# List of prime numbers

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List of prime numbers

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 500 primes are listed below, followed by lists of notable types of prime numbers in alphabetical order, giving their respective first terms.

## The first 500 prime numbers

The following table lists the first 500 primes; 20 consecutive primes in each of the 25 rows.[1]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
1-20 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71
21-40 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173
41-60 179 181 191 193 197 199 211 223 227 229 233 239 241 251 257 263 269 271 277 281
61-80 283 293 307 311 313 317 331 337 347 349 353 359 367 373 379 383 389 397 401 409
81-100 419 421 431 433 439 443 449 457 461 463 467 479 487 491 499 503 509 521 523 541
101-120 547 557 563 569 571 577 587 593 599 601 607 613 617 619 631 641 643 647 653 659
121-140 661 673 677 683 691 701 709 719 727 733 739 743 751 757 761 769 773 787 797 809
141-160 811 821 823 827 829 839 853 857 859 863 877 881 883 887 907 911 919 929 937 941
161-180 947 953 967 971 977 983 991 997 1009 1013 1019 1021 1031 1033 1039 1049 1051 1061 1063 1069
181-200 1087 1091 1093 1097 1103 1109 1117 1123 1129 1151 1153 1163 1171 1181 1187 1193 1201 1213 1217 1223
201-220 1229 1231 1237 1249 1259 1277 1279 1283 1289 1291 1297 1301 1303 1307 1319 1321 1327 1361 1367 1373
221-240 1381 1399 1409 1423 1427 1429 1433 1439 1447 1451 1453 1459 1471 1481 1483 1487 1489 1493 1499 1511
241-260 1523 1531 1543 1549 1553 1559 1567 1571 1579 1583 1597 1601 1607 1609 1613 1619 1621 1627 1637 1657
261-280 1663 1667 1669 1693 1697 1699 1709 1721 1723 1733 1741 1747 1753 1759 1777 1783 1787 1789 1801 1811
281-300 1823 1831 1847 1861 1867 1871 1873 1877 1879 1889 1901 1907 1913 1931 1933 1949 1951 1973 1979 1987
301-320 1993 1997 1999 2003 2011 2017 2027 2029 2039 2053 2063 2069 2081 2083 2087 2089 2099 2111 2113 2129
321-340 2131 2137 2141 2143 2153 2161 2179 2203 2207 2213 2221 2237 2239 2243 2251 2267 2269 2273 2281 2287
341-360 2293 2297 2309 2311 2333 2339 2341 2347 2351 2357 2371 2377 2381 2383 2389 2393 2399 2411 2417 2423
361-380 2437 2441 2447 2459 2467 2473 2477 2503 2521 2531 2539 2543 2549 2551 2557 2579 2591 2593 2609 2617
381-400 2621 2633 2647 2657 2659 2663 2671 2677 2683 2687 2689 2693 2699 2707 2711 2713 2719 2729 2731 2741
401-420 2749 2753 2767 2777 2789 2791 2797 2801 2803 2819 2833 2837 2843 2851 2857 2861 2879 2887 2897 2903
421-440 2909 2917 2927 2939 2953 2957 2963 2969 2971 2999 3001 3011 3019 3023 3037 3041 3049 3061 3067 3079
441-460 3083 3089 3109 3119 3121 3137 3163 3167 3169 3181 3187 3191 3203 3209 3217 3221 3229 3251 3253 3257
461-480 3259 3271 3299 3301 3307 3313 3319 3323 3329 3331 3343 3347 3359 3361 3371 3373 3389 3391 3407 3413
481-500 3433 3449 3457 3461 3463 3467 3469 3491 3499 3511 3517 3527 3529 3533 3539 3541 3547 3557 3559 3571

(sequence A000040 in OEIS).

The Goldbach conjecture verification project reports that it has computed all primes below 1018.[2] That means 24739954287740860 primes (roughly 2.5×1016), but they were not stored. There are known formulas to evaluate the prime-counting function (the number of primes below a given value) faster than computing the primes. This has been used to compute that there are 1925320391606803968923 primes (roughly 2×1021) below 1023. A different computation with a method assuming the Riemann hypothesis found that there are 18435599767349200867866 primes (roughly 2×1022) below 1024 if the Riemann hypothesis is true.[3]

## 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. A prime number is a number that cannot be divided by a number other than 1 and itself.

### Bell number primes

Primes that are the number of partitions of a set with n members.

2, 5, 877, 27644437, 35742549198872617291353508656626642567, 359334085968622831041960188598043661065388726959079837. The next term has 6539 digits. ()

### Carol primes

Of the form (2n − 1)2 − 2.

