651, 201487636602438195784363, 845100400152152934331135470251, 56713727820156410577229101238628035243 ()

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 ()

Wall–Sun–Sun primes

A prime p&nbsp;>&nbsp;5, if p divides the Fibonacci number <math>F_{p - \left(\frac\right)}</math>, where the Legendre symbol <math>\left(\frac\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>

, no Wall-Sun-Sun primes have been found below <math>2^{64}</math> (about <math>18\cdot 10^{18}</math>).

Wieferich primes

Primes p such that for fixed integer a > 1.

2^{p − 1} &equiv; 1 (mod p²): 1093, 3511 ()<br>

3^{p − 1} &equiv; 1 (mod p²): 11, 1006003 ()<br>

4^{p − 1} &equiv; 1 (mod p²): 1093, 3511<br>

5^{p − 1} &equiv; 1 (mod p²): 2, 20771, 40487, 53471161, 1645333507, 6692367337, 188748146801 ()<br>

6^{p − 1} &equiv; 1 (mod p²): 66161, 534851, 3152573 ()<br>

7^{p − 1} &equiv; 1 (mod p²): 5, 491531 ()<br>

8^{p − 1} &equiv; 1 (mod p²): 3, 1093, 3511<br>

9^{p − 1} &equiv; 1 (mod p²): 2, 11, 1006003<br>

10^{p − 1} &equiv; 1 (mod p²): 3, 487, 56598313 ()<br>

11^{p − 1} &equiv; 1 (mod p²): 71<br>

12^{p − 1} &equiv; 1 (mod p²): 2693, 123653 ()<br>

13^{p − 1} &equiv; 1 (mod p²): 2, 863, 1747591 ()