List of prime numbers - AbsoluteAstronomy.com

Euclid's theorem

Euclid's theorem is a fundamental statement in number theory that asserts that there are infinitely many prime numbers. There are several well-known proofs of the theorem.-Euclid's proof:...

, there are an infinite number of prime number

Prime number

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...

s. 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	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 11 (number) 11 is the natural number following 10 and preceding 12.Eleven is the first number which cannot be counted with a human's eight fingers and two thumbs additively. In English, it is the smallest positive integer requiring three syllables and the largest prime number with a single-morpheme name...	13 13 (number) 13 is the natural number after 12 and before 14. It is the smallest number with eight letters in its name spelled out in English. It is also the first of the teens – the numbers 13 through 19 – the ages of teenagers....	17 17 (number) 17 is the natural number following 16 and preceding 18. It is prime.In spoken English, the numbers 17 and 70 are sometimes confused because they sound similar. When carefully enunciated, they differ in which syllable is stressed: 17 vs 70...	19 19 (number) 19 is the natural number following 18 and preceding 20. It is a prime number.In English speech, the numbers 19 and 90 are often confused. When carefully enunciated, they differ in which syllable is stressed: 19 vs 90...	23 23 (number) 23 is the natural number following 22 and preceding 24.- In mathematics :Twenty-three is the ninth prime number, the smallest odd prime that is not a twin prime. Twenty-three is also the fifth factorial prime, the third Woodall prime...	29 29 (number) 29 is the natural number following 28 and preceding 30.-In mathematics:It is the tenth prime number, and also the fourth primorial prime. It forms a twin prime pair with thirty-one, which is also a primorial prime. Twenty-nine is also the sixth Sophie Germain prime. It is also the sum of three...	31 31 (number) 31 is the natural number following 30 and preceding 32.- In mathematics :Thirty-one is the third Mersenne prime as well as the fourth primorial prime, and together with twenty-nine, another primorial prime, it comprises a twin prime. As a Mersenne prime, 31 is related to the perfect number 496,...	37 37 (number) 37 is the natural number following 36 and preceding 38.-In mathematics:It is a prime number, the fifth lucky prime, the first irregular prime, the third unique prime and the third cuban prime of the form...	41 41 (number) 41 is the natural number following 40 and preceding 42.-In mathematics:Forty-one is the 13th smallest prime number. The next is forty-three, with which it comprises a twin prime...	43 43 (number) 43 is the natural number following 42 and preceding 44.- In mathematics :Forty-three is the 14th smallest prime number. The previous is forty-one, with which it comprises a twin prime, and the next is forty-seven. 43 is the smallest prime that is not a Chen prime...	47 47 (number) 47 is the natural number following 46 and preceding 48.-In mathematics:Forty-seven is the fifteenth prime number, a safe prime, the thirteenth supersingular prime, and the sixth Lucas prime. Forty-seven is a highly cototient number...	53 53 (number) 53 is the natural number following 52 and preceding 54.-In mathematics:Fifty-three is the 16th prime number. It is also an Eisenstein prime....	59 59 (number) 59 is the natural number following 58 and preceding 60.-In mathematics:Fifty-nine is the 17th smallest prime number. The next is sixty-one, with which it comprises a twin prime. 59 is an irregular prime, a safe prime and the 14th supersingular prime. It is an Eisenstein prime with no imaginary...	61 61 (number) 61 is the natural number following 60 and preceding 62.-In mathematics:It is the 18th prime number. The previous is 59, with which it comprises a twin prime. Sixty-one is a cuban prime of the form p = / , x = y + 1....	67 67 (number) 67 is the natural number following 66 and preceding 68. It is an odd number.-In mathematics:Sixty-seven is the 19th prime number , an irregular prime, a lucky prime, the sum of five consecutive primes , and a Heegner number.Since 18! + 1 is divisible by 67 but 67 is not one more than a multiple of...	71 71 (number) 71 is the natural number following 70 and preceding 72.-In mathematics:71 is the algebraic degree of Conway's constant, a remarkable number arising in the study of look-and-say sequences....
21–40	73 73 (number) 73 is the natural number following 72 and preceding 74. In English, it is the smallest integer with twelve letters in its spelled out name.- In mathematics :...	79 79 (number) Seventy-nine is the natural number following 78 and preceding 80.79 may represent:-In mathematics:An odd numberThe smallest number that can't be represented as a sum of fewer than 19 fourth powers*A strictly non-palindromic number...	83 83 (number) 83 is the natural number following 82 and preceding 84.-In mathematics:Eighty-three is the sum of three consecutive primes as well as the sum of five consecutive primes ....	89	97 97 (number) 97 is the natural number following 96 and preceding 98.-In mathematics:97 is the 25th prime number , following 89 and preceding 101. 97 is a Proth prime as it is 3 × 25 + 1.The numbers 97, 907, 9007, 90007 and 900007 are happy primes...	101 101 (number) 101 is the natural number following 100 and preceding 102.It is variously pronounced "one hundred and one" / "a hundred and one", "one hundred one" / "a hundred one", and "one oh one"...	103 103 (number) 103 is the natural number following 102 and preceding 104.-In mathematics:One hundred [and] three is the 27th prime number. The previous prime is 101, making them both twin primes...	107 107 (number) 107 is the natural number following 106 and preceding 108.-In mathematics:One hundred [and] seven is the 28th prime number. The next prime is 109, with which it comprises a twin prime, making 107 a Chen prime....	109 109 (number) 109 is the natural number following 108 and preceding 110.-In mathematics:One hundred [and] nine is the 29th prime number, and also a Chen prime. The previous prime is 107, making them both twin primes...	113 113 (number) 113 is the natural number following 112 and preceding 114.-In mathematics:One hundred [and] thirteen is the 30th prime number, following 109 and preceding 127, a Sophie Germain prime, a Chen prime and a Proth prime as it is a prime number of the form 7 × 24 + 1...	127 127 (number) 127 is the natural number following 126 and preceding 128.- In mathematics :*As a Mersenne prime, 127 is related to the perfect number 8128. 127 is also an exponent for the Mersenne prime 2127 - 1, making 127 a double Mersenne prime...	131 131 (number) 131 is the natural number following 130 and preceding 132.-In mathematics:131 is a Sophie Germain prime, the second 3-digit palindromic prime, and also a permutable prime with 113 and 311. It can be expressed as the sum of three consecutive primes, 131 = 41 + 43 + 47. 131 is an Eisenstein prime...	137 137 (number) 137 is the natural number following 136 and preceding 138.-In mathematics :One hundred [and] thirty-seven is the 33rd prime number; the next is 139, with which it comprises a twin prime, and thus 137 is a Chen prime. 137 is an Eisenstein prime with no imaginary part and a real part of the form 3n -...	139 139 (number) 139 is the natural number following 138 and preceding 140.-In mathematics:One hundred [and] thirty-nine is the 34th prime number, so it is divisible only by itself and 1. It is a twin prime with 137. Because 141 is a semiprime, 139 is a Chen prime...	149 149 (number) 149 is the natural number between 148 and 150. It is also a prime number.-In mathematics:*149 is the 35th prime number, and with the next prime number, 151, is a twin prime, thus 149 is a Chen prime. 149 is a strong prime in the sense that it is more than the arithmetic mean of its two neighboring...	151 151 (number) 151 is a natural number. It follows 150 and precedes 152.-In mathematics:151 is the 36th prime number, the previous is 149, with which it comprises a twin prime. 151 is also a palindromic prime. 151 is a centered decagonal number...	157 157 (number) 157 is the number equal to 100 + 50 + 7, following 156 and preceding 158.-In mathematics:* 157 is a prime number. The next prime is 163 and the previous prime is 151, which with 157 form a sexy prime triplet. Taking the arithmetic mean of those primes yields 157, thus it is a balanced prime.* In...	163 163 (number) 163 is the natural number following 162 and preceding 164.-In mathematics:163 is a strong prime in the sense that it is greater than the arithmetic mean of its two neighboring primes...	167 167 (number) 167 is the natural number following 166 and preceding 168.-In mathematics:* 167 is an odd number* 167 is a Chen prime, since the next odd number, 169, is a square of a prime...	173 173 (number) 173 is the natural number following 172 and preceding 174.-In mathematics:* 173 is an odd number* 173 is a balanced prime* 173 is a deficient number* 173 is an Eisenstein prime with no imaginary part* 173 is an Odious number* 173 is a prime number...
41–60	179 179 (number) 179 is the natural number following 178 and preceding 180.-In mathematics:* 179 is an odd number* 179 is a deficient number, as 1 is less than 179* 179 is a Gaussian number* 179 is an odious number* 179 is a square-free number...	181 181 (number) 181 is the natural number between 180 and 182.-In mathematics:* 181 is an odd number* 181 is a centered number 181 is a centered pentagonal number 181 is a centered 12-gonal number** 181 is a centered 18-gonal number...	191 191 (number) 191 is the natural number following 190 and preceding 192.-In mathematics:* 191 is an odd number* 191 is a centered 19-gonal number* 191 is a deficient number, as 1 is less than 191...	193 193 (number) 193 is the natural number following 192 and preceding 194.-In mathematics:* 193 is an odd number* 193 is a centered 32-gonal number* 193 is a deficient number, as 1 is less than 193* 193 is a happy number* 193 is a lucky number...	197 197 (number) 197 is the natural number following 196 and preceding 198.-In mathematics:* 197 is an odd number* 197 is a prime number 197 is a Chen prime 197 is an Eisenstein prime with no imaginary part 197 is a strong prime 197 is a twin prime with 199...	199 199 (number) 199 is the natural number following 198 and preceding 200.-In mathematics:* 199 is an odd number* 199 is a centered triangular number* 199 is a centered 33-gonal number* 199 is a deficient number, as 1 is less than 199* 199 is a Lucas number...	211 211 (number) 211 is the natural number between 210 and 212. It is also a prime number.-In mathematics:211 is an odd number.211 is a primorial prime, sum of three consecutive primes , Chen prime, centered decagonal prime, and self prime....	223 223 (number) 223 is the natural number between 222 and 224. It is also a prime number.-In mathematics:223 is a long prime, a strong prime, a lucky prime and a sexy prime .223 is the fourth Carol number and the third to be prime....	227 227 (number) 227 is the natural number between 226 and 228. It is also a prime number.-In mathematics:227 is a prime number, and a twin prime with 229 . 223 plus 4 is 227, so they are cousin primes...	229 229 (number) 229 is the natural number between 228 and 230. It is also a prime number.The North American telephone area code 229 is assigned to the area around the city of Albany in southwestern Georgia in North America....	233 233 (number) 233 is the natural number between 232 and 234. It is also a prime number.-In mathematics:233 is an irregular prime, a full reptend prime, a cousin prime, a Chen prime, a Fibonacci prime and a sexy prime. It is the 13th Fibonacci number. It is an Eisenstein prime of the form 3n - 1 with no...	239 239 (number) 239 is the natural number following 238 and preceding 240.-In mathematics:It is a prime number. The next is 241, with which it forms a pair of twin primes. 239 is a Sophie Germain prime and a Newman–Shanks–Williams prime...	241 241 (number) 241 is the natural number between 240 and 242. It is also a prime number.-In mathematics:241 is the larger of the twin primes . Prime twins are always of the form , with n = 1, 2, 3, ......	251 251 (number) 251 is the natural number between 250 and 252. It is also a prime number.-In mathematics:251 is a regular prime, an Eisenstein prime, a Chen prime, a Gaussian prime, a Sophie Germain prime, and a sexy prime....	257	263 263 (number) 263 is the natural number between 262 and 264. It is also a prime number.-In mathematics:263 is an irregular prime, an Eisenstein prime, a long prime, a Chen prime, a Gaussian prime, a happy prime, a sexy prime, a safe prime, and a Higgs prime....	269 269 (number) 269 is the natural number between 268 and 270. It is also a prime number.-In mathematics:269 is a regular prime, an Eisenstein prime with no imaginary part, a long prime, a Chen prime, a Pillai prime, a Pythagorean prime, a twin prime, a sexy prime, a Higgs prime, a strong prime, and a highly...	271	277 277 (number) 277 is the natural number following 276 and preceding 278.-In mathematics:277 is the fifty-ninth prime number, following 271 and preceding 281...	281
61–80	283	293	307	311 311 (number) 311 is the natural number after 310 and before 312.-In mathematics:311 is a twin prime with 313; an Eisenstein prime with no imaginary part and real part of the form 3n - 1; a Gaussian prime with no imaginary part and real part of the form 4n - 1; and a permutable prime with 113 and 131.It can be...	313 313 (number) 313 is an integer following 312 and preceding 314. 313 is a:* prime number* twin prime with 311* centered square number* long prime* pythagorean prime* regular prime* palindromic prime in both decimal and binary.* truncatable prime* happy number...	317	331	337	347	349	353 353 (number) 353 is the natural number between 352 and 354. It is a prime number.-In mathematics:353 is a palindromic prime, a cousin prime with 349, and a sexy prime with both 347 and 359. It is the first multidigit palindromic prime with all prime digits. Also, it is the smallest number whose 4th power is...	359 359 (number) Three hundred and fifty-nine is the number directly following 358 and directly preceding 360. 359 is a prime number.-359 in mathematics:359° is one degree less than a full revolution ; this makes it a reflex angle....	367	373	379	383	389	397	401	409
81–100	419 419 (number) 419 may refer to:419, the year 419 C.E.419 Advance-fee fraud, also known as the 419 scam, the Nigerian scam, and the Nigerian money transfer fraudArea code 419, which is the area code of the northwestern corner of Ohio...	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	593 593 (number) 593 is the natural number following 592 and preceding 594.-In mathematics:593 is an odd number. It is a prime number, an example of what Paul Erdős and Ernst G. Straus called a Good prime, or a prime whose square is greater than the product of its neighboring two primes. As such it is part of...	599	601	607	613	617	619 619 (number) 619 primarily refers to the year 619 AD. It may also refer to* The number 619* Area code 619 in San Diego, California, United States* Tiger Feint Kick variant in wrestling * USCGC Confidence Coast Guard Cutter Confidence...	631	641	643	647	653	659
121–140	661	673	677	683	691	701	709	719	727	733	739	743 743 (number) 743 is the natural number following 742 and preceding 744. It is a prime number.-In mathematics:743 is a Sophie Germain prime number, as well as a cousin prime....	751	757	761	769	773	787	797	809
141–160	811	821	823	827	829	839	853	857	859	863	877	881 881 (number) The number sequence 881 is...* a Paid Toll Free telephone number prefix in the USA* the Port of Los Angeles Long Wharf, California State Historic Landmark #881* the year AD 881 or the year 881 BC...	883	887	907	911 911 (number) 911 is the number following 910 and preceding 912.It is a prime number, a Sophie Germain prime and the sum of three consecutive primes . It is an Eisenstein prime with no imaginary part and real part of the form 3n − 1. Since 913 is a semiprime, 911 is a Chen prime...	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 1093 (number) 1093 is the natural number following 1092 and preceding 1094.1093 is a prime number. Together with 1091 and 1097, it forms a prime triplet. It is a happy prime and a star prime. It is also the smallest Wieferich prime. 1093 is a repunit prime in base 3 because:1093 = 1111111_3 = 3^6 + 3^5 +...	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 1289 (number) The number 1289 is the natural number following 1288 and preceding 1290.-In mathematics:The number 1289 is an odd prime number, following 1283 and preceding 1291...	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 1987 (number) 1987 is the natural number following 1986 and preceding 1988.-In mathematics:1987 is an odd number and the 300th prime number. It starts the sexy prime triplet . Being of the form 4n + 3, it is a Gaussian prime. It is a lucky number and therefore also a lucky prime...
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 3511 (number) 3511 is the natural number following 3510 and preceding 3512.3511 is a prime number, and is also an emirp.3511 is the largest known Wieferich prime – the only other being 1093....	3517	3527	3529	3533	3539	3541	3547	3557	3559	3571

.

The Goldbach conjecture

Goldbach's conjecture

Goldbach's conjecture is one of the oldest unsolved problems in number theory and in all of mathematics. It states:A Goldbach number is a number that can be expressed as the sum of two odd primes...

verification project reports that it has computed all primes below 10¹⁸. That means 24739954287740860 primes (roughly 2.5), 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) below 10²³. A different computation with a method assuming the Riemann hypothesis

Riemann hypothesis

In mathematics, the Riemann hypothesis, proposed by , is a conjecture about the location of the zeros of the Riemann zeta function which states that all non-trivial zeros have real part 1/2...

found that there are 18435599767349200867866 primes (roughly 2) below 10²⁴ if the Riemann hypothesis is true.

Lists of primes by type

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

Natural number

In mathematics, the natural numbers are the ordinary whole numbers used for counting and ordering . These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively...

(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

Partition of a set

In mathematics, a partition of a set X is a division of X into non-overlapping and non-empty "parts" or "blocks" or "cells" that cover all of X...

with n members.

