List of prime numbers

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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. ( A051131)

Carol primes

Of the form (2ⁿ − 1)² − 2.

7, 47, 223, 3967, 16127, 1046527, 16769023, 1073676287, 68718952447, 274876858367, 4398042316799, 1125899839733759, 18014398241046527, 1298074214633706835075030044377087 ( A091516)

Centered decagonal primes

Of the form 5(n² − n) + 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 ( A090562)

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  A069099)

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 ( A027862)

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 ( A125602)

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 ( 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 ( 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 ( A016114)

All repunit primes are circular.

Cousin primes

See also: #Twin primes, #Prime triplets, and #Prime quadruplets

(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) ( A023200,  A046132)

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 ( A002407)

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 ( A002648)

Cullen primes

Of the form n · 2ⁿ + 1.

3, 393050634124102232869567034555427371542904833 ( A050920)

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 ( A134996)

Double factorial primes

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

1, 2, 518, 33416, 37310, 52608 ( A080778)

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 ( A007749)

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  A077586)

As of 2011^[update], 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 ( A003627)

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 ( A006567)

Euclid primes

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

3, 7, 31, 211, 2311, 200560490131 ( A018239^[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 ( A088054)

Fermat primes

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

3, 5, 17, 257, 65537 ( A019434)

As of 2011^[update] these are the only known Fermat primes, and conjecturally the only Fermat primes.

Fibonacci primes

Primes in the Fibonacci sequence F₀ = 0, F₁ = 1, F_n = F_n-1 + F_n-2.

2, 3, 5, 13, 89, 233, 1597, 28657, 514229, 433494437, 2971215073, 99194853094755497, 1066340417491710595814572169, 19134702400093278081449423917 ( 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 ( A046066)

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 ( A002145)

Generalized Fermat primes base 10

Of the form 10ⁿ + 1, where n > 0.

11, 101

As of April 2011^[update], 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 ( A046850; another entry  A135995 is erroneous)

Good primes

Primes p_n for which p_n² > p_n−i × p_n+i for all 1 ≤ i ≤ n−1, where p_n 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 ( 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 ( A035497)

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 ( 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 ( A007459)

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 ( A105440)

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 ( A000928)

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 ( A007510)

Kynea primes

Of the form (2ⁿ + 1)² − 2.

7, 23, 79, 1087, 66047, 263167, 16785407, 1073807359, 17180131327, 68720001023, 4398050705407, 70368760954879, 18014398777917439, 18446744082299486207 ( A091514)

Left-truncatable primes

See also: #Right-truncatable primes and #Two-sided 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 ( A024785)

Leyland primes

Of the form x^y + y^x with 1 < x ≤ y.

17, 593, 32993, 2097593, 8589935681, 59604644783353249, 523347633027360537213687137, 43143988327398957279342419750374600193 ( A094133)

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 ( A001913)

Lucas primes

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

2,^[10] 3, 7, 11, 29, 47, 199, 521, 2207, 3571, 9349, 3010349, 54018521, 370248451, 6643838879, 119218851371, 5600748293801, 688846502588399, 32361122672259149 ( 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 ( A031157)

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  A002559)

Mersenne primes

Of the form 2ⁿ − 1.

3, 7, 31, 127, 8191, 131071, 524287, 2147483647, 2305843009213693951, 618970019642690137449562111, 162259276829213363391578010288127, 170141183460469231731687303715884105727 ( A000668)

As of 2011^[update], 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 2^p − 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 ( A000043)

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 ( 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 ( A071062)

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 ( A092832)

Newman–Shanks–Williams primes

Newman–Shanks–Williams numbers that are prime.

7, 41, 239, 9369319, 63018038201, 489133282872437279, 19175002942688032928599 ( A088165)

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... ( A065091)

All prime numbers except 2 are odd.

Padovan primes

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 ( A100891)

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 ( A002385)

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 ( A077798)

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 ( A049575)

Pell primes

Primes in the Pell number sequence P₀ = 0, P₁ = 1, P_n = 2P_n-1 + P_n-2.

2, 5, 29, 5741, 33461, 44560482149, 1746860020068409, 68480406462161287469, 13558774610046711780701, 4125636888562548868221559797461449 ( 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 ( A003459)

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 ( A074788)

Pierpont primes

Of the form 2^u3^v + 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 ( 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 ( A063980)

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 ( A119535)

Primorial primes

Of the form p_n# − 1 or p_n# + 1.

3, 5, 7, 29, 31, 211, 2309, 2311, 30029, 200560490131, 304250263527209, 23768741896345550770650537601358309 (union of  A057705 and  A018239^[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 ( 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 ( A002144)

Prime quadruplets

See also: #Cousin primes, #Twin primes, and #Prime triplets

(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) ( A007530,  A136720,  A136721,  A090258)

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 ( A106856)

Quartan primes

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

2, 17, 97, 257, 337, 641, 881 ( A002645)

Ramanujan primes

Integers R_n that are the smallest to give at least n primes from x/2 to x for all x ≥ R_n (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 ( A104272)

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 ( A007703)

Repunit primes

Primes containing only the decimal digit 1.

11, 1111111111111111111, 11111111111111111111111 ( A004022)

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

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

Right-truncatable primes

See also: #Left-truncatable primes and #Two-sided 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 ( A024770)

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 ( 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 ( A006378)

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) ( A023201,  A046117)

Smarandache–Wellin primes

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

2, 23, 2357 ( A069151)

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

Solinas primes

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

3, 5, 7, 11, 13 ( A165255)

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 ( A005384)

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 ( A083577)

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 ( A042978)

As of 2011^[update], 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 ( A006450)

Supersingular primes

There are exactly fifteen supersingular primes:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59, 71 ( A002267)

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 ( A163074)

Thabit number primes

Of the form 3 · 2ⁿ - 1.

