List of prime numbers

This page contains a list of the first 500 primes , as well as lists of some special types of primes.

First 500 Primes

2	3	five	7	eleven	13	17	nineteen	23	29th	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	211	223	227	229	233	239	241	251	257	263	269	271	277	281
283	293	307	311	313	317	331	337	347	349	353	359	367	373	379	383	389	397	401	409
419	421	431	433	439	443	449	457	461	463	467	479	487	491	499	503	509	521	523	541
547	557	563	569	571	577	587	593	599	601	607	613	617	619	631	641	643	647	653	659
661	673	677	683	691	701	709	719	727	733	739	743	751	757	761	769	773	787	797	809
811	821	823	827	829	839	853	857	859	863	877	881	883	887	907	911	919	929	937	941
947	953	967	971	977	983	991	997	1009	1013	1019	1021	1031	1033	1039	1049	1051	1061	1063	1069
1087	1091	1093	1097	1103	1109	1117	1123	1129	1151	1153	1163	1171	1181	1187	1193	1201	1213	1217	1223
1229	1231	1237	1249	1259	1277	1279	1283	1289	1291	1297	1301	1303	1307	1319	1321	1327	1361	1367	1373
1381	1399	1409	1423	1427	1429	1433	1439	1447	1451	1453	1459	1471	1481	1483	1487	1489	1493	1499	1511
1523	1531	1543	1549	1553	1559	1567	1571	1579	1583	1597	1601	1607	1609	1613	1619	1621	1627	1637	1657
1663	1667	1669	1693	1697	1699	1709	1721	1723	1733	1741	1747	1753	1759	1777	1783	1787	1789	1801	1811
1823	1831	1847	1861	1867	1871	1873	1877	1879	1889	1901	1907	1913	1931	1933	1949	1951	1973	1979	1987
1993	1997	1999	2003	2011	2017	2027	2029	2039	2053	2063	2069	2081	2083	2087	2089	2099	2111	2113	2129
2131	2137	2141	2143	2153	2161	2179	2203	2207	2213	2221	2237	2239	2243	2251	2267	2269	2273	2281	2287
2293	2297	2309	2311	2333	2339	2341	2347	2351	2357	2371	2377	2381	2383	2389	2393	2399	2411	2417	2423
2437	2441	2447	2459	2467	2473	2477	2503	2521	2531	2539	2543	2549	2551	2557	2579	2591	2593	2609	2617
2621	2633	2647	2657	2659	2663	2671	2677	2683	2687	2689	2693	2699	2707	2711	2713	2719	2729	2731	2741
2749	2753	2767	2777	2789	2791	2797	2801	2803	2819	2833	2837	2843	2851	2857	2861	2879	2887	2897	2903
2909	2917	2927	2939	2953	2957	2963	2969	2971	2999	3001	3011	3019	3023	3037	3041	3049	3061	3067	3079
3083	3089	3109	3119	3121	3137	3163	3167	3169	3181	3187	3191	3203	3209	3217	3221	3229	3251	3253	3257
3259	3271	3299	3301	3307	3313	3319	3323	3329	3331	3343	3347	3359	3361	3371	3373	3389	3391	3407	3413
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 problem verification project reports that all primes up to $10^{18}$ ${\ displaystyle 10 ^ {18}}$ . This amounts to 24,739,954,287,740,860 primes, but they were not saved. There are well-known formulas that allow you to calculate the number of primes (up to a given value) faster than calculating the primes themselves. This method was used to calculate what up $10^{23}$ ${\ displaystyle 10 ^ {23}}$ there are 1 925 320 391 606 803 968 923 primes.

Bell Primes

Prime numbers, which are the partition number of a set with $n$ ${\ displaystyle n}$ elements.

2, 5, 877, 27644437, 35742549198872617291353508656626642567, 359334085968622831041960188598043661065388726959079837. The following number has 6539 digits ^[1] . (sequence A051131 in OEIS )

Cubic primes

Prime numbers of the form ${\frac {x^{3}-y^{3}}{x-y}},x=y+1$ ${\ displaystyle {\ frac {x ^ {3} -y ^ {3}} {xy}}, x = y + 1}$

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 (sequence A002407 in OEIS ).

and ${\frac {x^{3}-y^{3}}{x-y}},x=y+2$ ${\ displaystyle {\ frac {x ^ {3} -y ^ {3}} {xy}}, x = y + 2}$

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

(sequence A002648 in OEIS ).

Super-simple numbers

Prime numbers located at the positions of a sequence of primes with prime numbers, i.e. 2nd, 3rd, 5th, etc.

The first members of a sequence of super-simple numbers: 3, 5, 11, 17, 31, 41, 59, 67, 83, 109, 127, 157, ... OEIS sequence : A006450

Simple, consisting of units

Numbers consisting of 2, 19, 23, 317, 1031, 49081, 86453, 109297, 270343 units are prime (sequence A004023 in OEIS ).

Simple, consisting of ones and zeros

In addition to primes consisting of only ones, one can also note primes consisting of ones and zeros. Within the first ten million, the following of these numbers are prime (sequence A020449 in OEIS ):

11, 101, 10111, 101111, 1011001, 1100101, etc.