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

Carol primes

Of the form

.

7, 47

47 (number)

47 is the natural number following 46 and preceding 48.-In mathematics:Forty-seven is the fifteenth prime number, a safe prime, the thirteenth supersingular prime, and the sixth Lucas prime. Forty-seven is a highly cototient number...

, 223

223 (number)

223 is the natural number between 222 and 224. It is also a prime number.-In mathematics:223 is a long prime, a strong prime, a lucky prime and a sexy prime .223 is the fourth Carol number and the third to be prime....

, 3967, 16127, 1046527, 16769023, 1073676287, 68718952447, 274876858367, 4398042316799, 1125899839733759, 18014398241046527, 1298074214633706835075030044377087

Centered decagonal
Centered decagonal number
A centered decagonal number is a centered figurate number that represents a decagon with a dot in the center and all other dots surrounding the center dot in successive decagonal layers...

primes

Of the form

.

11

11 (number)

11 is the natural number following 10 and preceding 12.Eleven is the first number which cannot be counted with a human's eight fingers and two thumbs additively. In English, it is the smallest positive integer requiring three syllables and the largest prime number with a single-morpheme name...

, 31

31 (number)

31 is the natural number following 30 and preceding 32.- In mathematics :Thirty-one is the third Mersenne prime as well as the fourth primorial prime, and together with twenty-nine, another primorial prime, it comprises a twin prime. As a Mersenne prime, 31 is related to the perfect number 496,...

, 61

61 (number)

61 is the natural number following 60 and preceding 62.-In mathematics:It is the 18th prime number. The previous is 59, with which it comprises a twin prime. Sixty-one is a cuban prime of the form p = / , x = y + 1....

, 101

101 (number)

101 is the natural number following 100 and preceding 102.It is variously pronounced "one hundred and one" / "a hundred and one", "one hundred one" / "a hundred one", and "one oh one"...

, 151

151 (number)

151 is a natural number. It follows 150 and precedes 152.-In mathematics:151 is the 36th prime number, the previous is 149, with which it comprises a twin prime. 151 is also a palindromic prime. 151 is a centered decagonal number...

, 211

211 (number)

211 is the natural number between 210 and 212. It is also a prime number.-In mathematics:211 is an odd number.211 is a primorial prime, sum of three consecutive primes , Chen prime, centered decagonal prime, and self prime....

, 281, 661, 911

911 (number)

911 is the number following 910 and preceding 912.It is a prime number, a Sophie Germain prime and the sum of three consecutive primes . It is an Eisenstein prime with no imaginary part and real part of the form 3n − 1. Since 913 is a semiprime, 911 is a Chen prime...

, 1051, 1201, 1361, 1531, 1901, 2311, 2531, 3001, 3251, 3511

3511 (number)

3511 is the natural number following 3510 and preceding 3512.3511 is a prime number, and is also an emirp.3511 is the largest known Wieferich prime – the only other being 1093....

, 4651, 5281, 6301, 6661, 7411, 9461, 9901, 12251, 13781, 14851, 15401, 18301, 18911, 19531, 20161, 22111, 24151, 24851, 25561, 27011, 27751

Centered heptagonal
Centered heptagonal number
A centered heptagonal number is a centered figurate number that represents a heptagon with a dot in the center and all other dots surrounding the center dot in successive heptagonal layers...

primes

Of the form (7n² − 7n + 2) / 2.

43

43 (number)

43 is the natural number following 42 and preceding 44.- In mathematics :Forty-three is the 14th smallest prime number. The previous is forty-one, with which it comprises a twin prime, and the next is forty-seven. 43 is the smallest prime that is not a Chen prime...

, 71

71 (number)

71 is the natural number following 70 and preceding 72.-In mathematics:71 is the algebraic degree of Conway's constant, a remarkable number arising in the study of look-and-say sequences....

, 197

197 (number)

197 is the natural number following 196 and preceding 198.-In mathematics:* 197 is an odd number* 197 is a prime number** 197 is a Chen prime** 197 is an Eisenstein prime with no imaginary part** 197 is a strong prime** 197 is a twin prime with 199...

, 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
Centered square number
In elementary number theory, a centered square number is a centered figurate number that gives the number of dots in a square with a dot in the center and all other dots surrounding the center dot in successive square layers. That is, each centered square number equals the number of dots within a...

primes

Of the form

.

5, 13

13 (number)

13 is the natural number after 12 and before 14. It is the smallest number with eight letters in its name spelled out in English. It is also the first of the teens – the numbers 13 through 19 – the ages of teenagers....

, 41

41 (number)

41 is the natural number following 40 and preceding 42.-In mathematics:Forty-one is the 13th smallest prime number. The next is forty-three, with which it comprises a twin prime...

, 61

61 (number)

, 113

113 (number)

113 is the natural number following 112 and preceding 114.-In mathematics:One hundred [and] thirteen is the 30th prime number, following 109 and preceding 127, a Sophie Germain prime, a Chen prime and a Proth prime as it is a prime number of the form 7 × 24 + 1...

, 181

181 (number)

181 is the natural number between 180 and 182.-In mathematics:* 181 is an odd number* 181 is a centered number** 181 is a centered pentagonal number** 181 is a centered 12-gonal number** 181 is a centered 18-gonal number...

, 313

313 (number)

313 is an integer following 312 and preceding 314. 313 is a:* prime number* twin prime with 311* centered square number* long prime* pythagorean prime* regular prime* palindromic prime in both decimal and binary.* truncatable prime* happy number...

, 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
Centered triangular number
A centered triangular number is a centered figurate number that represents a triangle with a dot in the center and all other dots surrounding the center in successive triangular layers...

primes

Of the form (3n² + 3n + 2) / 2.

19

19 (number)

19 is the natural number following 18 and preceding 20. It is a prime number.In English speech, the numbers 19 and 90 are often confused. When carefully enunciated, they differ in which syllable is stressed: 19 vs 90...

, 31

31 (number)

, 109

109 (number)

109 is the natural number following 108 and preceding 110.-In mathematics:One hundred [and] nine is the 29th prime number, and also a Chen prime. The previous prime is 107, making them both twin primes...

, 199

199 (number)

199 is the natural number following 198 and preceding 200.-In mathematics:* 199 is an odd number* 199 is a centered triangular number* 199 is a centered 33-gonal number* 199 is a deficient number, as 1 is less than 199* 199 is a Lucas number...

, 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 prime
Chen prime
A prime number p is called a Chen prime if p + 2 is either a prime or a product of two primes. The even number 2p + 2 therefore satisfies Chen's theorem....

s

p is prime and p + 2 is either a prime or semiprime

Semiprime

In mathematics, a semiprime is a natural number that is the product of two prime numbers. The first few semiprimes are 4, 6, 9, 10, 14, 15, 21, 22, 25, 26, ... ....

.

2, 3, 5, 7, 11

11 (number)

, 13

13 (number)

, 17

17 (number)

17 is the natural number following 16 and preceding 18. It is prime.In spoken English, the numbers 17 and 70 are sometimes confused because they sound similar. When carefully enunciated, they differ in which syllable is stressed: 17 vs 70...

, 19

19 (number)

, 23

23 (number)

23 is the natural number following 22 and preceding 24.- In mathematics :Twenty-three is the ninth prime number, the smallest odd prime that is not a twin prime. Twenty-three is also the fifth factorial prime, the third Woodall prime...

, 29

29 (number)

29 is the natural number following 28 and preceding 30.-In mathematics:It is the tenth prime number, and also the fourth primorial prime. It forms a twin prime pair with thirty-one, which is also a primorial prime. Twenty-nine is also the sixth Sophie Germain prime. It is also the sum of three...

, 31

31 (number)

, 37

37 (number)

37 is the natural number following 36 and preceding 38.-In mathematics:It is a prime number, the fifth lucky prime, the first irregular prime, the third unique prime and the third cuban prime of the form...

, 41

41 (number)

41 is the natural number following 40 and preceding 42.-In mathematics:Forty-one is the 13th smallest prime number. The next is forty-three, with which it comprises a twin prime...

, 47

47 (number)

, 53

53 (number)

53 is the natural number following 52 and preceding 54.-In mathematics:Fifty-three is the 16th prime number. It is also an Eisenstein prime....

, 59

59 (number)

59 is the natural number following 58 and preceding 60.-In mathematics:Fifty-nine is the 17th smallest prime number. The next is sixty-one, with which it comprises a twin prime. 59 is an irregular prime, a safe prime and the 14th supersingular prime. It is an Eisenstein prime with no imaginary...

, 67

67 (number)

67 is the natural number following 66 and preceding 68. It is an odd number.-In mathematics:Sixty-seven is the 19th prime number , an irregular prime, a lucky prime, the sum of five consecutive primes , and a Heegner number.Since 18! + 1 is divisible by 67 but 67 is not one more than a multiple of...

, 71

71 (number)

71 is the natural number following 70 and preceding 72.-In mathematics:71 is the algebraic degree of Conway's constant, a remarkable number arising in the study of look-and-say sequences....

, 83

83 (number)

83 is the natural number following 82 and preceding 84.-In mathematics:Eighty-three is the sum of three consecutive primes as well as the sum of five consecutive primes ....

, 89, 101

101 (number)

101 is the natural number following 100 and preceding 102.It is variously pronounced "one hundred and one" / "a hundred and one", "one hundred one" / "a hundred one", and "one oh one"...

, 107

107 (number)

107 is the natural number following 106 and preceding 108.-In mathematics:One hundred [and] seven is the 28th prime number. The next prime is 109, with which it comprises a twin prime, making 107 a Chen prime....

, 109

109 (number)

, 113

113 (number)

, 127

127 (number)

127 is the natural number following 126 and preceding 128.- In mathematics :*As a Mersenne prime, 127 is related to the perfect number 8128. 127 is also an exponent for the Mersenne prime 2127 - 1, making 127 a double Mersenne prime...

, 131

131 (number)

131 is the natural number following 130 and preceding 132.-In mathematics:131 is a Sophie Germain prime, the second 3-digit palindromic prime, and also a permutable prime with 113 and 311. It can be expressed as the sum of three consecutive primes, 131 = 41 + 43 + 47. 131 is an Eisenstein prime...

, 137

137 (number)

137 is the natural number following 136 and preceding 138.-In mathematics :One hundred [and] thirty-seven is the 33rd prime number; the next is 139, with which it comprises a twin prime, and thus 137 is a Chen prime. 137 is an Eisenstein prime with no imaginary part and a real part of the form 3n -...

, 139

139 (number)

139 is the natural number following 138 and preceding 140.-In mathematics:One hundred [and] thirty-nine is the 34th prime number, so it is divisible only by itself and 1. It is a twin prime with 137. Because 141 is a semiprime, 139 is a Chen prime...

, 149

149 (number)

149 is the natural number between 148 and 150. It is also a prime number.-In mathematics:*149 is the 35th prime number, and with the next prime number, 151, is a twin prime, thus 149 is a Chen prime. 149 is a strong prime in the sense that it is more than the arithmetic mean of its two neighboring...

, 157

157 (number)

157 is the number equal to 100 + 50 + 7, following 156 and preceding 158.-In mathematics:* 157 is a prime number. The next prime is 163 and the previous prime is 151, which with 157 form a sexy prime triplet. Taking the arithmetic mean of those primes yields 157, thus it is a balanced prime.* In...

, 167

167 (number)

167 is the natural number following 166 and preceding 168.-In mathematics:* 167 is an odd number* 167 is a Chen prime, since the next odd number, 169, is a square of a prime...

, 179

179 (number)

179 is the natural number following 178 and preceding 180.-In mathematics:* 179 is an odd number* 179 is a deficient number, as 1 is less than 179* 179 is a Gaussian number* 179 is an odious number* 179 is a square-free number...

, 181

181 (number)

, 191

191 (number)

191 is the natural number following 190 and preceding 192.-In mathematics:* 191 is an odd number* 191 is a centered 19-gonal number* 191 is a deficient number, as 1 is less than 191...

, 197

197 (number)

, 199

199 (number)

, 211

211 (number)

, 227

227 (number)

227 is the natural number between 226 and 228. It is also a prime number.-In mathematics:227 is a prime number, and a twin prime with 229 . 223 plus 4 is 227, so they are cousin primes...

, 233

233 (number)

233 is the natural number between 232 and 234. It is also a prime number.-In mathematics:233 is an irregular prime, a full reptend prime, a cousin prime, a Chen prime, a Fibonacci prime and a sexy prime. It is the 13th Fibonacci number. It is an Eisenstein prime of the form 3n - 1 with no...

, 239

239 (number)

239 is the natural number following 238 and preceding 240.-In mathematics:It is a prime number. The next is 241, with which it forms a pair of twin primes. 239 is a Sophie Germain prime and a Newman–Shanks–Williams prime...

, 251

251 (number)

251 is the natural number between 250 and 252. It is also a prime number.-In mathematics:251 is a regular prime, an Eisenstein prime, a Chen prime, a Gaussian prime, a Sophie Germain prime, and a sexy prime....

, 257, 263

263 (number)

263 is the natural number between 262 and 264. It is also a prime number.-In mathematics:263 is an irregular prime, an Eisenstein prime, a long prime, a Chen prime, a Gaussian prime, a happy prime, a sexy prime, a safe prime, and a Higgs prime....

, 269

269 (number)

269 is the natural number between 268 and 270. It is also a prime number.-In mathematics:269 is a regular prime, an Eisenstein prime with no imaginary part, a long prime, a Chen prime, a Pillai prime, a Pythagorean prime, a twin prime, a sexy prime, a Higgs prime, a strong prime, and a highly...

, 281, 293, 307, 311

311 (number)

311 is the natural number after 310 and before 312.-In mathematics:311 is a twin prime with 313; an Eisenstein prime with no imaginary part and real part of the form 3n - 1; a Gaussian prime with no imaginary part and real part of the form 4n - 1; and a permutable prime with 113 and 131.It can be...

, 317, 337, 347, 353

353 (number)

353 is the natural number between 352 and 354. It is a prime number.-In mathematics:353 is a palindromic prime, a cousin prime with 349, and a sexy prime with both 347 and 359. It is the first multidigit palindromic prime with all prime digits. Also, it is the smallest number whose 4th power is...

, 359

359 (number)

Three hundred and fifty-nine is the number directly following 358 and directly preceding 360. 359 is a prime number.-359 in mathematics:359° is one degree less than a full revolution ; this makes it a reflex angle....

, 379, 389, 401, 409

Circular prime
Circular prime
A circular prime is a prime number with the property that the number generated at each intermediate step when cyclically permuting its digits will be prime. For example 1193 is a circular prime, since 1931, 9311 and 3119 all are also prime...

s

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

11 (number)

, 13

13 (number)

, 17

17 (number)

, 31

31 (number)

, 37

37 (number)

, 71

71 (number)

71 is the natural number following 70 and preceding 72.-In mathematics:71 is the algebraic degree of Conway's constant, a remarkable number arising in the study of look-and-say sequences....

, 73

73 (number)

73 is the natural number following 72 and preceding 74. In English, it is the smallest integer with twelve letters in its spelled out name.- In mathematics :...

, 79

79 (number)

Seventy-nine is the natural number following 78 and preceding 80.79 may represent:-In mathematics:*An odd number*The smallest number that can't be represented as a sum of fewer than 19 fourth powers*A strictly non-palindromic number...

, 97

97 (number)

97 is the natural number following 96 and preceding 98.-In mathematics:97 is the 25th prime number , following 89 and preceding 101. 97 is a Proth prime as it is 3 × 25 + 1.The numbers 97, 907, 9007, 90007 and 900007 are happy primes...

, 113

113 (number)

, 131

131 (number)

, 197

197 (number)

, 199

199 (number)

, 311

311 (number)

, 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

Repunit

In recreational mathematics, a repunit is a number like 11, 111, or 1111 that contains only the digit 1. The term stands for repeated unit and was coined in 1966 by Albert H. Beiler...

primes are circular.

Cousin prime
Cousin prime
In mathematics, cousin primes are prime numbers that differ by four; compare this with twin primes, pairs of prime numbers that differ by two, and sexy primes, pairs of prime numbers that differ by six....

s

(p, p + 4) are both prime.

(3, 7), (7, 11

11 (number)

), (13

13 (number)

, 17

17 (number)

), (19

19 (number)

, 23

23 (number)

), (37

37 (number)

, 41

41 (number)

41 is the natural number following 40 and preceding 42.-In mathematics:Forty-one is the 13th smallest prime number. The next is forty-three, with which it comprises a twin prime...

), (43

43 (number)

, 47

47 (number)

), (67

67 (number)

, 71

71 (number)

71 is the natural number following 70 and preceding 72.-In mathematics:71 is the algebraic degree of Conway's constant, a remarkable number arising in the study of look-and-say sequences....

), (79

79 (number)

, 83

83 (number)

83 is the natural number following 82 and preceding 84.-In mathematics:Eighty-three is the sum of three consecutive primes as well as the sum of five consecutive primes ....