7, 47, 223, 3967, 16127, 1046527, 16769023, 1073676287, 68718952447, 274876858367, 4398042316799, 1125899839733759, 18014398241046527, 1298074214633706835075030044377087 ()

### Centered decagonal primes

Of the form 5(n2n) + 1.

11, 31, 61, 101, 151, 211, 281, 661, 911, 1051, 1201, 1361, 1531, 1901, 2311, 2531, 3001, 3251, 3511, 4651, 5281, 6301, 6661, 7411, 9461, 9901, 12251, 13781, 14851, 15401, 18301, 18911, 19531, 20161, 22111, 24151, 24851, 25561, 27011, 27751 ()

### Centered heptagonal primes

Of the form (7n2 − 7n + 2) / 2.

43, 71, 197, 463, 547, 953, 1471, 1933, 2647, 2843, 3697, 4663, 5741, 8233, 9283, 10781, 11173, 12391, 14561, 18397, 20483, 29303, 29947, 34651, 37493, 41203, 46691, 50821, 54251, 56897, 57793, 65213, 68111, 72073, 76147, 84631, 89041, 93563 (primes in )

### Centered square primes

Of the form n2 + (n + 1)2.

5, 13, 41, 61, 113, 181, 313, 421, 613, 761, 1013, 1201, 1301, 1741, 1861, 2113, 2381, 2521, 3121, 3613, 4513, 5101, 7321, 8581, 9661, 9941, 10513, 12641, 13613, 14281, 14621, 15313, 16381, 19013, 19801, 20201, 21013, 21841, 23981, 24421, 26681 ()

### Centered triangular primes

Of the form (3n2 + 3n + 2) / 2.

19, 31, 109, 199, 409, 571, 631, 829, 1489, 1999, 2341, 2971, 3529, 4621, 4789, 7039, 7669, 8779, 9721, 10459, 10711, 13681, 14851, 16069, 16381, 17659, 20011, 20359, 23251, 25939, 27541, 29191, 29611, 31321, 34429, 36739, 40099, 40591, 42589 ()

### Chen primes

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 ()

### 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 ()

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 ()

All repunit primes are circular.

### Cousin primes

(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) (, )

### Cuban primes

Of the form $\tfrac{x^3-y^3}{x-y},\ x=y+1.\,$

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 ()

Of the form $\tfrac{x^3-y^3}{x-y},\ x=y+2.\,$

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 ()

### Cullen primes

Of the form n · 2n + 1.

3, 393050634124102232869567034555427371542904833 ()

### 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 ()

### Double factorial primes

Of the form n!! + 1. Values of n:

1, 2, 518, 33416, 37310, 52608 ()

Note that n = 0 and n = 1 produce the same prime, namely 2.

Of the form n!! − 1. Values of n:

3, 4, 6, 8, 16, 26, 64, 82, 90, 118, 194, 214, 728, 842, 888, 2328, 3326, 6404, 8670, 9682, 27056, 44318 ()

### Double Mersenne primes

A subset of Mersenne primes: of the form $2^{(2^p-1)}-1$ for prime p.

7, 127, 2147483647, 170141183460469231731687303715884105727 (primes in )

As of 2011, these are the only known double Mersenne primes, and probably the only double Mersenne primes.

### Eisenstein primes without imaginary part

Eisenstein integers that are irreducible and real numbers (primes of 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 ()

### Emirps

Primes which become a different prime when their decimal digits are reversed.

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

Of the form pn# + 1 (a subset of primorial primes).

3, 7, 31, 211, 2311, 200560490131 ([4])

### Even prime

Of the form 2n; n = 1, 2, 3, 4, ...

2

The only even prime is 2.
2 is therefore sometimes called "the oddest prime" as a pun on the non-mathematical meaning of "odd".[5]

### Factorial primes

Of the form n! − 1 or n! + 1.

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

### Fermat primes

Of the form $2^{2^n} + 1$.

3, 5, 17, 257, 65537 ()

As of 2011 these are the only known Fermat primes, and conjecturally the only Fermat primes.

### 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 ()

### 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 ()

### Gaussian primes

Prime elements of the Gaussian integers (primes of 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 ()

### Generalized Fermat primes base 10

Of the form 10n + 1, where n > 0.