Simple Palindromes

Palindromes are numbers that are read the same way from right to left and left to right, for example, 30103. Among these numbers, primes are also found. It is clear that any simple palindrome consists of an odd number of digits (except for the number 11), since any palindrome with an even number of digits is always divided by 11. The first simple palindromes are the following numbers:

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, 15451, 15551, 16061, 16361, 16561, 16661, 17471, 17971, 18181, ... (sequence A002385 in OEIS ).

Wilson Primes

Prime numbers $p$ ${\ displaystyle p}$ for which $(p-1)!+1$ ${\ displaystyle (p-1)! + 1}$ divided entirely into $p^{2}$ ${\ displaystyle p ^ {2}}$ .

Well-known Wilson simple: 5, 13, 563 (sequence A007540 in OEIS ).

Wilson's other simple ones are unknown. Guaranteed there are no other Wilson primes smaller than 2⋅10 ¹³ ^[2] .

Wolstenholm Primes

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

Only these numbers are known up to a billion: 16843, 2124679 (sequence A088164 in OEIS )

Carol Primes

Prime numbers of the form $(2^{n}-1)^{2}-2$ ${\ displaystyle (2 ^ {n} -1) ^ {2} -2}$ .

7, 47, 223, 3967, 16127, 1046527, 16769023, 1073676287, 68718952447, 274876858367, 4398042316799, 1125899839733759, 18014398241046527, 1298074214633706835075030044377087 (sequence A0915151166167161687

Cullen Primes

Prime numbers of the form $n2^{n}+1$ ${\ displaystyle n2 ^ {n} +1}$ .

All known Cullen numbers correspond to $n$ ${\ displaystyle n}$ equal to:

1, 141, 4713, 5795, 6611, 18496, 32292, 32469, 59656, 90825, 262419, 361275, 481899, 1354828, 6328548, 6679881 A005849 sequence in OEIS .

There is speculation that there are infinitely many Cullen primes.

Markov

Prime numbers $p$ ${\ displaystyle p}$ for which there are integers $x$ ${\ displaystyle x}$ and $y$ ${\ displaystyle y}$ such that $x^{2}+y^{2}+p^{2}=3xyp$ ${\ displaystyle x ^ {2} + y ^ {2} + p ^ {2} = 3xyp}$ .

2, 5, 13, 29, 89, 233, 433, 1597, 2897, 5741, 7561, 28657, 33461, 43261, 96557, 426389, 514229 (sequence A178444 in OEIS )

Mersenne Primes

Prime numbers of the form $2^{n}-1$ ${\ displaystyle 2 ^ {n} -1}$ . The first 12 numbers:

3, 7, 31, 127, 8191, 131071, 524287, 2147483647 , 2305843009213693951, 618970019642690137449562111, 162259276829213363391578010288127, 1701411834604692317316873037158841027 .

Newman-Shanks-Williams Primes

The Newman-Shanks-Williams (NSW) prime is called the prime $p$ ${\ displaystyle p}$ , which can be written as:

S_{2m+1}={\frac {\left(1+{\sqrt {2}}\right)^{2m+1}+\left(1-{\sqrt {2}}\right)^{2m+1}}{2}}.

{\ displaystyle S_ {2m + 1} = {\ frac {\ left (1 + {\ sqrt {2}} \ right) ^ {2m + 1} + \ left (1 - {\ sqrt {2}} \ right ) ^ {2m + 1}} {2}}.}

The first few NSW primes: 7, 41, 239, 9369319, 63018038201, 489133282872437279, 19175002942688032928599, 123426017006182806728593424683999798008235734137469123231828679 (sequence A088165 in OEIS .

Proth Primes

Prime numbers of the form $P=k\cdot 2^{n}+1$ ${\ displaystyle P = k \ cdot 2 ^ {n} +1}$ , and $k$ ${\ displaystyle k}$ odd and $2^{n}>k$ ${\ displaystyle 2 ^ {n}> k}$ (sequence A080076 in OEIS ).

Sophie Germain Primes

Prime numbers $p$ ${\ displaystyle p}$ such that $2p+1$ ${\ displaystyle 2p + 1}$ also simple.

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 (sequence A005384 in OEIS ).

Farm Primes

These are primes of the form $2^{2^{n}}+1$ ${\ displaystyle 2 ^ {2 ^ {n}} + 1}$ .

Fermat's known primes: 3, 5, 17, 257, 65537 (sequence A019434 in OEIS ).

Fibonacci Primes

Prime numbers in the Fibonacci sequence F ₀ = 0, F ₁ = 1, F _n = F _{n −1} + F _{n −2} .

2 , 3 , 5 , 13 , 89 , 233, 1597, 28657, 514229, 433494437, 2971215073, 99194853094755497, 1066340417491710595814572169, 19134702400093278081449423917 (sequence A005478 in OEIS )

Chen prime numbers

Such prime numbers $p$ ${\ displaystyle p}$ , what $p+2$ ${\ displaystyle p + 2}$ either simple or semi-simple :

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 (sequence A109611 in OEIS ).