), (97

97 (number)

, 101

101 (number)

101 is the natural number following 100 and preceding 102.It is variously pronounced "one hundred and one" / "a hundred and one", "one hundred one" / "a hundred one", and "one oh one"...

), (103

103 (number)

103 is the natural number following 102 and preceding 104.-In mathematics:One hundred [and] three is the 27th prime number. The previous prime is 101, making them both twin primes...

, 107

107 (number)

), (109

109 (number)

, 113

113 (number)

), (127

127 (number)

, 131

131 (number)

), (163

163 (number)

163 is the natural number following 162 and preceding 164.-In mathematics:163 is a strong prime in the sense that it is greater than the arithmetic mean of its two neighboring primes...

, 167

167 (number)

167 is the natural number following 166 and preceding 168.-In mathematics:* 167 is an odd number* 167 is a Chen prime, since the next odd number, 169, is a square of a prime...

), (193

193 (number)

193 is the natural number following 192 and preceding 194.-In mathematics:* 193 is an odd number* 193 is a centered 32-gonal number* 193 is a deficient number, as 1 is less than 193* 193 is a happy number* 193 is a lucky number...

, 197

197 (number)

), (223

223 (number)

, 227

227 (number)

227 is the natural number between 226 and 228. It is also a prime number.-In mathematics:227 is a prime number, and a twin prime with 229 . 223 plus 4 is 227, so they are cousin primes...

), (229

229 (number)

229 is the natural number between 228 and 230. It is also a prime number.The North American telephone area code 229 is assigned to the area around the city of Albany in southwestern Georgia in North America....

, 233

233 (number)

), (277

277 (number)

277 is the natural number following 276 and preceding 278.-In mathematics:277 is the fifty-ninth prime number, following 271 and preceding 281...

, 281)

Cuban primes

Of the form

7, 19

19 (number)

, 37

37 (number)

, 61

61 (number)

, 127

127 (number)

, 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

13 (number)

, 109

109 (number)

, 193

193 (number)

, 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
Cullen number
In mathematics, a Cullen number is a natural number of the form n · 2n + 1 . Cullen numbers were first studied by Fr. James Cullen in 1905...

primes

Of the form n · 2ⁿ + 1.

3, 393050634124102232869567034555427371542904833

Dihedral prime
Dihedral prime
A dihedral prime or dihedral calculator prime is a prime number that still reads like itself or another prime number when read in a seven-segment display, regardless of orientation , and surface...

s

Primes that remain prime when read upside down or mirrored in a seven-segment display

Seven-segment display

A seven-segment display , or seven-segment indicator, is a form of electronic display device for displaying decimal numerals that is an alternative to the more complex dot-matrix displays...

.

2, 5, 11

11 (number)

, 101

101 (number)

101 is the natural number following 100 and preceding 102.It is variously pronounced "one hundred and one" / "a hundred and one", "one hundred one" / "a hundred one", and "one oh one"...

, 181

181 (number)

, 1181, 1811, 18181, 108881, 110881, 118081, 120121, 121021, 121151, 150151, 151051, 151121, 180181, 180811, 181081

Double factorial primes

Of the form

. 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

. Values of n:

3, 4, 6, 8, 16

16 (number)

16 is the natural number following 15 and preceding 17. 16 is a composite number, and a square number, being 42 = 4 × 4. It is the smallest number with exactly five divisors, its proper divisors being , , and ....

, 26

26 (number)

26 is the natural number following 25 and preceding 27.- In mathematics :26 is the only positive integer that is one greater than a square and one less than a cube .A rhombicuboctahedron has twenty-six sides....

, 64

64 (number)

64 is the natural number following 63 and preceding 65.-In mathematics:Sixty-four is the square of 8, the cube of 4, and the sixth power of 2. It is the smallest number with exactly seven divisors. It is the lowest positive power of two that is adjacent to neither a Mersenne prime nor a Fermat...

, 82

82 (number)

82 is the natural number following 81 and preceding 83.-In mathematics:Eighty-two is the twenty-third biprime and the twelfth of the form...

, 90

90 (number)

90 is the natural number preceded by 89 and followed by 91.- In mathematics :*Because 90 is the sum of its unitary divisors , it is a unitary perfect number, and because it is equal to the sum of a subset of its divisors, it is a semiperfect number. 90 is a pronic number. But it is also a...

, 118

118 (number)

118 is the natural number following 117 and preceding 119.-In mathematics:There is no answer to the equation φ = 118, making 118 a nontotient.118 is the smallest n such that the range n, n + 1, .....

, 194

194 (number)

194 is the natural number following 193 and preceding 195.-In mathematics:* 194 is an even number* 194 is a composite number* 194 is a deficient number, as 100 is less than 194...

, 214

214 (number)

214 is the natural number following 213 and preceding 215.-In mathematics:*214 is a composite number and a 37-gonal number.*214!! − 1 is a 205-digit prime number.*The 11th perfect number 2106× has 214 divisors....

, 728, 842, 888, 2328, 3326, 6404, 8670, 9682, 27056, 44318

Double Mersenne primes

A subset of Mersenne primes: of the form

for prime p.

7, 127

127 (number)

, 2147483647

2147483647

The number 2,147,483,647 is the eighth Mersenne prime, equal to 231 − 1. It is one of only four known double Mersenne primes....

, 170141183460469231731687303715884105727 (primes in )

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

Eisenstein prime
Eisenstein prime
In mathematics, an Eisenstein prime is an Eisenstein integerz = a + b\,\omega\qquadthat is irreducible in the ring-theoretic sense: its only Eisenstein divisors are the units , a + bω itself and its associates.The associates and the complex conjugate...

s without imaginary part

Eisenstein integer

In mathematics, Eisenstein integers , also known as Eulerian integers , are complex numbers of the formz = a + b\omega \,\!where a and b are integers and...

s that are irreducible

Irreducible element

In abstract algebra, a non-zero non-unit element in an integral domain is said to be irreducible if it is not a product of two non-units.Irreducible elements should not be confused with prime elements...

and real numbers (primes of form 3n − 1).

2, 5, 11

11 (number)

, 17

17 (number)

, 23

23 (number)

, 29

29 (number)

, 41

41 (number)

41 is the natural number following 40 and preceding 42.-In mathematics:Forty-one is the 13th smallest prime number. The next is forty-three, with which it comprises a twin prime...

, 47

47 (number)

, 53

53 (number)

53 is the natural number following 52 and preceding 54.-In mathematics:Fifty-three is the 16th prime number. It is also an Eisenstein prime....

, 59

59 (number)

, 71

71 (number)

71 is the natural number following 70 and preceding 72.-In mathematics:71 is the algebraic degree of Conway's constant, a remarkable number arising in the study of look-and-say sequences....

, 83

83 (number)

83 is the natural number following 82 and preceding 84.-In mathematics:Eighty-three is the sum of three consecutive primes as well as the sum of five consecutive primes ....

, 89, 101

101 (number)

101 is the natural number following 100 and preceding 102.It is variously pronounced "one hundred and one" / "a hundred and one", "one hundred one" / "a hundred one", and "one oh one"...

, 107

107 (number)

, 113

113 (number)

, 131

131 (number)

, 137

137 (number)

, 149

149 (number)

, 167

167 (number)

167 is the natural number following 166 and preceding 168.-In mathematics:* 167 is an odd number* 167 is a Chen prime, since the next odd number, 169, is a square of a prime...

, 173

173 (number)

173 is the natural number following 172 and preceding 174.-In mathematics:* 173 is an odd number* 173 is a balanced prime* 173 is a deficient number* 173 is an Eisenstein prime with no imaginary part* 173 is an Odious number* 173 is a prime number...

, 179

179 (number)

, 191

191 (number)

191 is the natural number following 190 and preceding 192.-In mathematics:* 191 is an odd number* 191 is a centered 19-gonal number* 191 is a deficient number, as 1 is less than 191...

, 197

197 (number)

, 227

227 (number)

227 is the natural number between 226 and 228. It is also a prime number.-In mathematics:227 is a prime number, and a twin prime with 229 . 223 plus 4 is 227, so they are cousin primes...

, 233

233 (number)

, 239

239 (number)

, 251

251 (number)

, 257, 263

263 (number)

, 269

269 (number)

, 281, 293, 311

311 (number)

, 317, 347, 353

353 (number)

, 359

359 (number)

, 383, 389, 401

Emirp
Emirp
An emirp is a prime number that results in a different prime when its digits are reversed. This definition excludes the related palindromic primes. Emirps are also called reversible primes....

s

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

13

13 (number)

, 17

17 (number)

, 31

31 (number)

, 37

37 (number)

, 71

71 (number)

71 is the natural number following 70 and preceding 72.-In mathematics:71 is the algebraic degree of Conway's constant, a remarkable number arising in the study of look-and-say sequences....

, 73

73 (number)

73 is the natural number following 72 and preceding 74. In English, it is the smallest integer with twelve letters in its spelled out name.- In mathematics :...

, 79

79 (number)

, 97

97 (number)

, 107

107 (number)

, 113

113 (number)

, 149

149 (number)

, 157

157 (number)

, 167

167 (number)

167 is the natural number following 166 and preceding 168.-In mathematics:* 167 is an odd number* 167 is a Chen prime, since the next odd number, 169, is a square of a prime...

, 179

179 (number)

, 199

199 (number)

, 311

311 (number)

, 337, 347, 359

359 (number)

, 389, 701, 709, 733, 739, 743

743 (number)

743 is the natural number following 742 and preceding 744. It is a prime number.-In mathematics:743 is a Sophie Germain prime number, as well as a cousin prime....

, 751, 761, 769, 907, 937, 941, 953, 967, 971, 983, 991

Euclid
Euclid number
In mathematics, Euclid numbers are integers of the form En = pn# + 1, where pn# is the primorial of pn which is the nth prime. They are named after the ancient Greek mathematician Euclid....

primes

Of the form p_n#

Primorial

In mathematics, and more particularly in number theory, primorial is a function from natural numbers to natural numbers similar to the factorial function, but rather than multiplying successive positive integers, only successive prime numbers are multiplied...

+ 1 (a subset of primorial primes).

3, 7, 31

31 (number)

, 211

211 (number)

, 2311, 200560490131

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

Even and odd numbers

In mathematics, the parity of an object states whether it is even or odd.This concept begins with integers. An even number is an integer that is "evenly divisible" by 2, i.e., divisible by 2 without remainder; an odd number is an integer that is not evenly divisible by 2...

Factorial prime
Factorial prime
A factorial prime is a prime number that is one less or one more than a factorial . The first few factorial primes are:n! − 1 is prime for :n! + 1 is prime for :...

s

Of the form n!

Factorial

In mathematics, the factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n...

− 1 or n! + 1.

2, 3, 5, 7, 23

23 (number)

, 719, 5039, 39916801, 479001599, 87178291199, 10888869450418352160768000001, 265252859812191058636308479999999, 263130836933693530167218012159999999, 8683317618811886495518194401279999999

Fermat primes

Of the form

.

3, 5, 17

17 (number)

, 257, 65537

these are the only known Fermat primes, and conjecturally the only Fermat primes.

Fibonacci prime
Fibonacci prime
A Fibonacci prime is a Fibonacci number that is prime, a type of integer sequence prime.The first Fibonacci primes are :-Known Fibonacci primes:It is not known if there are infinitely many Fibonacci primes...

s

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

2, 3, 5, 13

13 (number)

, 89, 233

233 (number)

, 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

13 (number)

, 17

17 (number)

, 19

19 (number)

, 23

23 (number)

, 37

37 (number)

, 47

47 (number)

, 59

59 (number)

, 61

61 (number)

, 67

67 (number)

, 71

71 (number)

71 is the natural number following 70 and preceding 72.-In mathematics:71 is the algebraic degree of Conway's constant, a remarkable number arising in the study of look-and-say sequences....

, 79

79 (number)

, 89, 101

101 (number)

101 is the natural number following 100 and preceding 102.It is variously pronounced "one hundred and one" / "a hundred and one", "one hundred one" / "a hundred one", and "one oh one"...

, 103

103 (number)

103 is the natural number following 102 and preceding 104.-In mathematics:One hundred [and] three is the 27th prime number. The previous prime is 101, making them both twin primes...

, 107

107 (number)

, 109

109 (number)

, 127

127 (number)

, 151

151 (number)

, 157

157 (number)

, 163

163 (number)

163 is the natural number following 162 and preceding 164.-In mathematics:163 is a strong prime in the sense that it is greater than the arithmetic mean of its two neighboring primes...

, 167

167 (number)

167 is the natural number following 166 and preceding 168.-In mathematics:* 167 is an odd number* 167 is a Chen prime, since the next odd number, 169, is a square of a prime...

, 191

191 (number)

191 is the natural number following 190 and preceding 192.-In mathematics:* 191 is an odd number* 191 is a centered 19-gonal number* 191 is a deficient number, as 1 is less than 191...

, 197

197 (number)

, 199

199 (number)

, 223

223 (number)

, 229

229 (number)

, 233

233 (number)

, 239

239 (number)

, 271, 277

277 (number)

277 is the natural number following 276 and preceding 278.-In mathematics:277 is the fifty-ninth prime number, following 271 and preceding 281...

, 283, 293, 307, 311

311 (number)

, 313

313 (number)

, 331, 353

353 (number)

, 373, 379, 383, 397

Gaussian prime
Gaussian integer
In number theory, a Gaussian integer is a complex number whose real and imaginary part are both integers. The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain, usually written as Z[i]. The Gaussian integers are a special case of the quadratic...

s

Prime element

In abstract algebra, an element p of a commutative ring R is said to be prime if it is not zero, not a unit and whenever p divides ab for some a and b in R, then p divides a or p divides b...

s of the Gaussian integers (primes of form 4n + 3).

3, 7, 11

11 (number)

, 19

19 (number)

, 23

23 (number)

, 31

31 (number)

, 43

43 (number)

, 47

47 (number)

, 59

59 (number)

, 67

67 (number)

, 71

71 (number)

71 is the natural number following 70 and preceding 72.-In mathematics:71 is the algebraic degree of Conway's constant, a remarkable number arising in the study of look-and-say sequences....

, 79

79 (number)

, 83

83 (number)

83 is the natural number following 82 and preceding 84.-In mathematics:Eighty-three is the sum of three consecutive primes as well as the sum of five consecutive primes ....

, 103

103 (number)

103 is the natural number following 102 and preceding 104.-In mathematics:One hundred [and] three is the 27th prime number. The previous prime is 101, making them both twin primes...

, 107

107 (number)

, 127

127 (number)

, 131

131 (number)

, 139

139 (number)

, 151

151 (number)

, 163

163 (number)

163 is the natural number following 162 and preceding 164.-In mathematics:163 is a strong prime in the sense that it is greater than the arithmetic mean of its two neighboring primes...

, 167

167 (number)

167 is the natural number following 166 and preceding 168.-In mathematics:* 167 is an odd number* 167 is a Chen prime, since the next odd number, 169, is a square of a prime...

, 179

179 (number)

, 191

191 (number)

191 is the natural number following 190 and preceding 192.-In mathematics:* 191 is an odd number* 191 is a centered 19-gonal number* 191 is a deficient number, as 1 is less than 191...

, 199

199 (number)

, 211

211 (number)

, 223

223 (number)

, 227

227 (number)

227 is the natural number between 226 and 228. It is also a prime number.-In mathematics:227 is a prime number, and a twin prime with 229 . 223 plus 4 is 227, so they are cousin primes...

, 239

239 (number)

, 251

251 (number)

, 263

263 (number)

, 271, 283, 307, 311

311 (number)

, 331, 347, 359

359 (number)

, 367, 379, 383, 419

419 (number)

419 may refer to:*419, the year 419 C.E.*419 *Advance-fee fraud, also known as the 419 scam, the Nigerian scam, and the Nigerian money transfer fraud*Area code 419, which is the area code of the northwestern corner of Ohio...

, 431, 439, 443, 463, 467, 479, 487, 491, 499, 503

Generalized Fermat primes base 10

Of the form 10ⁿ + 1, where n > 0.

11

11 (number)

, 101

101 (number)

101 is the natural number following 100 and preceding 102.It is variously pronounced "one hundred and one" / "a hundred and one", "one hundred one" / "a hundred one", and "one oh one"...

, these are the only known generalized Fermat primes in base 10.

Genocchi number primes

17 (number)

The only positive prime Genocchi number is 17.

Gilda's primes

Gilda's numbers that are prime.

29

29 (number)

, 683, 997, 2207, 30571351

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

11 (number)

, 17

17 (number)

, 29

29 (number)

, 37

37 (number)

, 41

41 (number)

41 is the natural number following 40 and preceding 42.-In mathematics:Forty-one is the 13th smallest prime number. The next is forty-three, with which it comprises a twin prime...