As of April 2011, these are the only known generalized Fermat primes in base 10.[6]

### Genocchi number primes

17

The only positive prime Genocchi number is 17.[7]

### Gilda's primes

Gilda's numbers that are prime.[8]

29, 683, 997, 2207, 30571351 (; another entry is erroneous)

### Good primes

Primes pn for which pn2 > pn−i × pn+i for all 1 ≤ in−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 ()

### 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 ()

### Harmonic primes

Primes p for which there are no solutions to $H_k \equiv 0 \pmod{p}$ and $H_k \equiv -\omega\,\!_p \pmod{p}$ for $1 \leq k \leq p-2$.[9]

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 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 ()

### Highly cototient number 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 ()

### Irregular primes

Odd primes p which 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, 617, 619 ()

### 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 ()

### Kynea primes

Of the form (2n + 1)2 − 2.

7, 23, 79, 1087, 66047, 263167, 16785407, 1073807359, 17180131327, 68720001023, 4398050705407, 70368760954879, 18014398777917439, 18446744082299486207 ()

### Left-truncatable primes

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 ()

### Leyland primes

Of the form xy + yx with 1 < xy.

17, 593, 32993, 2097593, 8589935681, 59604644783353249, 523347633027360537213687137, 43143988327398957279342419750374600193 ()

### Long primes

Primes p for which, in a given base b, $\frac{b^{p-1}-1}{p}$ 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 ()

### Lucas primes

Primes in the Lucas number sequence L0 = 2, L1 = 1, Ln = Ln-1 + Ln-2.

2,[10] 3, 7, 11, 29, 47, 199, 521, 2207, 3571, 9349, 3010349, 54018521, 370248451, 6643838879, 119218851371, 5600748293801, 688846502588399, 32361122672259149 ()

### 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 ()

### Markov primes

Primes p for which there exist integers x and y such that x2 + y2 + p2 = 3xyp.

2, 5, 13, 29, 89, 233, 433, 1597, 2897, 5741, 7561, 28657, 33461, 43261, 96557, 426389, 514229, 1686049, 2922509, 3276509, 94418953, 321534781, 433494437, 780291637, 1405695061, 2971215073, 19577194573, 25209506681 (primes in )

### Mersenne primes

Of the form 2n − 1.

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

As of 2011, there are 47 known Mersenne primes (The 47th discovered is actually the 46th in size). The 13th, 14th, and 47th (based upon size), respectively, have 157, 183, and 12,978,189 digits.

### 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 ()

### Mills primes

Of the form $\lfloor \theta^{3^{n}}\;\rfloor$, 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 ()

### Motzkin primes

Primes that are the number of different ways of drawing non-intersecting chords on a circle between n points.

2, 127, 15511, 953467954114363 ()

### Newman–Shanks–Williams primes

Newman–Shanks–Williams numbers that are prime.

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

### Odd primes

Of the form 2n - 1.

3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199... ()

All prime numbers except 2 are odd.

Primes in the Padovan sequence P(0) = P(1) = P(2) = 1, P(n) = P(n − 2) + P(n − 3).

2, 3, 5, 7, 37, 151, 3329, 23833, 13091204281, 3093215881333057, 1363005552434666078217421284621279933627102780881053358473 ()

### 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 $\frac{a \big( 10^m-1 \big)}{9} \pm b \times 10^{\frac{m}{2}}$.[11]

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 numbers 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 P0 = 0, P1 = 1, Pn = 2Pn-1 + Pn-2.

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 ()

It seems likely that all further permutable primes are repunits, i.e. contain only the digit 1.

### 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 ()

### 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 ()

### 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 ()

### 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 pn# − 1 or pn# + 1.

3, 5, 7, 29, 31, 211, 2309, 2311, 30029, 200560490131, 304250263527209, 23768741896345550770650537601358309 (union of and [4])

### 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 ()

### 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 ()

(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) (, , , )

### Primes of binary quadratic form

Of the form x2 + xy + 2y2, with $x, y \in \mathbb{N}$.

2, 11, 23, 37, 43, 53, 71, 79, 107, 109, 127, 137, 149, 151, 163, 193, 197, 211, 233, 239, 263, 281, 317, 331, 337, 373, 389, 401, 421, 431, 443, 463, 487, 491, 499, 541, 547, 557, 569, 599, 613, 617, 641, 653, 659, 673, 683, 739, 743, 751, 757, 809, 821 ()

### Quartan primes

Of the form x4 + y4, where x > 0, y > 0.

2, 17, 97, 257, 337, 641, 881 ()

### Ramanujan primes

Integers Rn that are the smallest to give at least n primes from x/2 to x for all xRn (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 ()

### Regular primes

Primes p which 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 ()

### Repunit primes

Primes containing only the decimal digit 1.