2, 5, 11, 23, 47, 191, 383, 6143, 786431, 51539607551, 824633720831, 26388279066623, 108086391056891903, 55340232221128654847, 226673591177742970257407 ( A007505)

Prime triplets

See also: #Cousin primes, #Twin primes, and #Prime quadruplets

(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) ( A007529,  A098414,  A098415)

Twin primes

See also: #Cousin primes, #Prime triplets, and #Prime quadruplets

(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) ( A001359,  A006512)

Two-sided primes

See also: #Right-truncatable primes and #Left-truncatable 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 ( A020994)

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 ( A068820)

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 ( A040017)

Wagstaff primes

Of the form (2ⁿ + 1) / 3.

3, 11, 43, 683, 2731, 43691, 174763, 2796203, 715827883, 2932031007403, 768614336404564651, 201487636602438195784363, 845100400152152934331135470251, 56713727820156410577229101238628035243 ( A000979)

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 ( A000978)

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^[update], 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  A001190)

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 ( A050249)

Wieferich primes

Primes p for which p² divides 2^{p − 1} − 1.

1093, 3511 ( A001220)

As of 2011^[update], these are the only known Wieferich primes.

Wieferich primes base 3 (Mirimanoff primes)

Primes p for which p² divides 3^{p − 1} − 1.

11, 1006003 ( A014127)

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

Wieferich primes base 5

Primes p for which p² divides 5^{p − 1} − 1

2, 20771, 40487, 53471161, 1645333507, 6692367337, 188748146801 ( A123692)

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

Wieferich primes base 6

Primes p for which p² divides 6^{p − 1} − 1

66161, 534851, 3152573

Wieferich primes base 7

Primes p for which p² divides 7^{p − 1} − 1

5, 491531 ( A123693)

Wieferich primes base 10

Primes p for which p² divides 10^{p − 1} − 1

3, 487, 56598313 ( A045616)

Wieferich primes base 11

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

71

Wieferich primes base 12

Primes p for which p² divides 12^{p − 1} − 1

2693, 123653 ( A111027)

Wieferich primes base 13

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

863, 1747591 ( A128667)

Wieferich primes base 17

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

3, 46021, 48947

Wieferich primes base 19

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

3, 7, 13, 43, 137, 63061489 ( A090968)

Wilson primes

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

5, 13, 563 ( A007540)

As of 2011^[update], 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 ( A088164)

As of 2011^[update], these are the only known Wolstenholme primes.

Woodall primes

Of the form n · 2ⁿ − 1.

7, 23, 383, 32212254719, 2833419889721787128217599, 195845982777569926302400511, 4776913109852041418248056622882488319 ( A050918)

See also

• Illegal prime
• Largest known prime
• List of numbers
• Prime gap
• Probable prime
• Pseudoprime
• Strobogrammatic prime
• Strong prime
• Wieferich pair

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. http://openlibrary.org/books/OL16553580M/List_of_prime_numbers_from_1_to_10_006_721. 
2. ^ Tomás Oliveira e Silva, Goldbach conjecture verification.
3. ^ Jens Franke (29 July 2010). "Conditional Calculation of pi(10²⁴)". http://primes.utm.edu/notes/pi(10%5E24).html. Retrieved 2011-05-17. 
4. ^ ^{a b}  A018239 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!. http://primes.utm.edu/curios/page.php?short=101. Retrieved 1 April 2011. 
7. ^ Weisstein, Eric W., "Genocchi Number" from MathWorld.
8. ^ Russo, F., A Set of New Samarandache Functions, Sequences and Conjectures in Number Theory, pp. 73–74, http://fs.gallup.unm.edu//Felice-Russo-book1.pdf 
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. http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.56.7026&rep=rep1&type=pdf. 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 A_{n − k − 1}B₁A_k, especially 9_{n − k − 1}8₁9_k". Journal of Recreational Mathematics 28 (1): 1–9. 
12. ^ Ribenboim, P.. The new book of prime number records. New York: Springer-Verlag. p. 347. ISBN 0387944575. http://books.google.de/books?id=72eg8bFw40kC&printsec=frontcover&dq=ribenboim&hl=de&ei=PoJATZvqO4WU4Qamg-n-Ag&sa=X&oi=book_result&ct=result&resnum=1&ved=0CDEQ6AEwAA#v=onepage&q&f=false. 
13. ^ "Mirimanoff's Congruence: Other Congruences". http://www.museumstuff.com/learn/topics/Mirimanoff%27s_congruence::sub::Other_Congruences. Retrieved 26 January 2011. 
14. ^ Gallot, Y.; Moree, P.; Zudilin, W. (2011). "The Erdös-Moser equation 1^k + 2^k +...+ (m-1)^k = m^k revisited using continued fractions". Mathematics of Computation (American Mathematical Society) 80: 1221–1237. arXiv:0907.1356. doi:10.1090/S0025-5718-2010-02439-1. http://www.mpim-bonn.mpg.de/preprints/send?bid=4053. 
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. http://www.scribd.com/doc/35180646/Ribenboim-Die-Welt-der-Primzahlen. 

External links

• Lists of Primes at the Prime Pages.
• Interface to a list of the first 98 million primes (primes less than 2,000,000,000)
• Weisstein, Eric W., "Prime Number Sequences" from MathWorld.
• Selected prime related sequences in OEIS.
• Fischer, R. Thema: Fermatquotient B^(P-1) == 1 (mod P^2) (German) (Lists Wieferich primes in all bases up to 1052)