, 53

53 (number)

53 is the natural number following 52 and preceding 54.-In mathematics:Fifty-three is the 16th prime number. It is also an Eisenstein prime....

, 59

59 (number)

, 67

67 (number)

, 71

71 (number)

71 is the natural number following 70 and preceding 72.-In mathematics:71 is the algebraic degree of Conway's constant, a remarkable number arising in the study of look-and-say sequences....

, 97

97 (number)

, 101

101 (number)

101 is the natural number following 100 and preceding 102.It is variously pronounced "one hundred and one" / "a hundred and one", "one hundred one" / "a hundred one", and "one oh one"...

, 127

127 (number)

, 149

149 (number)

, 179

179 (number)

, 191

191 (number)

191 is the natural number following 190 and preceding 192.-In mathematics:* 191 is an odd number* 191 is a centered 19-gonal number* 191 is a deficient number, as 1 is less than 191...

, 223

223 (number)

, 227

227 (number)

227 is the natural number between 226 and 228. It is also a prime number.-In mathematics:227 is a prime number, and a twin prime with 229 . 223 plus 4 is 227, so they are cousin primes...

, 251

251 (number)

, 257, 269

269 (number)

, 307

Happy primes
Happy number
A happy number is defined by the following process. Starting with any positive integer, replace the number by the sum of the squares of its digits, and repeat the process until the number equals 1 , or it loops endlessly in a cycle which does not include 1...

Happy numbers that are prime.

7, 13

13 (number)

, 19

19 (number)

, 23

23 (number)

, 31

31 (number)

, 79

79 (number)

, 97

97 (number)

, 103

103 (number)

103 is the natural number following 102 and preceding 104.-In mathematics:One hundred [and] three is the 27th prime number. The previous prime is 101, making them both twin primes...

, 109

109 (number)

, 139

139 (number)

, 167

167 (number)

167 is the natural number following 166 and preceding 168.-In mathematics:* 167 is an odd number* 167 is a Chen prime, since the next odd number, 169, is a square of a prime...

, 193

193 (number)

, 239

239 (number)

, 263

263 (number)

, 293, 313

313 (number)

, 331, 367, 379, 383, 397, 409, 487, 563, 617, 653, 673, 683, 709, 739, 761, 863, 881

881 (number)

The number sequence 881 is...* a Paid Toll Free telephone number prefix in the USA* the Port of Los Angeles Long Wharf, California State Historic Landmark #881* the year AD 881 or the year 881 BC...

, 907, 937, 1009, 1033, 1039, 1093

1093 (number)

1093 is the natural number following 1092 and preceding 1094.1093 is a prime number. Together with 1091 and 1097, it forms a prime triplet. It is a happy prime and a star prime. It is also the smallest Wieferich prime. 1093 is a repunit prime in base 3 because:1093 = 1111111_3 = 3^6 + 3^5 +...

Harmonic primes

Primes p for which there are no solutions to

and

for

.

5, 13

13 (number)

, 17

17 (number)

, 23

23 (number)

, 41

41 (number)

41 is the natural number following 40 and preceding 42.-In mathematics:Forty-one is the 13th smallest prime number. The next is forty-three, with which it comprises a twin prime...

, 67

67 (number)

, 73

73 (number)

73 is the natural number following 72 and preceding 74. In English, it is the smallest integer with twelve letters in its spelled out name.- In mathematics :...

, 79

79 (number)

, 107

107 (number)

, 113

113 (number)

, 139

139 (number)

, 149

149 (number)

, 157

157 (number)

, 179

179 (number)

, 191

191 (number)

191 is the natural number following 190 and preceding 192.-In mathematics:* 191 is an odd number* 191 is a centered 19-gonal number* 191 is a deficient number, as 1 is less than 191...

, 193

193 (number)

, 223

223 (number)

, 239

239 (number)

, 241

241 (number)

241 is the natural number between 240 and 242. It is also a prime number.-In mathematics:241 is the larger of the twin primes . Prime twins are always of the form , with n = 1, 2, 3, ......

, 251

251 (number)

, 263

263 (number)

, 277

277 (number)

277 is the natural number following 276 and preceding 278.-In mathematics:277 is the fifty-ninth prime number, following 271 and preceding 281...

, 281, 293, 307, 311

311 (number)

, 317, 331, 337, 349

Higgs prime
Higgs prime
A Higgs prime is a prime number with a totient that evenly divides the square of the product of the smaller Higgs primes...

s for squares

Primes p for which p − 1 divides the square of the product of all earlier terms.

2, 3, 5, 7, 11

11 (number)

, 13

13 (number)

, 19

19 (number)

, 23

23 (number)

, 29

29 (number)

, 31

31 (number)

, 37

37 (number)

, 43

43 (number)

, 47

47 (number)

, 53

53 (number)

53 is the natural number following 52 and preceding 54.-In mathematics:Fifty-three is the 16th prime number. It is also an Eisenstein prime....

, 59

59 (number)

, 61

61 (number)

, 67

67 (number)

, 71

71 (number)

71 is the natural number following 70 and preceding 72.-In mathematics:71 is the algebraic degree of Conway's constant, a remarkable number arising in the study of look-and-say sequences....

, 79

79 (number)

, 101

101 (number)

101 is the natural number following 100 and preceding 102.It is variously pronounced "one hundred and one" / "a hundred and one", "one hundred one" / "a hundred one", and "one oh one"...

, 107

107 (number)

, 127

127 (number)

, 131

131 (number)

, 139

139 (number)

, 149

149 (number)

, 151

151 (number)

, 157

157 (number)

, 173

173 (number)

, 181

181 (number)

, 191

191 (number)

191 is the natural number following 190 and preceding 192.-In mathematics:* 191 is an odd number* 191 is a centered 19-gonal number* 191 is a deficient number, as 1 is less than 191...

, 197

197 (number)

, 199

199 (number)

, 211

211 (number)

, 223

223 (number)

, 229

229 (number)

, 263

263 (number)

, 269

269 (number)

, 277

277 (number)

277 is the natural number following 276 and preceding 278.-In mathematics:277 is the fifty-ninth prime number, following 271 and preceding 281...

, 283, 311

311 (number)

, 317, 331, 347, 349

Highly cototient number
Highly cototient number
In number theory, a branch of mathematics, a highly cototient number is a positive integer k which is above one and has more solutions to the equation...

primes

Primes that are a cototient more often than any integer below it except 1.

2, 23

23 (number)

, 47

47 (number)

, 59

59 (number)

, 83

83 (number)

83 is the natural number following 82 and preceding 84.-In mathematics:Eighty-three is the sum of three consecutive primes as well as the sum of five consecutive primes ....

, 89, 113

113 (number)

, 167

167 (number)

167 is the natural number following 166 and preceding 168.-In mathematics:* 167 is an odd number* 167 is a Chen prime, since the next odd number, 169, is a square of a prime...

, 269

269 (number)

, 389, 419

419 (number)

, 509, 659, 839, 1049, 1259, 1889

Irregular primes

Odd primes p which divide the class number of the p-th cyclotomic field

Cyclotomic field

In number theory, a cyclotomic field is a number field obtained by adjoining a complex primitive root of unity to Q, the field of rational numbers...

.

37

37 (number)

, 59

59 (number)

, 67

67 (number)

, 101

101 (number)

101 is the natural number following 100 and preceding 102.It is variously pronounced "one hundred and one" / "a hundred and one", "one hundred one" / "a hundred one", and "one oh one"...

, 103

103 (number)

103 is the natural number following 102 and preceding 104.-In mathematics:One hundred [and] three is the 27th prime number. The previous prime is 101, making them both twin primes...

, 131

131 (number)

, 149

149 (number)

, 157

157 (number)

, 233

233 (number)

, 257, 263

263 (number)

, 271, 283, 293, 307, 311

311 (number)

, 347, 353

353 (number)

, 379, 389, 401, 409, 421, 433, 461, 463, 467, 491, 523, 541, 547, 557, 577, 587, 593

593 (number)

593 is the natural number following 592 and preceding 594.-In mathematics:593 is an odd number. It is a prime number, an example of what Paul Erdős and Ernst G. Straus called a Good prime, or a prime whose square is greater than the product of its neighboring two primes. As such it is part of...

, 607, 613, 617, 619

Isolated primes
Twin prime
A twin prime is a prime number that differs from another prime number by two. Except for the pair , this is the smallest possible difference between two primes. Some examples of twin prime pairs are , , , , and...

Primes p such that neither p − 2 nor p + 2 is prime.

2, 23

23 (number)

, 37

37 (number)

, 47

47 (number)

, 53

53 (number)

53 is the natural number following 52 and preceding 54.-In mathematics:Fifty-three is the 16th prime number. It is also an Eisenstein prime....

, 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

.

7, 23

23 (number)

, 79

79 (number)

, 1087, 66047, 263167, 16785407, 1073807359, 17180131327, 68720001023, 4398050705407, 70368760954879, 18014398777917439, 18446744082299486207

Left-truncatable primes
Truncatable prime
In number theory, a left-truncatable prime is a prime number which, in a given base, contains no 0, and if the leading digit is successively removed, then all resulting numbers are prime. For example 9137, since 9137, 137, 37 and 7 are all prime...

Primes that remain prime when the leading decimal digit is successively removed.

2, 3, 5, 7, 13

13 (number)

, 17

17 (number)

, 23

23 (number)

, 37

37 (number)

, 43

43 (number)

, 47

47 (number)

, 53

53 (number)

53 is the natural number following 52 and preceding 54.-In mathematics:Fifty-three is the 16th prime number. It is also an Eisenstein prime....

, 67

67 (number)

, 73

73 (number)

73 is the natural number following 72 and preceding 74. In English, it is the smallest integer with twelve letters in its spelled out name.- In mathematics :...

, 83

83 (number)

83 is the natural number following 82 and preceding 84.-In mathematics:Eighty-three is the sum of three consecutive primes as well as the sum of five consecutive primes ....

, 97

97 (number)

, 113

113 (number)

, 137

137 (number)

, 167

167 (number)

167 is the natural number following 166 and preceding 168.-In mathematics:* 167 is an odd number* 167 is a Chen prime, since the next odd number, 169, is a square of a prime...

, 173

173 (number)

, 197

197 (number)

, 223

223 (number)

, 283, 313

313 (number)

, 317, 337, 347, 353

353 (number)

, 367, 373, 383, 397, 443, 467, 523, 547, 613, 617, 643, 647, 653, 673, 683

Leyland
Leyland number
In number theory, a Leyland number is a number of the form xy + yx, where x and y are integers greater than 1. The first few Leyland numbers are...

primes

Of the form x^y + y^x with 1 < x ≤ y.

17

17 (number)

, 593

593 (number)

, 32993, 2097593, 8589935681, 59604644783353249, 523347633027360537213687137, 43143988327398957279342419750374600193

Long primes

Primes p for which, in a given base b,

gives a cyclic number

Cyclic number

A cyclic number is an integer in which cyclic permutations of the digits are successive multiples of the number. The most widely known is 142857:For example:Multiples of these fractions exhibit cyclic permutation:...

. They are also called full reptend primes. Primes p for base 10:

7, 17

17 (number)

, 19

19 (number)

, 23

23 (number)

, 29

29 (number)

, 47

47 (number)

, 59

59 (number)

, 61

61 (number)

, 97

97 (number)

, 109

109 (number)

, 113

113 (number)

, 131

131 (number)

, 149

149 (number)

, 167

167 (number)

167 is the natural number following 166 and preceding 168.-In mathematics:* 167 is an odd number* 167 is a Chen prime, since the next odd number, 169, is a square of a prime...

, 179

179 (number)

, 181

181 (number)

, 193

193 (number)

, 223

223 (number)

, 229

229 (number)

, 233

233 (number)

, 257, 263

263 (number)

, 269

269 (number)

, 313

313 (number)

, 337, 367, 379, 383, 389, 419

419 (number)

, 433, 461, 487, 491, 499, 503, 509, 541, 571, 577, 593

593 (number)

Lucas primes
Lucas number
The Lucas numbers are an integer sequence named after the mathematician François Édouard Anatole Lucas , who studied both that sequence and the closely related Fibonacci numbers...

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

2, 3, 7, 11

11 (number)

, 29

29 (number)

, 47

47 (number)

, 199

199 (number)

, 521, 2207, 3571, 9349, 3010349, 54018521, 370248451, 6643838879, 119218851371, 5600748293801, 688846502588399, 32361122672259149

Lucky primes
Lucky number
In number theory, a lucky number is a natural number in a set which is generated by a "sieve" similar to the Sieve of Eratosthenes that generates the primes.Begin with a list of integers starting with 1:...

Lucky numbers that are prime.

3, 7, 13

13 (number)

, 31

31 (number)

, 37

37 (number)

, 43

43 (number)

, 67

67 (number)

, 73

73 (number)

73 is the natural number following 72 and preceding 74. In English, it is the smallest integer with twelve letters in its spelled out name.- In mathematics :...

, 79

79 (number)

, 127

127 (number)

, 151

151 (number)

, 163

163 (number)

163 is the natural number following 162 and preceding 164.-In mathematics:163 is a strong prime in the sense that it is greater than the arithmetic mean of its two neighboring primes...

, 193

193 (number)

, 211

211 (number)

, 223

223 (number)

, 241

241 (number)

241 is the natural number between 240 and 242. It is also a prime number.-In mathematics:241 is the larger of the twin primes . Prime twins are always of the form , with n = 1, 2, 3, ......

, 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
Markov number
A Markov number or Markoff number is a positive integer x, y or z that is part of a solution to the Markov Diophantine equationx^2 + y^2 + z^2 = 3xyz,\,studied by .The first few Markov numbers are...

primes

Primes p for which there exist integers x and y such that

.

2, 5, 13

13 (number)

, 29

29 (number)

, 89, 233

233 (number)

, 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 prime
Mersenne prime
In mathematics, a Mersenne number, named after Marin Mersenne , is a positive integer that is one less than a power of two: M_p=2^p-1.\,...

s

Of the form 2ⁿ − 1.

3, 7, 31

31 (number)

, 127

127 (number)

, 8191, 131071, 524287, 2147483647

2147483647

The number 2,147,483,647 is the eighth Mersenne prime, equal to 231 − 1. It is one of only four known double Mersenne primes....

, 2305843009213693951, 618970019642690137449562111, 162259276829213363391578010288127, 170141183460469231731687303715884105727

, 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

13 (number)

, 17

17 (number)

, 19

19 (number)

, 31

31 (number)

, 61

61 (number)

, 89, 107

107 (number)

, 127

127 (number)

, 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

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

2, 11

11 (number)

, 1361, 2521008887, 16022236204009818131831320183

Minimal primes
Minimal prime (number theory)
In recreational number theory, a minimal prime is a prime number for which there is no shorter subsequence of its digits in a given base that form a prime. In base 10 there are exactly 26 minimal primes:...

Primes for which there is no shorter sub-sequence

Subsequence

In mathematics, a subsequence is a sequence that can be derived from another sequence by deleting some elements without changing the order of the remaining elements...

of the decimal digits that form a prime. There are exactly 26 minimal primes:

2, 3, 5, 7, 11

11 (number)

, 19

19 (number)

, 41

41 (number)

41 is the natural number following 40 and preceding 42.-In mathematics:Forty-one is the 13th smallest prime number. The next is forty-three, with which it comprises a twin prime...

, 61

61 (number)

, 89, 409, 449, 499, 881

881 (number)

The number sequence 881 is...* a Paid Toll Free telephone number prefix in the USA* the Port of Los Angeles Long Wharf, California State Historic Landmark #881* the year AD 881 or the year 881 BC...

, 991, 6469, 6949, 9001, 9049, 9649, 9949, 60649, 666649, 946669, 60000049, 66000049, 66600049

Motzkin
Motzkin number
In mathematics, a Motzkin number for a given number n is the number of different ways of drawing non-intersecting chords on a circle between n points. The Motzkin numbers have very diverse applications in geometry, combinatorics and number theory...

primes

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

2, 127

127 (number)

, 15511, 953467954114363

Newman–Shanks–Williams primes

Newman–Shanks–Williams numbers that are prime.

7, 41

41 (number)

41 is the natural number following 40 and preceding 42.-In mathematics:Forty-one is the 13th smallest prime number. The next is forty-three, with which it comprises a twin prime...

, 239

239 (number)

, 9369319, 63018038201, 489133282872437279, 19175002942688032928599

Odd primes

Of the form 2n – 1.

3, 5, 7, 11

11 (number)

, 13

13 (number)

, 17

17 (number)

, 19

19 (number)

, 23

23 (number)

, 29

29 (number)

, 31

31 (number)

, 37

37 (number)

, 41

41 (number)

41 is the natural number following 40 and preceding 42.-In mathematics:Forty-one is the 13th smallest prime number. The next is forty-three, with which it comprises a twin prime...

, 43

43 (number)

, 47

47 (number)

, 53

53 (number)

53 is the natural number following 52 and preceding 54.-In mathematics:Fifty-three is the 16th prime number. It is also an Eisenstein prime....