11, 1111111111111111111, 11111111111111111111111 ()

The next have 317 and 1031 digits.

### Primes in residue classes

Of form a · n + d for fixed a and d. Also called primes congruent to d modulo a.

Three cases have their own entry: 2n+1 are the odd primes, 4n+1 are Pythagorean primes, 4n+3 are the integer Gaussian primes.

2n+1: 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53 ()
4n+1: 5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101, 109, 113, 137 ()
4n+3: 3, 7, 11, 19, 23, 31, 43, 47, 59, 67, 71, 79, 83, 103, 107 ()
6n+1: 7, 13, 19, 31, 37, 43, 61, 67, 73, 79, 97, 103, 109, 127, 139 ()
6n+5: 5, 11, 17, 23, 29, 41, 47, 53, 59, 71, 83, 89, 101, 107, 113 ()
8n+1: 17, 41, 73, 89, 97, 113, 137, 193, 233, 241, 257, 281, 313, 337, 353 ()
8n+3: 3, 11, 19, 43, 59, 67, 83, 107, 131, 139, 163, 179, 211, 227, 251 ()
8n+5: 5, 13, 29, 37, 53, 61, 101, 109, 149, 157, 173, 181, 197, 229, 269 ()
8n+7: 7, 23, 31, 47, 71, 79, 103, 127, 151, 167, 191, 199, 223, 239, 263 ()
10n+1: 11, 31, 41, 61, 71, 101, 131, 151, 181, 191, 211, 241, 251, 271, 281 ()
10n+3: 3, 13, 23, 43, 53, 73, 83, 103, 113, 163, 173, 193, 223, 233, 263 ()
10n+7: 7, 17, 37, 47, 67, 97, 107, 127, 137, 157, 167, 197, 227, 257, 277 ()
10n+9: 19, 29, 59, 79, 89, 109, 139, 149, 179, 199, 229, 239, 269, 349, 359 ()
...

10n+d (d = 1, 3, 7, 9) are primes ending in the decimal digit d.

### Right-truncatable primes

Primes that remain prime when the last 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 ()

### Safe primes

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 ()

### 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 ()

### Sexy primes

Where (p, p + 6) are both prime, both p and p + 6 are sexy primes.

(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) (, )

### Smarandache–Wellin primes

Primes which are the concatenation of the first n primes written in decimal.

2, 23, 2357 ()

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

### Solinas primes

Of the form 2a ± 2b ± 1, where 0 < b < a.

3, 5, 7, 11, 13 ()

### Sophie Germain primes

p and 2p + 1 are both 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 ()

### Star primes

Of the form 6n(n - 1) + 1.

13, 37, 73, 181, 337, 433, 541, 661, 937, 1093, 2053, 2281, 2521, 3037, 3313, 5581, 5953, 6337, 6733, 7561, 7993, 8893, 10333, 10837, 11353, 12421, 12973, 13537, 15913, 18481 ()

### 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 ()

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

### Super-primes

Primes with a prime index 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 ()

### Swinging primes

Primes which are within 1 of a swinging factorial: n≀ ±1.

2, 3, 5, 7, 19, 29, 31, 71, 139, 251, 631, 3433, 12011 ()

### Thabit number primes

Of the form 3 · 2n - 1.

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

### Prime triplets

(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) (, , )

### Twin primes

(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) (, )

### Two-sided primes

Primes which 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 ()

### Ulam number primes

Ulam numbers that are prime.

2, 3, 11, 13, 47, 53, 97, 131, 197, 241, 409, 431, 607, 673, 739, 751, 983, 991, 1103, 1433, 1489, 1531, 1553, 1709, 1721, 2371, 2393, 2447, 2633, 2789, 2833, 2897 ()

### Unique primes

Primes p for which the period length of 1/p is unique (no other prime gives the same).

3, 11, 37, 101, 9091, 9901, 333667, 909091, 99990001, 999999000001, 9999999900000001, 909090909090909091, 1111111111111111111, 11111111111111111111111, 900900900900990990990991 ()

### Wagstaff primes

Of the form (2n + 1) / 3.

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

n values:

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 > 5 is called a Wall-Sun-Sun prime if p² divides the Fibonacci number $F_{p - \left(\frac{{p}}{{5}}\right)}$, where the Legendre symbol $\left(\frac{{p}}{{5}}\right)$ is defined as

$\left(\frac{p}{5}\right) = \begin{cases} 1 &\textrm{if}\;p \equiv \pm1 \pmod 5\\ -1 &\textrm{if}\;p \equiv \pm2 \pmod 5. \end{cases}$

As of 2011, no Wall-Sun-Sun primes are known.