Pell Primes

In number theory, Pell numbers mean an infinite sequence of integers that are denominators of suitable fractions for the square root of 2. This sequence of approximations starts with 1/1, 3/2, 7/5, 17/12, and 41/29, so the sequence Pell numbers start at 1, 2, 5, 12, and 29. The first few Pell primes are 2, 5, 29, 5741, ... (sequence A086383 in OEIS ).

Prime numbers in the form $n^{4}+1$ ${\ displaystyle n ^ {4} +1}$ ${\ displaystyle n ^ {4} +1}$

^[3] ^[4]

2 , 17 , 257 , 1297, 65537 , 160001, 331777, 614657, 1336337, 4477457, 5308417, 8503057, 9834497, 29986577, 40960001, 45212177, 59969537, 65610001, 126247697, 193877777, 303595977, 5665657658, 568658 sequence A037896 in OEIS ).

Balanced Primes

Prime numbers, which are the arithmetic mean of the previous prime number and the following prime number:

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 (sequence A006562 in OEIS ).

Unique primes

Prime numbers $p$ ${\ displaystyle p}$ , the length of the periodic fraction of which ${\frac {1}{p}}$ ${\ displaystyle {\ frac {1} {p}}}$ unique (no other prime gives the same):

3, 11, 37, 101, 9091, 9901, 333667, 909091, 99990001, 999999000001, 9999999900000001, 909090909090909091, 1111111111111111111, 11111111111111111111111, 900900900900990990990991 (sequence A040017 in OE .

Factorial simple

These are primes of the form $n!\pm 1$ ${\ displaystyle n! \ pm 1}$ for some $n\in {\mathbb {N} }$ ${\ displaystyle n \ in {\ mathbb {N}}}$ :

2, 3, 5, 7, 23, 719, 5039, 39916801, 479001599, 87178291199, 10888869450418352160768000001, 265252859812191058636308479999999, 263130836933693530167218012159999999, 86833179998999199198999999191919818919

Primary primes

Prime numbers of the form p # ± 1 :

p _n # - 1 is simple for n = 2, 3, 5, 6, 13, 24, ... the sequence A057704 in OEIS

p _n # + 1 is simple for n = 1, 2, 3, 4, 5, 11, ... the sequence A014545 in OEIS

Centered Square Primes

Type numbers $n^{2}+(n+1)^{2}$ ${\ displaystyle n ^ {2} + (n + 1) ^ {2}}$ :

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 (sequence A027862 in OEIS ).

Centered Triangular Primes

Type numbers $(3n^{2}+3n+2)/2$ ${\ displaystyle (3n ^ {2} + 3n + 2) / 2}$ :

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 (sequence A125602 in OEIS ).

Centered Heptagonal Primes

Type numbers $(7n^{2}-7n+2)/2$ ${\ displaystyle (7n ^ {2} -7n + 2) / 2}$ :

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 (sequence A069099 in OEIS ).

Centered Decagonal Primes

Prime numbers that can be represented as $5(n^{2}-n)+1$ ${\ displaystyle 5 (n ^ {2} -n) +1}$ :

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 (sequence A090562 in OEIS ).

Notes

↑ 93074010508593618333 ... (6499 other digits) ... 83885253703080601131 , The Largest Known Primes - primes.utm.edu
↑ A Search for Wilson primes
↑ Lal, M. Primes of the Form n ⁴ + 1 (English) // Mathematics of Computation : journal. - American Mathematical Society , 1967. - Vol. 21 . - P. 245-247 . - ISSN 1088-6842 . - DOI : 10.1090 / S0025-5718-1967-0222007-9 .
↑ Bohman, J. New primes of the form n ⁴ + 1 (English) // BIT Numerical Mathematics : journal. - Springer, 1973. - Vol. 13 , no. 3 . - P. 370-372 . - ISSN 1572-9125 . - DOI : 10.1007 / BF01951947 .

Literature

Henry S. Warren, Jr. Chapter 16. Formulas for primes // Algorithmic tricks for programmers = Hacker's Delight. - M .: Williams , 2007 .-- 288 p. - ISBN 0-201-91465-4 .

Links

Lists of prime and factorized compound numbers

[primes_utm1-1] 93074010508593618333 ... (6499 other digits) ... 83885253703080601131 , The Largest Known Primes - primes.utm.edu

[2] A Search for Wilson primes

[3] Lal, M. Primes of the Form n ⁴ + 1 (English) // Mathematics of Computation : journal. - American Mathematical Society , 1967. - Vol. 21 . - P. 245-247 . - ISSN 1088-6842 . - DOI : 10.1090 / S0025-5718-1967-0222007-9 .

[4] Bohman, J. New primes of the form n ⁴ + 1 (English) // BIT Numerical Mathematics : journal. - Springer, 1973. - Vol. 13 , no. 3 . - P. 370-372 . - ISSN 1572-9125 . - DOI : 10.1007 / BF01951947 .