, 59

59 (number)

, 61

61 (number)

, 67

67 (number)

, 71

71 (number)

71 is the natural number following 70 and preceding 72.-In mathematics:71 is the algebraic degree of Conway's constant, a remarkable number arising in the study of look-and-say sequences....

, 73

73 (number)

73 is the natural number following 72 and preceding 74. In English, it is the smallest integer with twelve letters in its spelled out name.- In mathematics :...

, 79

79 (number)

, 83

83 (number)

83 is the natural number following 82 and preceding 84.-In mathematics:Eighty-three is the sum of three consecutive primes as well as the sum of five consecutive primes ....

, 89, 97

97 (number)

, 101

101 (number)

101 is the natural number following 100 and preceding 102.It is variously pronounced "one hundred and one" / "a hundred and one", "one hundred one" / "a hundred one", and "one oh one"...

, 103

103 (number)

103 is the natural number following 102 and preceding 104.-In mathematics:One hundred [and] three is the 27th prime number. The previous prime is 101, making them both twin primes...

, 107

107 (number)

, 109

109 (number)

, 113

113 (number)

, 127

127 (number)

, 131

131 (number)

, 137

137 (number)

, 139

139 (number)

, 149

149 (number)

, 151

151 (number)

, 157

157 (number)

, 163

163 (number)

163 is the natural number following 162 and preceding 164.-In mathematics:163 is a strong prime in the sense that it is greater than the arithmetic mean of its two neighboring primes...

, 167

167 (number)

167 is the natural number following 166 and preceding 168.-In mathematics:* 167 is an odd number* 167 is a Chen prime, since the next odd number, 169, is a square of a prime...

, 173

173 (number)

, 179

179 (number)

, 181

181 (number)

, 191

191 (number)

191 is the natural number following 190 and preceding 192.-In mathematics:* 191 is an odd number* 191 is a centered 19-gonal number* 191 is a deficient number, as 1 is less than 191...

, 193

193 (number)

, 197

197 (number)

, 199

199 (number)

...

All prime numbers except 2 are odd.

Padovan
Padovan sequence
The Padovan sequence is the sequence of integers P defined by the initial valuesP=P=P=1,and the recurrence relationP=P+P.The first few values of P are...

primes

Primes in the Padovan sequence

.

2, 3, 5, 7, 37

37 (number)

, 151

151 (number)

, 3329, 23833, 13091204281, 3093215881333057, 1363005552434666078217421284621279933627102780881053358473

Palindromic prime
Palindromic prime
A palindromic prime is a prime number that is also a palindromic number. Palindromicity depends on the base of the numbering system and its writing conventions, while primality is independent of such concerns...

s

Primes that remain the same when their decimal digits are read backwards.

2, 3, 5, 7, 11

11 (number)

, 101

101 (number)

101 is the natural number following 100 and preceding 102.It is variously pronounced "one hundred and one" / "a hundred and one", "one hundred one" / "a hundred one", and "one oh one"...

, 131

131 (number)

, 151

151 (number)

, 181

181 (number)

, 191

191 (number)

191 is the natural number following 190 and preceding 192.-In mathematics:* 191 is an odd number* 191 is a centered 19-gonal number* 191 is a deficient number, as 1 is less than 191...

, 313

313 (number)

, 353

353 (number)

, 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

.

101

101 (number)

101 is the natural number following 100 and preceding 102.It is variously pronounced "one hundred and one" / "a hundred and one", "one hundred one" / "a hundred one", and "one oh one"...

, 131

131 (number)

, 151

151 (number)

, 181

181 (number)

, 191

191 (number)

191 is the natural number following 190 and preceding 192.-In mathematics:* 191 is an odd number* 191 is a centered 19-gonal number* 191 is a deficient number, as 1 is less than 191...

, 313

313 (number)

, 353

353 (number)

, 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
Partition (number theory)
In number theory and combinatorics, a partition of a positive integer n, also called an integer partition, is a way of writing n as a sum of positive integers. Two sums that differ only in the order of their summands are considered to be the same partition; if order matters then the sum becomes a...

primes

Partition numbers that are prime.

2, 3, 5, 7, 11

11 (number)

, 101

101 (number)

101 is the natural number following 100 and preceding 102.It is variously pronounced "one hundred and one" / "a hundred and one", "one hundred one" / "a hundred one", and "one oh one"...

, 17977, 10619863, 6620830889, 80630964769, 228204732751, 1171432692373, 1398341745571, 10963707205259, 15285151248481, 10657331232548839, 790738119649411319, 18987964267331664557

Pell
Pell number
In mathematics, the Pell numbers are an infinite sequence of integers that have been known since ancient times, the denominators of the closest rational approximations to the square root of 2. This sequence of approximations begins 1/1, 3/2, 7/5, 17/12, and 41/29, so the sequence of Pell numbers...

primes

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

2, 5, 29

29 (number)

, 5741, 33461, 44560482149, 1746860020068409, 68480406462161287469, 13558774610046711780701, 4125636888562548868221559797461449

Permutable prime
Permutable prime
A permutable prime is a prime number, which, in a given base, can have its digits' positions switched through any permutation and still spell a prime number. H. E...

s

Any permutation of the decimal digits is a prime.

2, 3, 5, 7, 11

11 (number)

, 13

13 (number)

, 17

17 (number)

, 31

31 (number)

, 37

37 (number)

, 71

71 (number)

71 is the natural number following 70 and preceding 72.-In mathematics:71 is the algebraic degree of Conway's constant, a remarkable number arising in the study of look-and-say sequences....

, 73

73 (number)

73 is the natural number following 72 and preceding 74. In English, it is the smallest integer with twelve letters in its spelled out name.- In mathematics :...

, 79

79 (number)

, 97

97 (number)

, 113

113 (number)

, 131

131 (number)

, 199

199 (number)

, 311

311 (number)

, 337, 373, 733, 919, 991, 1111111111111111111, 11111111111111111111111

It seems likely that all further permutable primes are repunit

Repunit

In recreational mathematics, a repunit is a number like 11, 111, or 1111 that contains only the digit 1. The term stands for repeated unit and was coined in 1966 by Albert H. Beiler...

s, i.e. contain only the digit 1.

Perrin
Perrin number
In mathematics, the Perrin numbers are defined by the recurrence relationandThe sequence of Perrin numbers starts withThe number of different maximal independent sets in an n-vertex cycle graph is counted by the nth Perrin number.-History:...

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

17 (number)

, 29

29 (number)

, 277

277 (number)

277 is the natural number following 276 and preceding 278.-In mathematics:277 is the fifty-ninth prime number, following 271 and preceding 281...

, 367, 853, 14197, 43721, 1442968193, 792606555396977, 187278659180417234321, 66241160488780141071579864797

Pierpont prime
Pierpont prime
A Pierpont prime is a prime number of the form2^u 3^v + 1\,for some nonnegative integers u and v. They are named after the mathematician James Pierpont....

s

Of the form

for some integer

Integer

The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...

s u,v ≥ 0.

These are also class 1- primes.

2, 3, 5, 7, 13

13 (number)

, 17

17 (number)

, 19

19 (number)

, 37

37 (number)

, 73

73 (number)

73 is the natural number following 72 and preceding 74. In English, it is the smallest integer with twelve letters in its spelled out name.- In mathematics :...

, 97

97 (number)

, 109

109 (number)

, 163

163 (number)

163 is the natural number following 162 and preceding 164.-In mathematics:163 is a strong prime in the sense that it is greater than the arithmetic mean of its two neighboring primes...

, 193

193 (number)

, 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

23 (number)

, 29

29 (number)

, 59

59 (number)

, 61

61 (number)

, 67

67 (number)

, 71

71 (number)

71 is the natural number following 70 and preceding 72.-In mathematics:71 is the algebraic degree of Conway's constant, a remarkable number arising in the study of look-and-say sequences....

, 79

79 (number)

, 83

83 (number)

83 is the natural number following 82 and preceding 84.-In mathematics:Eighty-three is the sum of three consecutive primes as well as the sum of five consecutive primes ....

, 109

109 (number)

, 137

137 (number)

, 139

139 (number)

, 149

149 (number)

, 193

193 (number)

, 227

227 (number)

227 is the natural number between 226 and 228. It is also a prime number.-In mathematics:227 is a prime number, and a twin prime with 229 . 223 plus 4 is 227, so they are cousin primes...

, 233

233 (number)

, 239

239 (number)

, 251

251 (number)

, 257, 269

269 (number)

, 271, 277

277 (number)

277 is the natural number following 276 and preceding 278.-In mathematics:277 is the fifty-ninth prime number, following 271 and preceding 281...

, 293, 307, 311

311 (number)

, 317, 359

359 (number)

, 379, 383, 389, 397, 401, 419

419 (number)

, 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

13 (number)

, 37

37 (number)

, 107

107 (number)

, 113

113 (number)

, 137

137 (number)

, 1013, 1237, 1367, 10079

Primorial prime
Primorial prime
In mathematics, primorial primes are prime numbers of the form pn# ± 1, where:The first few primorial primes are, the largest known primorial prime is 843301# - 1 with 365,851 digits, found in 2010 by the PrimeGrid project....

s

Of the form p_n#

Primorial

− 1 or p_n# + 1.

3, 5, 7, 29

29 (number)

, 31

31 (number)

, 211

211 (number)

, 2309, 2311, 30029, 200560490131, 304250263527209, 23768741896345550770650537601358309 (union of and )

Proth primes

Of the form k · 2ⁿ + 1 with odd k and k < 2ⁿ.

3, 5, 13

13 (number)

, 17

17 (number)

, 41

41 (number)

41 is the natural number following 40 and preceding 42.-In mathematics:Forty-one is the 13th smallest prime number. The next is forty-three, with which it comprises a twin prime...

, 97

97 (number)

, 113

113 (number)

, 193

193 (number)

, 241

241 (number)

241 is the natural number between 240 and 242. It is also a prime number.-In mathematics:241 is the larger of the twin primes . Prime twins are always of the form , with n = 1, 2, 3, ......

, 257, 353

353 (number)

, 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 prime
Pythagorean prime
A Pythagorean prime is prime number of the form 4n + 1. These are exactly the primes that can be the hypotenuse of a Pythagorean triangle.The first few Pythagorean primes are...

s

Of the form 4n + 1.

5, 13

13 (number)

, 17

17 (number)

, 29

29 (number)

, 37

37 (number)

, 41

41 (number)

41 is the natural number following 40 and preceding 42.-In mathematics:Forty-one is the 13th smallest prime number. The next is forty-three, with which it comprises a twin prime...

, 53

53 (number)

53 is the natural number following 52 and preceding 54.-In mathematics:Fifty-three is the 16th prime number. It is also an Eisenstein prime....

, 61

61 (number)

, 73

73 (number)

73 is the natural number following 72 and preceding 74. In English, it is the smallest integer with twelve letters in its spelled out name.- In mathematics :...

, 89, 97

97 (number)

, 101

101 (number)

101 is the natural number following 100 and preceding 102.It is variously pronounced "one hundred and one" / "a hundred and one", "one hundred one" / "a hundred one", and "one oh one"...

, 109

109 (number)

, 113

113 (number)

, 137

137 (number)

, 149

149 (number)

, 157

157 (number)

, 173

173 (number)

, 181

181 (number)

, 193

193 (number)

, 197

197 (number)

, 229

229 (number)

, 233

233 (number)

, 241

241 (number)

241 is the natural number between 240 and 242. It is also a prime number.-In mathematics:241 is the larger of the twin primes . Prime twins are always of the form , with n = 1, 2, 3, ......

, 257, 269

269 (number)

, 277

277 (number)

277 is the natural number following 276 and preceding 278.-In mathematics:277 is the fifty-ninth prime number, following 271 and preceding 281...

, 281, 293, 313

313 (number)

, 317, 337, 349, 353

353 (number)

, 373, 389, 397, 401, 409, 421, 433, 449

Prime quadruplets

(p, p+2, p+6, p+8) are all prime.

(5, 7, 11

11 (number)

, 13

13 (number)

), (11, 13, 17

17 (number)

, 19

19 (number)

), (101

101 (number)

101 is the natural number following 100 and preceding 102.It is variously pronounced "one hundred and one" / "a hundred and one", "one hundred one" / "a hundred one", and "one oh one"...

, 103

103 (number)

103 is the natural number following 102 and preceding 104.-In mathematics:One hundred [and] three is the 27th prime number. The previous prime is 101, making them both twin primes...

, 107

107 (number)

, 109

109 (number)

), (191

191 (number)

191 is the natural number following 190 and preceding 192.-In mathematics:* 191 is an odd number* 191 is a centered 19-gonal number* 191 is a deficient number, as 1 is less than 191...

, 193

193 (number)

, 197

197 (number)

, 199

199 (number)

), (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

, with

.

2, 11

11 (number)

, 23

23 (number)

, 37

37 (number)

, 43

43 (number)

, 53

53 (number)

53 is the natural number following 52 and preceding 54.-In mathematics:Fifty-three is the 16th prime number. It is also an Eisenstein prime....

, 71

71 (number)

71 is the natural number following 70 and preceding 72.-In mathematics:71 is the algebraic degree of Conway's constant, a remarkable number arising in the study of look-and-say sequences....

, 79

79 (number)

, 107

107 (number)

, 109

109 (number)

, 127

127 (number)

, 137

137 (number)

, 149

149 (number)

, 151

151 (number)

, 163

163 (number)

163 is the natural number following 162 and preceding 164.-In mathematics:163 is a strong prime in the sense that it is greater than the arithmetic mean of its two neighboring primes...

, 193

193 (number)

, 197

197 (number)

, 211

211 (number)

, 233

233 (number)

, 239

239 (number)

, 263

263 (number)

, 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

743 (number)

743 is the natural number following 742 and preceding 744. It is a prime number.-In mathematics:743 is a Sophie Germain prime number, as well as a cousin prime....

, 751, 757, 809, 821

Quartan primes

Of the form

, where

.

2, 17

17 (number)

, 97

97 (number)

, 257, 337, 641, 881

881 (number)

The number sequence 881 is...* a Paid Toll Free telephone number prefix in the USA* the Port of Los Angeles Long Wharf, California State Historic Landmark #881* the year AD 881 or the year 881 BC...

Ramanujan prime
Ramanujan prime
In mathematics, a Ramanujan prime is a prime number that satisfies a result proven by Srinivasa Ramanujan relating to the prime-counting function.-Origins and definition:...

s

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

11 (number)

, 17

17 (number)

, 29

29 (number)

, 41

41 (number)

41 is the natural number following 40 and preceding 42.-In mathematics:Forty-one is the 13th smallest prime number. The next is forty-three, with which it comprises a twin prime...

, 47

47 (number)

, 59

59 (number)

, 67

67 (number)

, 71

71 (number)

71 is the natural number following 70 and preceding 72.-In mathematics:71 is the algebraic degree of Conway's constant, a remarkable number arising in the study of look-and-say sequences....

, 97

97 (number)

, 101

101 (number)

101 is the natural number following 100 and preceding 102.It is variously pronounced "one hundred and one" / "a hundred and one", "one hundred one" / "a hundred one", and "one oh one"...

, 107

107 (number)

, 127

127 (number)

, 149

149 (number)

, 151

151 (number)

, 167

167 (number)

167 is the natural number following 166 and preceding 168.-In mathematics:* 167 is an odd number* 167 is a Chen prime, since the next odd number, 169, is a square of a prime...

, 179

179 (number)

, 181

181 (number)

, 227

227 (number)

227 is the natural number between 226 and 228. It is also a prime number.-In mathematics:227 is a prime number, and a twin prime with 229 . 223 plus 4 is 227, so they are cousin primes...

, 229

229 (number)

, 233

233 (number)

, 239

239 (number)

, 241

241 (number)

241 is the natural number between 240 and 242. It is also a prime number.-In mathematics:241 is the larger of the twin primes . Prime twins are always of the form , with n = 1, 2, 3, ......

, 263

263 (number)

, 269

269 (number)

, 281, 307, 311

311 (number)

, 347, 349, 367, 373, 401, 409, 419

419 (number)

, 431, 433, 439, 461, 487, 491

Regular primes

Primes p which do not divide the class number of the p-th cyclotomic field

Cyclotomic field

In number theory, a cyclotomic field is a number field obtained by adjoining a complex primitive root of unity to Q, the field of rational numbers...

.

3, 5, 7, 11

11 (number)

, 13

13 (number)

, 17

17 (number)

, 19

19 (number)

, 23

23 (number)

, 29

29 (number)

, 31

31 (number)

, 41

41 (number)

41 is the natural number following 40 and preceding 42.-In mathematics:Forty-one is the 13th smallest prime number. The next is forty-three, with which it comprises a twin prime...

, 43

43 (number)

, 47

47 (number)

, 53

53 (number)

53 is the natural number following 52 and preceding 54.-In mathematics:Fifty-three is the 16th prime number. It is also an Eisenstein prime....