### Wedderburn-Etherington number primes

Wedderburn-Etherington numbers that are prime.

2, 3, 11, 23, 983, 2179, 24631, 3626149, 253450711, 596572387 (primes in )

### Weakly prime numbers

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 ()

### Wieferich primes

Primes p for which p2 divides 2p − 1 − 1.

1093, 3511 ()

As of 2011, these are the only known Wieferich primes.

#### Wieferich primes base 3 (Mirimanoff primes)

Primes p for which p2 divides 3p − 1 − 1.

11, 1006003 ()

As of January 2011, these are the only known Mirimanoff primes.[12][13][14]

#### Wieferich primes base 5

Primes p for which p2 divides 5p − 1 − 1

2, 20771, 40487, 53471161, 1645333507, 6692367337, 188748146801 ()

As of February 2011, these are the only known base 5 Wieferich primes.[15]

#### Wieferich primes base 6

Primes p for which p2 divides 6p − 1 − 1

66161, 534851, 3152573

#### Wieferich primes base 7

Primes p for which p2 divides 7p − 1 − 1

5, 491531 ()

#### Wieferich primes base 10

Primes p for which p2 divides 10p − 1 − 1

3, 487, 56598313 ()

#### Wieferich primes base 11

Primes p for which p2 divides 11p − 1 − 1[16]

71

#### Wieferich primes base 12

Primes p for which p2 divides 12p − 1 − 1

2693, 123653 ()

#### Wieferich primes base 13

Primes p for which p2 divides 13p − 1 − 1[16]

863, 1747591 ()

#### Wieferich primes base 17

Primes p for which p2 divides 17p − 1 − 1[16]

3, 46021, 48947

#### Wieferich primes base 19

Primes p for which p2 divides 19p − 1 − 1[16]

3, 7, 13, 43, 137, 63061489 ()

### Wilson primes

Primes p for which p2 divides (p − 1)! + 1.

5, 13, 563 ()

As of 2011, these are the only known Wilson primes.

### Wolstenholme primes

Primes p for which the binomial coefficient ${{2p-1}\choose{p-1}} \equiv 1 \pmod{p^4}.$

16843, 2124679 ()

As of 2011, these are the only known Wolstenholme primes.

### Woodall primes

Of the form n · 2n − 1.

7, 23, 383, 32212254719, 2833419889721787128217599, 195845982777569926302400511, 4776913109852041418248056622882488319 ()

## Notes

1. ^ Lehmer, D. N. (1982). List of prime numbers from 1 to 10,006,721. 165. Washington D.C.: Carnegie Institution of Washington. OL16553580M.
2. ^ Tomás Oliveira e Silva, Goldbach conjecture verification.
3. ^ Jens Franke (29 July 2010). "Conditional Calculation of pi(1024)". Retrieved 2011-05-17.
4. ^ a b includes 2 = empty product of first 0 primes plus 1, but 2 is excluded in this list.
5. ^ http://mathworld.wolfram.com/OddPrime.html
6. ^ Caldwell, C.; Honaker, Jr., G. L.. "101". Prime Curios!. Retrieved 1 April 2011.
7. ^
8. ^
9. ^ Boyd, D. W. (1994). "A p-adic Study of the Partial Sums of the Harmonic Series". Experimental Mathematics (A K Peters, Ltd.) 3 (4): 292–293. doi:10.1.1.56.7026. Retrieved 2011-05-13.
10. ^ It varies whether L0 = 2 is included in the Lucas numbers.
11. ^ Caldwell, C.; Dubner, H. (1996-97). "The near repdigit primes Ank − 1B1Ak, especially 9nk − 1819k". Journal of Recreational Mathematics 28 (1): 1–9.
12. ^
13. ^
14. ^ Gallot, Y.; Moree, P.; Zudilin, W. (2011). "The Erdös-Moser equation 1k + 2k +...+ (m-1)k = mk revisited using continued fractions". Mathematics of Computation (American Mathematical Society) 80: 1221–1237. arXiv:0907.1356. doi:10.1090/S0025-5718-2010-02439-1.
15. ^ Dorais, F. G., Klyve, D. W. Near Wieferich primes up to 6.7×1015 page 6
16. ^ a b c d Ribenboim, P. (2006). Die Welt der Primzahlen. Berlin: Springer. p. 240. ISBN 3540342834.

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