, 61

61 (number)

, 71

71 (number)

71 is the natural number following 70 and preceding 72.-In mathematics:71 is the algebraic degree of Conway's constant, a remarkable number arising in the study of look-and-say sequences....

, 73

73 (number)

73 is the natural number following 72 and preceding 74. In English, it is the smallest integer with twelve letters in its spelled out name.- In mathematics :...

, 79

79 (number)

, 83

83 (number)

83 is the natural number following 82 and preceding 84.-In mathematics:Eighty-three is the sum of three consecutive primes as well as the sum of five consecutive primes ....

, 89, 97

97 (number)

, 107

107 (number)

, 109

109 (number)

, 113

113 (number)

, 127

127 (number)

, 137

137 (number)

, 139

139 (number)

, 151

151 (number)

, 163

163 (number)

163 is the natural number following 162 and preceding 164.-In mathematics:163 is a strong prime in the sense that it is greater than the arithmetic mean of its two neighboring primes...

, 167

167 (number)

167 is the natural number following 166 and preceding 168.-In mathematics:* 167 is an odd number* 167 is a Chen prime, since the next odd number, 169, is a square of a prime...

, 173

173 (number)

, 179

179 (number)

, 181

181 (number)

, 191

191 (number)

191 is the natural number following 190 and preceding 192.-In mathematics:* 191 is an odd number* 191 is a centered 19-gonal number* 191 is a deficient number, as 1 is less than 191...

, 193

193 (number)

, 197

197 (number)

, 199

199 (number)

, 211

211 (number)

, 223

223 (number)

, 227

227 (number)

227 is the natural number between 226 and 228. It is also a prime number.-In mathematics:227 is a prime number, and a twin prime with 229 . 223 plus 4 is 227, so they are cousin primes...

, 229

229 (number)

, 239

239 (number)

, 241

241 (number)

241 is the natural number between 240 and 242. It is also a prime number.-In mathematics:241 is the larger of the twin primes . Prime twins are always of the form , with n = 1, 2, 3, ......

, 251

251 (number)

, 269

269 (number)

, 277

277 (number)

277 is the natural number following 276 and preceding 278.-In mathematics:277 is the fifty-ninth prime number, following 271 and preceding 281...

, 281

Repunit
Repunit
In recreational mathematics, a repunit is a number like 11, 111, or 1111 that contains only the digit 1. The term stands for repeated unit and was coined in 1966 by Albert H. Beiler...

primes

Primes containing only the decimal digit 1.

11

11 (number)

, 1111111111111111111, 11111111111111111111111

The next have 317 and 1031 digits.

Primes in residue classes
Dirichlet's theorem on arithmetic progressions
In number theory, Dirichlet's theorem, also called the Dirichlet prime number theorem, states that for any two positive coprime integers a and d, there are infinitely many primes of the form a + nd, where n ≥ 0. In other words, there are infinitely many primes which are...

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

Modular arithmetic

In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" after they reach a certain value—the modulus....

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

11 (number)

, 13

13 (number)

, 17

17 (number)

, 19

19 (number)

, 23

23 (number)

, 29

29 (number)

, 31

31 (number)

, 37

37 (number)

, 41

41 (number)

41 is the natural number following 40 and preceding 42.-In mathematics:Forty-one is the 13th smallest prime number. The next is forty-three, with which it comprises a twin prime...

, 43

43 (number)

, 47

47 (number)

, 53

53 (number)

53 is the natural number following 52 and preceding 54.-In mathematics:Fifty-three is the 16th prime number. It is also an Eisenstein prime....

4n+1: 5, 13, 17, 29, 37, 41, 53, 61

61 (number)

, 73

73 (number)

73 is the natural number following 72 and preceding 74. In English, it is the smallest integer with twelve letters in its spelled out name.- In mathematics :...

, 89, 97

97 (number)

, 101

101 (number)

101 is the natural number following 100 and preceding 102.It is variously pronounced "one hundred and one" / "a hundred and one", "one hundred one" / "a hundred one", and "one oh one"...

, 109

109 (number)

, 113

113 (number)

, 137

137 (number)

4n+3: 3, 7, 11, 19, 23, 31, 43, 47, 59

59 (number)

, 67

67 (number)

, 71

71 (number)

71 is the natural number following 70 and preceding 72.-In mathematics:71 is the algebraic degree of Conway's constant, a remarkable number arising in the study of look-and-say sequences....

, 79

79 (number)

, 83

83 (number)

83 is the natural number following 82 and preceding 84.-In mathematics:Eighty-three is the sum of three consecutive primes as well as the sum of five consecutive primes ....

, 103

103 (number)

103 is the natural number following 102 and preceding 104.-In mathematics:One hundred [and] three is the 27th prime number. The previous prime is 101, making them both twin primes...

, 107

107 (number)

6n+1: 7, 13, 19, 31, 37, 43, 61, 67, 73, 79, 97, 103, 109, 127

127 (number)

, 139

139 (number)

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

193 (number)

, 233

233 (number)

, 241

241 (number)

241 is the natural number between 240 and 242. It is also a prime number.-In mathematics:241 is the larger of the twin primes . Prime twins are always of the form , with n = 1, 2, 3, ......

, 257, 281, 313

313 (number)

, 337, 353

353 (number)

8n+3: 3, 11, 19, 43, 59, 67, 83, 107, 131

131 (number)

, 139, 163

163 (number)

163 is the natural number following 162 and preceding 164.-In mathematics:163 is a strong prime in the sense that it is greater than the arithmetic mean of its two neighboring primes...

, 179

179 (number)

, 211

211 (number)

, 227

227 (number)

227 is the natural number between 226 and 228. It is also a prime number.-In mathematics:227 is a prime number, and a twin prime with 229 . 223 plus 4 is 227, so they are cousin primes...

, 251

251 (number)

8n+5: 5, 13, 29, 37, 53, 61, 101, 109, 149

149 (number)

, 157

157 (number)

, 173

173 (number)

, 181

181 (number)

, 197

197 (number)

, 229

229 (number)

, 269

269 (number)

8n+7: 7, 23, 31, 47, 71, 79, 103, 127, 151

151 (number)

, 167

167 (number)

167 is the natural number following 166 and preceding 168.-In mathematics:* 167 is an odd number* 167 is a Chen prime, since the next odd number, 169, is a square of a prime...

, 191

191 (number)

191 is the natural number following 190 and preceding 192.-In mathematics:* 191 is an odd number* 191 is a centered 19-gonal number* 191 is a deficient number, as 1 is less than 191...

, 199

199 (number)

, 223

223 (number)

, 239

239 (number)

, 263

263 (number)

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

277 (number)

277 is the natural number following 276 and preceding 278.-In mathematics:277 is the fifty-ninth prime number, following 271 and preceding 281...

10n+9: 19, 29, 59, 79, 89, 109, 139, 149, 179, 199, 229, 239, 269, 349, 359

359 (number)

...

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

Right-truncatable primes
Truncatable prime
In number theory, a left-truncatable prime is a prime number which, in a given base, contains no 0, and if the leading digit is successively removed, then all resulting numbers are prime. For example 9137, since 9137, 137, 37 and 7 are all prime...

Primes that remain prime when the last decimal digit is successively removed.

2, 3, 5, 7, 23

23 (number)

, 29

29 (number)

, 31

31 (number)

, 37

37 (number)

, 53

53 (number)

53 is the natural number following 52 and preceding 54.-In mathematics:Fifty-three is the 16th prime number. It is also an Eisenstein prime....

, 59

59 (number)

, 71

71 (number)

71 is the natural number following 70 and preceding 72.-In mathematics:71 is the algebraic degree of Conway's constant, a remarkable number arising in the study of look-and-say sequences....

, 73

73 (number)

73 is the natural number following 72 and preceding 74. In English, it is the smallest integer with twelve letters in its spelled out name.- In mathematics :...

, 79

79 (number)

, 233

233 (number)

, 239

239 (number)

, 293, 311

311 (number)

, 313

313 (number)

, 317, 373, 379, 593

593 (number)

, 599, 719, 733, 739, 797, 2333, 2339, 2393, 2399, 2939, 3119, 3137, 3733, 3739, 3793, 3797

Safe prime
Safe prime
A safe prime is a prime number of the form 2p + 1, where p is also a prime. The first few safe primes are...

s

p and (p-1) / 2 are both prime.

5, 7, 11

11 (number)

, 23

23 (number)

, 47

47 (number)

, 59

59 (number)

, 83

83 (number)

83 is the natural number following 82 and preceding 84.-In mathematics:Eighty-three is the sum of three consecutive primes as well as the sum of five consecutive primes ....

, 107

107 (number)

, 167

167 (number)

167 is the natural number following 166 and preceding 168.-In mathematics:* 167 is an odd number* 167 is a Chen prime, since the next odd number, 169, is a square of a prime...

, 179

179 (number)

, 227

227 (number)

227 is the natural number between 226 and 228. It is also a prime number.-In mathematics:227 is a prime number, and a twin prime with 229 . 223 plus 4 is 227, so they are cousin primes...

, 263

263 (number)

, 347, 359

359 (number)

, 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
Self number
A self number, Colombian number or Devlali number is an integer which, in a given base, cannot be generated by any other integer added to the sum of that other integer's digits. For example, 21 is not a self number, since it can be generated by the sum of 15 and the digits comprising 15, that is,...

in base 10

Primes that cannot be generated by any integer added to the sum of its decimal digits.

3, 5, 7, 31

31 (number)

, 53

53 (number)

53 is the natural number following 52 and preceding 54.-In mathematics:Fifty-three is the 16th prime number. It is also an Eisenstein prime....

, 97

97 (number)

, 211

211 (number)

, 233

233 (number)

, 277

277 (number)

277 is the natural number following 276 and preceding 278.-In mathematics:277 is the fifty-ninth prime number, following 271 and preceding 281...

, 367, 389, 457, 479, 547, 569, 613, 659, 727, 839, 883, 929, 1021, 1087, 1109, 1223, 1289

1289 (number)

The number 1289 is the natural number following 1288 and preceding 1290.-In mathematics:The number 1289 is an odd prime number, following 1283 and preceding 1291...

, 1447, 1559, 1627, 1693, 1783, 1873

Sexy prime
Sexy prime
In mathematics, a sexy prime is a prime number that differs from another prime number by six. For example, the numbers 5 and 11 are both sexy primes, because they differ by 6...

s

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

(5, 11

11 (number)

), (7, 13

13 (number)

), (11, 17

17 (number)

), (13, 19

19 (number)

), (17, 23

23 (number)

), (23, 29

29 (number)

), (31

31 (number)

, 37

37 (number)

), (37, 43

43 (number)

), (41

41 (number)

41 is the natural number following 40 and preceding 42.-In mathematics:Forty-one is the 13th smallest prime number. The next is forty-three, with which it comprises a twin prime...

, 47

47 (number)

), (47, 53

53 (number)

53 is the natural number following 52 and preceding 54.-In mathematics:Fifty-three is the 16th prime number. It is also an Eisenstein prime....

), (53, 59

59 (number)

), (61

61 (number)

, 67

67 (number)

), (67, 73

73 (number)

73 is the natural number following 72 and preceding 74. In English, it is the smallest integer with twelve letters in its spelled out name.- In mathematics :...

), (73, 79

79 (number)

), (83

83 (number)

83 is the natural number following 82 and preceding 84.-In mathematics:Eighty-three is the sum of three consecutive primes as well as the sum of five consecutive primes ....

, 89), (97

97 (number)

, 103

103 (number)

103 is the natural number following 102 and preceding 104.-In mathematics:One hundred [and] three is the 27th prime number. The previous prime is 101, making them both twin primes...

), (101

101 (number)

101 is the natural number following 100 and preceding 102.It is variously pronounced "one hundred and one" / "a hundred and one", "one hundred one" / "a hundred one", and "one oh one"...

, 107

107 (number)

), (103, 109

109 (number)

), (107, 113

113 (number)

), (131

131 (number)

, 137

137 (number)

), (151

151 (number)

, 157

157 (number)

), (157, 163

163 (number)

163 is the natural number following 162 and preceding 164.-In mathematics:163 is a strong prime in the sense that it is greater than the arithmetic mean of its two neighboring primes...

), (167

167 (number)

167 is the natural number following 166 and preceding 168.-In mathematics:* 167 is an odd number* 167 is a Chen prime, since the next odd number, 169, is a square of a prime...

, 173

173 (number)

), (173, 179

179 (number)

), (191

191 (number)

191 is the natural number following 190 and preceding 192.-In mathematics:* 191 is an odd number* 191 is a centered 19-gonal number* 191 is a deficient number, as 1 is less than 191...

, 197

197 (number)

), (193

193 (number)

, 199

199 (number)

)

Smarandache–Wellin primes

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

2, 23

23 (number)

, 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 2^a ± 2^b ± 1, where 0 < b < a.

3, 5, 7, 11

11 (number)

, 13

13 (number)

Sophie Germain prime
Sophie Germain prime
In number theory, a prime number p is a Sophie Germain prime if 2p + 1 is also prime. For example, 23 is a Sophie Germain prime because it is a prime and 2 × 23 + 1 = 47, and 47 is also a prime number...

s

p and 2p + 1 are both prime.

2, 3, 5, 11

11 (number)

, 23

23 (number)

, 29

29 (number)

, 41

41 (number)

41 is the natural number following 40 and preceding 42.-In mathematics:Forty-one is the 13th smallest prime number. The next is forty-three, with which it comprises a twin prime...

, 53

53 (number)

53 is the natural number following 52 and preceding 54.-In mathematics:Fifty-three is the 16th prime number. It is also an Eisenstein prime....

, 83

83 (number)

83 is the natural number following 82 and preceding 84.-In mathematics:Eighty-three is the sum of three consecutive primes as well as the sum of five consecutive primes ....

, 89, 113

113 (number)

, 131

131 (number)

, 173

173 (number)

, 179

179 (number)

, 191

191 (number)

191 is the natural number following 190 and preceding 192.-In mathematics:* 191 is an odd number* 191 is a centered 19-gonal number* 191 is a deficient number, as 1 is less than 191...

, 233

233 (number)

, 239

239 (number)

, 251

251 (number)

, 281, 293, 359

359 (number)

, 419

419 (number)

, 431, 443, 491, 509, 593

593 (number)

, 641, 653, 659, 683, 719, 743

743 (number)

743 is the natural number following 742 and preceding 744. It is a prime number.-In mathematics:743 is a Sophie Germain prime number, as well as a cousin prime....

, 761, 809, 911

911 (number)

, 953

Star
Star number
A star number is a centered figurate number that represents a centered hexagram, such as the one that Chinese checkers is played on.The nth star number is given by the formula 6n + 1...

primes

Of the form 6n(n – 1) + 1.

13

13 (number)

, 37

37 (number)

, 73

73 (number)

73 is the natural number following 72 and preceding 74. In English, it is the smallest integer with twelve letters in its spelled out name.- In mathematics :...

, 181

181 (number)

, 337, 433, 541, 661, 937, 1093

1093 (number)

, 2053, 2281, 2521, 3037, 3313, 5581, 5953, 6337, 6733, 7561, 7993, 8893, 10333, 10837, 11353, 12421, 12973, 13537, 15913, 18481

Stern prime
Stern prime
A Stern prime, named for Moritz Abraham Stern, is a prime number that is not the sum of a smaller prime and twice the square of a nonzero integer. Or, to put it algebraically, if for a prime q there is no smaller prime p and nonzero integer b such that q = p + 2b², then q is a...

s

Primes that are not the sum of a smaller prime and twice the square of a nonzero integer.

2, 3, 17

17 (number)

, 137

137 (number)

, 227

227 (number)

227 is the natural number between 226 and 228. It is also a prime number.-In mathematics:227 is a prime number, and a twin prime with 229 . 223 plus 4 is 227, so they are cousin primes...

, 977, 1187, 1493

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

Super-prime
Super-prime
Super-prime numbers are the subsequence of prime numbers that occupy prime-numbered positions within the sequence of all prime numbers. The subsequence beginsThat is, if p denotes the ith prime number, the numbers in this sequence are those of the form p...

s

Primes with a prime index in the sequence of prime numbers (the 2nd, 3rd, 5th, ... prime).

3, 5, 11

11 (number)

, 17

17 (number)

, 31

31 (number)

, 41

41 (number)

41 is the natural number following 40 and preceding 42.-In mathematics:Forty-one is the 13th smallest prime number. The next is forty-three, with which it comprises a twin prime...

, 59

59 (number)

, 67

67 (number)

, 83

83 (number)

83 is the natural number following 82 and preceding 84.-In mathematics:Eighty-three is the sum of three consecutive primes as well as the sum of five consecutive primes ....

, 109

109 (number)

, 127

127 (number)

, 157

157 (number)

, 179

179 (number)

, 191

191 (number)

191 is the natural number following 190 and preceding 192.-In mathematics:* 191 is an odd number* 191 is a centered 19-gonal number* 191 is a deficient number, as 1 is less than 191...

, 211

211 (number)

, 241

241 (number)

241 is the natural number between 240 and 242. It is also a prime number.-In mathematics:241 is the larger of the twin primes . Prime twins are always of the form , with n = 1, 2, 3, ......

, 277

277 (number)

277 is the natural number following 276 and preceding 278.-In mathematics:277 is the fifty-ninth prime number, following 271 and preceding 281...

, 283, 331, 353

353 (number)

, 367, 401, 431, 461, 509, 547, 563, 587, 599, 617, 709, 739, 773, 797, 859, 877, 919, 967, 991

Supersingular prime
Supersingular prime (moonshine theory)
In the mathematical branch of moonshine theory, a supersingular prime is a certain type of prime number.Namely, a supersingular prime is a prime divisor of the order of the Monster group M, the largest of the sporadic simple groups...

s

There are exactly fifteen supersingular primes:

2, 3, 5, 7, 11

11 (number)

, 13

13 (number)

, 17

17 (number)

, 19

19 (number)

, 23

23 (number)

, 29

29 (number)

, 31

31 (number)

, 41

41 (number)

41 is the natural number following 40 and preceding 42.-In mathematics:Forty-one is the 13th smallest prime number. The next is forty-three, with which it comprises a twin prime...

, 47

47 (number)

, 59

59 (number)

, 71

71 (number)

71 is the natural number following 70 and preceding 72.-In mathematics:71 is the algebraic degree of Conway's constant, a remarkable number arising in the study of look-and-say sequences....

Swinging primes

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

2, 3, 5, 7, 19

19 (number)

, 29

29 (number)

, 31

31 (number)

, 71

71 (number)

71 is the natural number following 70 and preceding 72.-In mathematics:71 is the algebraic degree of Conway's constant, a remarkable number arising in the study of look-and-say sequences....

, 139

139 (number)

, 251

251 (number)

, 631, 3433, 12011

Thabit number
Thabit number
In number theory, a Thabit number, Thâbit ibn Kurrah number, or 321 number is an integer of the form 3·2n−1 for a non-negative integer n...

primes

Of the form 3 · 2ⁿ – 1.

2, 5, 11

11 (number)

, 23

23 (number)

, 47

47 (number)

, 191

191 (number)

191 is the natural number following 190 and preceding 192.-In mathematics:* 191 is an odd number* 191 is a centered 19-gonal number* 191 is a deficient number, as 1 is less than 191...

, 383, 6143, 786431, 51539607551, 824633720831, 26388279066623, 108086391056891903, 55340232221128654847, 226673591177742970257407

Prime triplet
Prime triplet
In mathematics, a prime triplet is a set of three prime numbers of the form or...

s

(p, p+2, p+6) or (p, p+4, p+6) are all prime.

(5, 7, 11

11 (number)

), (7, 11, 13

13 (number)

), (11, 13, 17

17 (number)

), (13, 17, 19

19 (number)

), (17, 19, 23

23 (number)

), (37

37 (number)

, 41

41 (number)

41 is the natural number following 40 and preceding 42.-In mathematics:Forty-one is the 13th smallest prime number. The next is forty-three, with which it comprises a twin prime...

, 43

43 (number)

), (41, 43, 47

47 (number)

), (67

67 (number)

, 71

71 (number)

71 is the natural number following 70 and preceding 72.-In mathematics:71 is the algebraic degree of Conway's constant, a remarkable number arising in the study of look-and-say sequences....

, 73

73 (number)

73 is the natural number following 72 and preceding 74. In English, it is the smallest integer with twelve letters in its spelled out name.- In mathematics :...

), (97

97 (number)

, 101

101 (number)

101 is the natural number following 100 and preceding 102.It is variously pronounced "one hundred and one" / "a hundred and one", "one hundred one" / "a hundred one", and "one oh one"...

, 103

103 (number)

103 is the natural number following 102 and preceding 104.-In mathematics:One hundred [and] three is the 27th prime number. The previous prime is 101, making them both twin primes...

), (101, 103, 107

107 (number)

), (103, 107, 109

109 (number)

), (107, 109, 113

113 (number)

), (191

191 (number)

191 is the natural number following 190 and preceding 192.-In mathematics:* 191 is an odd number* 191 is a centered 19-gonal number* 191 is a deficient number, as 1 is less than 191...

, 193

193 (number)

, 197

197 (number)

), (193, 197, 199

199 (number)

), (223

223 (number)

, 227

227 (number)

227 is the natural number between 226 and 228. It is also a prime number.-In mathematics:227 is a prime number, and a twin prime with 229 . 223 plus 4 is 227, so they are cousin primes...

, 229

229 (number)

), (227, 229, 233

233 (number)

), (277

277 (number)

277 is the natural number following 276 and preceding 278.-In mathematics:277 is the fifty-ninth prime number, following 271 and preceding 281...

, 281, 283), (307, 311

311 (number)

, 313

313 (number)

), (311, 313, 317), (347, 349, 353

353 (number)

)

Twin prime
Twin prime
A twin prime is a prime number that differs from another prime number by two. Except for the pair , this is the smallest possible difference between two primes. Some examples of twin prime pairs are , , , , and...

s

(p, p + 2) are both prime.

(3, 5), (5, 7), (11

11 (number)

, 13

13 (number)

), (17

17 (number)

, 19

19 (number)

), (29

29 (number)

, 31

31 (number)

), (41

41 (number)

41 is the natural number following 40 and preceding 42.-In mathematics:Forty-one is the 13th smallest prime number. The next is forty-three, with which it comprises a twin prime...

, 43

43 (number)

), (59

59 (number)

, 61

61 (number)

), (71

71 (number)

71 is the natural number following 70 and preceding 72.-In mathematics:71 is the algebraic degree of Conway's constant, a remarkable number arising in the study of look-and-say sequences....

, 73

73 (number)

73 is the natural number following 72 and preceding 74. In English, it is the smallest integer with twelve letters in its spelled out name.- In mathematics :...

), (101

101 (number)

101 is the natural number following 100 and preceding 102.It is variously pronounced "one hundred and one" / "a hundred and one", "one hundred one" / "a hundred one", and "one oh one"...

, 103

103 (number)

103 is the natural number following 102 and preceding 104.-In mathematics:One hundred [and] three is the 27th prime number. The previous prime is 101, making them both twin primes...

), (107

107 (number)

, 109

109 (number)

), (137

137 (number)

, 139

139 (number)

), (149

149 (number)

, 151

151 (number)

), (179

179 (number)

, 181

181 (number)

), (191

191 (number)

191 is the natural number following 190 and preceding 192.-In mathematics:* 191 is an odd number* 191 is a centered 19-gonal number* 191 is a deficient number, as 1 is less than 191...

, 193

193 (number)

), (197

197 (number)

, 199

199 (number)

), (227

227 (number)

227 is the natural number between 226 and 228. It is also a prime number.-In mathematics:227 is a prime number, and a twin prime with 229 . 223 plus 4 is 227, so they are cousin primes...

, 229

229 (number)

), (239

239 (number)

, 241

241 (number)

241 is the natural number between 240 and 242. It is also a prime number.-In mathematics:241 is the larger of the twin primes . Prime twins are always of the form , with n = 1, 2, 3, ......

), (269

269 (number)

, 271), (281, 283), (311

311 (number)

, 313

313 (number)

), (347, 349), (419

419 (number)

, 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

23 (number)

, 37

37 (number)

, 53

53 (number)

53 is the natural number following 52 and preceding 54.-In mathematics:Fifty-three is the 16th prime number. It is also an Eisenstein prime....

, 73

73 (number)

73 is the natural number following 72 and preceding 74. In English, it is the smallest integer with twelve letters in its spelled out name.- In mathematics :...

, 313

313 (number)

, 317, 373, 797, 3137, 3797, 739397

Ulam number primes

Ulam numbers that are prime.

2, 3, 11

11 (number)

, 13

13 (number)

, 47

47 (number)

, 53

53 (number)

53 is the natural number following 52 and preceding 54.-In mathematics:Fifty-three is the 16th prime number. It is also an Eisenstein prime....

, 97

97 (number)

, 131

131 (number)

, 197

197 (number)

, 241

241 (number)

241 is the natural number between 240 and 242. It is also a prime number.-In mathematics:241 is the larger of the twin primes . Prime twins are always of the form , with n = 1, 2, 3, ......

, 409, 431, 607, 673, 739, 751, 983, 991, 1103, 1433, 1489, 1531, 1553, 1709, 1721, 2371, 2393, 2447, 2633, 2789, 2833, 2897

Unique prime
Unique prime
In number theory, a unique prime is a certain kind of prime number. A prime p ≠ 2, 5 is called unique if there is no other prime q such that the period length of the decimal expansion of its reciprocal, 1 / p, is equivalent to the period length of the reciprocal of q, 1 / q...

s

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

3, 11

11 (number)

, 37

37 (number)

, 101

101 (number)

101 is the natural number following 100 and preceding 102.It is variously pronounced "one hundred and one" / "a hundred and one", "one hundred one" / "a hundred one", and "one oh one"...

, 9091, 9901, 333667, 909091, 99990001, 999999000001, 9999999900000001, 909090909090909091, 1111111111111111111, 11111111111111111111111, 900900900900990990990991

Wagstaff primes

Of the form (2ⁿ + 1) / 3.

3, 11

11 (number)

, 43

43 (number)

, 683, 2731, 43691, 174763, 2796203, 715827883, 2932031007403, 768614336404564651, 201487636602438195784363, 845100400152152934331135470251, 56713727820156410577229101238628035243

n values:

3, 5, 7, 11, 13

13 (number)

, 17

17 (number)

, 19

19 (number)

, 23

23 (number)

, 31

31 (number)

, 43, 61

61 (number)

, 79

79 (number)

, 101

101 (number)

101 is the natural number following 100 and preceding 102.It is variously pronounced "one hundred and one" / "a hundred and one", "one hundred one" / "a hundred one", and "one oh one"...

, 127

127 (number)

, 167

167 (number)

167 is the natural number following 166 and preceding 168.-In mathematics:* 167 is an odd number* 167 is a Chen prime, since the next odd number, 169, is a square of a prime...

, 191

191 (number)

191 is the natural number following 190 and preceding 192.-In mathematics:* 191 is an odd number* 191 is a centered 19-gonal number* 191 is a deficient number, as 1 is less than 191...

, 199

199 (number)

, 313

313 (number)

, 347, 701, 1709, 2617, 3539, 5807, 10501, 10691, 11279, 12391, 14479, 42737, 83339, 95369, 117239, 127031, 138937, 141079, 267017, 269987, 374321

Wall-Sun-Sun prime
Wall-Sun-Sun prime
In number theory, a Wall–Sun–Sun prime or Fibonacci–Wieferich prime is a certain kind of prime number which is conjectured to exist although none are known.- Definition :...

s

A prime p > 5 is called a Wall-Sun-Sun prime if p² divides the Fibonacci number

Fibonacci number

In mathematics, the Fibonacci numbers are the numbers in the following integer sequence:0,\;1,\;1,\;2,\;3,\;5,\;8,\;13,\;21,\;34,\;55,\;89,\;144,\; \ldots\; ....

, where the Legendre symbol

Legendre symbol

In number theory, the Legendre symbol is a multiplicative function with values 1, −1, 0 that is a quadratic character modulo a prime number p: its value on a quadratic residue mod p is 1 and on a quadratic non-residue is −1....

is defined as

, no Wall-Sun-Sun primes are known.

Wedderburn-Etherington number
Wedderburn-Etherington number
In graph theory, the Wedderburn–Etherington numbers, named for Ivor Malcolm Haddon Etherington and Joseph Wedderburn, count how many weak binary trees can be constructed: that is, the number of trees for which each graph vertex is adjacent to no more than three other such vertices, for a...

primes

Wedderburn-Etherington numbers that are prime.

2, 3, 11

11 (number)

, 23

23 (number)

, 983, 2179, 24631, 3626149, 253450711, 596572387 (primes in )

Weakly prime number
Weakly prime number
In number theory, a prime number is called weakly prime if it becomes composite when any one of its digits is changed to every single other digit. Decimal digits are usually assumed.The first weakly prime numbers are:...

s

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 prime
Wieferich prime
In number theory, a Wieferich prime is a prime number p such that p2 divides 2p − 1 − 1, therefore connecting these primes with Fermat's little theorem, which states that every odd prime p divides 2p − 1 − 1...

s

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

1093

1093 (number)

, 3511

3511 (number)

3511 is the natural number following 3510 and preceding 3512.3511 is a prime number, and is also an emirp.3511 is the largest known Wieferich prime – the only other being 1093....

, these are the only known Wieferich primes.

Wieferich primes base 3 (Mirimanoff primes)

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

11

11 (number)

, 1006003

, these are the only known Mirimanoff primes.

Wieferich primes base 5

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

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

, these are the only known base 5 Wieferich primes.

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

Wieferich primes base 10

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

3, 487, 56598313

Wieferich primes base 11

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

71

71 (number)

71 is the natural number following 70 and preceding 72.-In mathematics:71 is the algebraic degree of Conway's constant, a remarkable number arising in the study of look-and-say sequences....

Wieferich primes base 12

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

2693, 123653

Wieferich primes base 13

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

863, 1747591

Wieferich primes base 17

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

3, 46021, 48947

Wieferich primes base 19

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

3, 7, 13

13 (number)

, 43

43 (number)

, 137

137 (number)

, 63061489

Wilson prime
Wilson prime
A Wilson prime, named after English mathematician John Wilson, is a prime number p such that p2 divides ! + 1, where "!" denotes the factorial function; compare this with Wilson's theorem, which states that every prime p divides ! + 1.The only known Wilson primes are 5, 13, and...

s

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

5, 13

13 (number)

, 563

, these are the only known Wilson primes.

Wolstenholme prime
Wolstenholme prime
In number theory, a Wolstenholme prime is a special type of prime number satisfying a stronger version of Wolstenholme's theorem. Wolstenholme's theorem is a congruence relation satisfied by all prime numbers greater than 7...

s

Primes p for which the binomial coefficient

Binomial coefficient

In mathematics, binomial coefficients are a family of positive integers that occur as coefficients in the binomial theorem. They are indexed by two nonnegative integers; the binomial coefficient indexed by n and k is usually written \tbinom nk , and it is the coefficient of the x k term in...

16843, 2124679

, these are the only known Wolstenholme primes.

Woodall
Woodall number
In number theory, a Woodall number is any natural number of the formfor some natural number n. The first few Woodall numbers are:Woodall numbers were first studied by Allan J. C. Cunningham and H. J. Woodall in 1917, inspired by James Cullen's earlier study of the similarly-defined Cullen numbers...

primes

Of the form n · 2ⁿ − 1.

7, 23

23 (number)

, 383, 32212254719, 2833419889721787128217599, 195845982777569926302400511, 4776913109852041418248056622882488319

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)
Selected prime related sequences in OEIS
On-Line Encyclopedia of Integer Sequences
The On-Line Encyclopedia of Integer Sequences , also cited simply as Sloane's, is an online database of integer sequences, created and maintained by N. J. A. Sloane, a researcher at AT&T Labs...

.
Fischer, R. Thema: Fermatquotient B^(P-1) 1 (mod P^2) (Lists Wieferich primes in all bases up to 1052)

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The first 500 prime numbers

Lists of primes by type

Bell number primes

Carol primes

Centered decagonalCentered decagonal numberA centered decagonal number is a centered figurate number that represents a decagon with a dot in the center and all other dots surrounding the center dot in successive decagonal layers... primes

Centered heptagonalCentered heptagonal numberA centered heptagonal number is a centered figurate number that represents a heptagon with a dot in the center and all other dots surrounding the center dot in successive heptagonal layers... primes

Centered triangularCentered triangular numberA centered triangular number is a centered figurate number that represents a triangle with a dot in the center and all other dots surrounding the center in successive triangular layers... primes

Chen primeChen primeA prime number p is called a Chen prime if p + 2 is either a prime or a product of two primes. The even number 2p + 2 therefore satisfies Chen's theorem....s

Circular primeCircular primeA circular prime is a prime number with the property that the number generated at each intermediate step when cyclically permuting its digits will be prime. For example 1193 is a circular prime, since 1931, 9311 and 3119 all are also prime...s

Cousin primeCousin primeIn mathematics, cousin primes are prime numbers that differ by four; compare this with twin primes, pairs of prime numbers that differ by two, and sexy primes, pairs of prime numbers that differ by six....s

Cuban primes

CullenCullen numberIn mathematics, a Cullen number is a natural number of the form n · 2n + 1 . Cullen numbers were first studied by Fr. James Cullen in 1905... primes

Dihedral primeDihedral primeA dihedral prime or dihedral calculator prime is a prime number that still reads like itself or another prime number when read in a seven-segment display, regardless of orientation , and surface...s

Double factorial primes

Double Mersenne primes

EmirpEmirpAn emirp is a prime number that results in a different prime when its digits are reversed. This definition excludes the related palindromic primes. Emirps are also called reversible primes....s

EuclidEuclid numberIn mathematics, Euclid numbers are integers of the form En = pn# + 1, where pn# is the primorial of pn which is the nth prime. They are named after the ancient Greek mathematician Euclid.... primes

Even prime

Factorial primeFactorial primeA factorial prime is a prime number that is one less or one more than a factorial . The first few factorial primes are:n! − 1 is prime for :n! + 1 is prime for :...s

Fermat primes

Fibonacci primeFibonacci primeA Fibonacci prime is a Fibonacci number that is prime, a type of integer sequence prime.The first Fibonacci primes are :-Known Fibonacci primes:It is not known if there are infinitely many Fibonacci primes...s

Fortunate primes

Generalized Fermat primes base 10

Genocchi number primes

Gilda's primes

Good primes

Happy primesHappy numberA happy number is defined by the following process. Starting with any positive integer, replace the number by the sum of the squares of its digits, and repeat the process until the number equals 1 , or it loops endlessly in a cycle which does not include 1...

Harmonic primes

Higgs primeHiggs primeA Higgs prime is a prime number with a totient that evenly divides the square of the product of the smaller Higgs primes...s for squares

Highly cototient numberHighly cototient numberIn number theory, a branch of mathematics, a highly cototient number is a positive integer k which is above one and has more solutions to the equation... primes

Irregular primes

Isolated primesTwin primeA twin prime is a prime number that differs from another prime number by two. Except for the pair , this is the smallest possible difference between two primes. Some examples of twin prime pairs are , , , , and...

Kynea primes

Left-truncatable primesTruncatable primeIn number theory, a left-truncatable prime is a prime number which, in a given base, contains no 0, and if the leading digit is successively removed, then all resulting numbers are prime. For example 9137, since 9137, 137, 37 and 7 are all prime...

LeylandLeyland numberIn number theory, a Leyland number is a number of the form xy + yx, where x and y are integers greater than 1. The first few Leyland numbers are... primes

Long primes

Lucas primesLucas numberThe Lucas numbers are an integer sequence named after the mathematician François Édouard Anatole Lucas , who studied both that sequence and the closely related Fibonacci numbers...

Lucky primesLucky numberIn number theory, a lucky number is a natural number in a set which is generated by a "sieve" similar to the Sieve of Eratosthenes that generates the primes.Begin with a list of integers starting with 1:...

MarkovMarkov numberA Markov number or Markoff number is a positive integer x, y or z that is part of a solution to the Markov Diophantine equationx^2 + y^2 + z^2 = 3xyz,\,studied by .The first few Markov numbers are... primes

Mersenne primeMersenne primeIn mathematics, a Mersenne number, named after Marin Mersenne , is a positive integer that is one less than a power of two: M_p=2^p-1.\,...s

Mersenne prime exponents

Mills primes

Minimal primesMinimal prime (number theory)In recreational number theory, a minimal prime is a prime number for which there is no shorter subsequence of its digits in a given base that form a prime. In base 10 there are exactly 26 minimal primes:...

MotzkinMotzkin numberIn mathematics, a Motzkin number for a given number n is the number of different ways of drawing non-intersecting chords on a circle between n points. The Motzkin numbers have very diverse applications in geometry, combinatorics and number theory... primes

Newman–Shanks–Williams primes

Odd primes

PadovanPadovan sequenceThe Padovan sequence is the sequence of integers P defined by the initial valuesP=P=P=1,and the recurrence relationP=P+P.The first few values of P are... primes

Palindromic primePalindromic primeA palindromic prime is a prime number that is also a palindromic number. Palindromicity depends on the base of the numbering system and its writing conventions, while primality is independent of such concerns...s

Palindromic wing primes

Permutable primePermutable primeA permutable prime is a prime number, which, in a given base, can have its digits' positions switched through any permutation and still spell a prime number. H. E...s

PerrinPerrin numberIn mathematics, the Perrin numbers are defined by the recurrence relationandThe sequence of Perrin numbers starts withThe number of different maximal independent sets in an n-vertex cycle graph is counted by the nth Perrin number.-History:... primes

Pierpont primePierpont primeA Pierpont prime is a prime number of the form2^u 3^v + 1\,for some nonnegative integers u and v. They are named after the mathematician James Pierpont....s

Pillai primes

Primeval primes

Primorial primePrimorial primeIn mathematics, primorial primes are prime numbers of the form pn# ± 1, where:The first few primorial primes are, the largest known primorial prime is 843301# - 1 with 365,851 digits, found in 2010 by the PrimeGrid project....s

Proth primes

Pythagorean primePythagorean primeA Pythagorean prime is prime number of the form 4n + 1. These are exactly the primes that can be the hypotenuse of a Pythagorean triangle.The first few Pythagorean primes are...s

Prime quadruplets

Primes of binary quadratic form

Quartan primes

Ramanujan primeRamanujan primeIn mathematics, a Ramanujan prime is a prime number that satisfies a result proven by Srinivasa Ramanujan relating to the prime-counting function.-Origins and definition:...s

Regular primes

RepunitRepunitIn recreational mathematics, a repunit is a number like 11, 111, or 1111 that contains only the digit 1. The term stands for repeated unit and was coined in 1966 by Albert H. Beiler... primes

Right-truncatable primesTruncatable primeIn number theory, a left-truncatable prime is a prime number which, in a given base, contains no 0, and if the leading digit is successively removed, then all resulting numbers are prime. For example 9137, since 9137, 137, 37 and 7 are all prime...

Safe primeSafe primeA safe prime is a prime number of the form 2p + 1, where p is also a prime. The first few safe primes are...s

Sexy primeSexy primeIn mathematics, a sexy prime is a prime number that differs from another prime number by six. For example, the numbers 5 and 11 are both sexy primes, because they differ by 6...s

Smarandache–Wellin primes

Solinas primes

Sophie Germain primeSophie Germain primeIn number theory, a prime number p is a Sophie Germain prime if 2p + 1 is also prime. For example, 23 is a Sophie Germain prime because it is a prime and 2 × 23 + 1 = 47, and 47 is also a prime number...s

StarStar numberA star number is a centered figurate number that represents a centered hexagram, such as the one that Chinese checkers is played on.The nth star number is given by the formula 6n + 1... primes

Super-primeSuper-primeSuper-prime numbers are the subsequence of prime numbers that occupy prime-numbered positions within the sequence of all prime numbers. The subsequence beginsThat is, if p denotes the ith prime number, the numbers in this sequence are those of the form p...s

Supersingular primeSupersingular prime (moonshine theory)In the mathematical branch of moonshine theory, a supersingular prime is a certain type of prime number.Namely, a supersingular prime is a prime divisor of the order of the Monster group M, the largest of the sporadic simple groups...s

Swinging primes

Thabit numberThabit numberIn number theory, a Thabit number, Thâbit ibn Kurrah number, or 321 number is an integer of the form 3·2n−1 for a non-negative integer n... primes

Prime tripletPrime tripletIn mathematics, a prime triplet is a set of three prime numbers of the form or...s

Twin primeTwin primeA twin prime is a prime number that differs from another prime number by two. Except for the pair , this is the smallest possible difference between two primes. Some examples of twin prime pairs are , , , , and...s

Two-sided primes

Ulam number primes

Wagstaff primes

Wall-Sun-Sun primeWall-Sun-Sun primeIn number theory, a Wall–Sun–Sun prime or Fibonacci–Wieferich prime is a certain kind of prime number which is conjectured to exist although none are known.- Definition :...s

Centered decagonal
Centered decagonal number
A centered decagonal number is a centered figurate number that represents a decagon with a dot in the center and all other dots surrounding the center dot in successive decagonal layers...

primes

Centered heptagonal
Centered heptagonal number
A centered heptagonal number is a centered figurate number that represents a heptagon with a dot in the center and all other dots surrounding the center dot in successive heptagonal layers...

primes

Centered triangular
Centered triangular number
A centered triangular number is a centered figurate number that represents a triangle with a dot in the center and all other dots surrounding the center in successive triangular layers...

primes

Chen prime
Chen prime
A prime number p is called a Chen prime if p + 2 is either a prime or a product of two primes. The even number 2p + 2 therefore satisfies Chen's theorem....

s

Circular prime
Circular prime
A circular prime is a prime number with the property that the number generated at each intermediate step when cyclically permuting its digits will be prime. For example 1193 is a circular prime, since 1931, 9311 and 3119 all are also prime...

s

Cousin prime
Cousin prime
In mathematics, cousin primes are prime numbers that differ by four; compare this with twin primes, pairs of prime numbers that differ by two, and sexy primes, pairs of prime numbers that differ by six....

s

Cullen
Cullen number
In mathematics, a Cullen number is a natural number of the form n · 2n + 1 . Cullen numbers were first studied by Fr. James Cullen in 1905...

primes

Dihedral prime
Dihedral prime
A dihedral prime or dihedral calculator prime is a prime number that still reads like itself or another prime number when read in a seven-segment display, regardless of orientation , and surface...

s

Emirp
Emirp
An emirp is a prime number that results in a different prime when its digits are reversed. This definition excludes the related palindromic primes. Emirps are also called reversible primes....

s

Euclid
Euclid number
In mathematics, Euclid numbers are integers of the form En = pn# + 1, where pn# is the primorial of pn which is the nth prime. They are named after the ancient Greek mathematician Euclid....

primes

Factorial prime
Factorial prime
A factorial prime is a prime number that is one less or one more than a factorial . The first few factorial primes are:n! − 1 is prime for :n! + 1 is prime for :...

s

Fibonacci prime
Fibonacci prime
A Fibonacci prime is a Fibonacci number that is prime, a type of integer sequence prime.The first Fibonacci primes are :-Known Fibonacci primes:It is not known if there are infinitely many Fibonacci primes...

s

Happy primes
Happy number
A happy number is defined by the following process. Starting with any positive integer, replace the number by the sum of the squares of its digits, and repeat the process until the number equals 1 , or it loops endlessly in a cycle which does not include 1...

Higgs prime
Higgs prime
A Higgs prime is a prime number with a totient that evenly divides the square of the product of the smaller Higgs primes...

s for squares

Highly cototient number
Highly cototient number
In number theory, a branch of mathematics, a highly cototient number is a positive integer k which is above one and has more solutions to the equation...

primes

Isolated primes
Twin prime
A twin prime is a prime number that differs from another prime number by two. Except for the pair , this is the smallest possible difference between two primes. Some examples of twin prime pairs are , , , , and...

Left-truncatable primes
Truncatable prime
In number theory, a left-truncatable prime is a prime number which, in a given base, contains no 0, and if the leading digit is successively removed, then all resulting numbers are prime. For example 9137, since 9137, 137, 37 and 7 are all prime...

Leyland
Leyland number
In number theory, a Leyland number is a number of the form xy + yx, where x and y are integers greater than 1. The first few Leyland numbers are...

primes

Lucas primes
Lucas number
The Lucas numbers are an integer sequence named after the mathematician François Édouard Anatole Lucas , who studied both that sequence and the closely related Fibonacci numbers...

Lucky primes
Lucky number
In number theory, a lucky number is a natural number in a set which is generated by a "sieve" similar to the Sieve of Eratosthenes that generates the primes.Begin with a list of integers starting with 1:...

Markov
Markov number
A Markov number or Markoff number is a positive integer x, y or z that is part of a solution to the Markov Diophantine equationx^2 + y^2 + z^2 = 3xyz,\,studied by .The first few Markov numbers are...

primes

Mersenne prime
Mersenne prime
In mathematics, a Mersenne number, named after Marin Mersenne , is a positive integer that is one less than a power of two: M_p=2^p-1.\,...

s

Minimal primes
Minimal prime (number theory)
In recreational number theory, a minimal prime is a prime number for which there is no shorter subsequence of its digits in a given base that form a prime. In base 10 there are exactly 26 minimal primes:...

Motzkin
Motzkin number
In mathematics, a Motzkin number for a given number n is the number of different ways of drawing non-intersecting chords on a circle between n points. The Motzkin numbers have very diverse applications in geometry, combinatorics and number theory...

primes

Padovan
Padovan sequence
The Padovan sequence is the sequence of integers P defined by the initial valuesP=P=P=1,and the recurrence relationP=P+P.The first few values of P are...

primes

Palindromic prime
Palindromic prime
A palindromic prime is a prime number that is also a palindromic number. Palindromicity depends on the base of the numbering system and its writing conventions, while primality is independent of such concerns...

s

Permutable prime
Permutable prime
A permutable prime is a prime number, which, in a given base, can have its digits' positions switched through any permutation and still spell a prime number. H. E...

s

Perrin
Perrin number
In mathematics, the Perrin numbers are defined by the recurrence relationandThe sequence of Perrin numbers starts withThe number of different maximal independent sets in an n-vertex cycle graph is counted by the nth Perrin number.-History:...

primes

Pierpont prime
Pierpont prime
A Pierpont prime is a prime number of the form2^u 3^v + 1\,for some nonnegative integers u and v. They are named after the mathematician James Pierpont....

s

Primorial prime
Primorial prime
In mathematics, primorial primes are prime numbers of the form pn# ± 1, where:The first few primorial primes are, the largest known primorial prime is 843301# - 1 with 365,851 digits, found in 2010 by the PrimeGrid project....

s

Pythagorean prime
Pythagorean prime
A Pythagorean prime is prime number of the form 4n + 1. These are exactly the primes that can be the hypotenuse of a Pythagorean triangle.The first few Pythagorean primes are...

s

Ramanujan prime
Ramanujan prime
In mathematics, a Ramanujan prime is a prime number that satisfies a result proven by Srinivasa Ramanujan relating to the prime-counting function.-Origins and definition:...

s

Repunit
Repunit
In recreational mathematics, a repunit is a number like 11, 111, or 1111 that contains only the digit 1. The term stands for repeated unit and was coined in 1966 by Albert H. Beiler...

primes

Right-truncatable primes
Truncatable prime
In number theory, a left-truncatable prime is a prime number which, in a given base, contains no 0, and if the leading digit is successively removed, then all resulting numbers are prime. For example 9137, since 9137, 137, 37 and 7 are all prime...

Safe prime
Safe prime
A safe prime is a prime number of the form 2p + 1, where p is also a prime. The first few safe primes are...

s

Sexy prime
Sexy prime
In mathematics, a sexy prime is a prime number that differs from another prime number by six. For example, the numbers 5 and 11 are both sexy primes, because they differ by 6...

s

Sophie Germain prime
Sophie Germain prime
In number theory, a prime number p is a Sophie Germain prime if 2p + 1 is also prime. For example, 23 is a Sophie Germain prime because it is a prime and 2 × 23 + 1 = 47, and 47 is also a prime number...

s

Star
Star number
A star number is a centered figurate number that represents a centered hexagram, such as the one that Chinese checkers is played on.The nth star number is given by the formula 6n + 1...

primes

Super-prime
Super-prime
Super-prime numbers are the subsequence of prime numbers that occupy prime-numbered positions within the sequence of all prime numbers. The subsequence beginsThat is, if p denotes the ith prime number, the numbers in this sequence are those of the form p...

s

Supersingular prime
Supersingular prime (moonshine theory)
In the mathematical branch of moonshine theory, a supersingular prime is a certain type of prime number.Namely, a supersingular prime is a prime divisor of the order of the Monster group M, the largest of the sporadic simple groups...

s

Thabit number
Thabit number
In number theory, a Thabit number, Thâbit ibn Kurrah number, or 321 number is an integer of the form 3·2n−1 for a non-negative integer n...

primes

Prime triplet
Prime triplet
In mathematics, a prime triplet is a set of three prime numbers of the form or...

s

Twin prime
Twin prime
A twin prime is a prime number that differs from another prime number by two. Except for the pair , this is the smallest possible difference between two primes. Some examples of twin prime pairs are , , , , and...

s

Wall-Sun-Sun prime
Wall-Sun-Sun prime
In number theory, a Wall–Sun–Sun prime or Fibonacci–Wieferich prime is a certain kind of prime number which is conjectured to exist although none are known.- Definition :...

s

Weakly prime number
Weakly prime number
In number theory, a prime number is called weakly prime if it becomes composite when any one of its digits is changed to every single other digit. Decimal digits are usually assumed.The first weakly prime numbers are:...

s

Wieferich prime
Wieferich prime
In number theory, a Wieferich prime is a prime number p such that p2 divides 2p − 1 − 1, therefore connecting these primes with Fermat's little theorem, which states that every odd prime p divides 2p − 1 − 1...

s

Wolstenholme prime
Wolstenholme prime
In number theory, a Wolstenholme prime is a special type of prime number satisfying a stronger version of Wolstenholme's theorem. Wolstenholme's theorem is a congruence relation satisfied by all prime numbers greater than 7...

s