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prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...

s. A prime number (or ''prime'') is a

natural number In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...

greater than 1 that has no positive

divisor In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a '' multiple'' of m. An integer n is divisible or evenly divisibl ...

s other than 1 and itself. By

Euclid's theorem Euclid's theorem is a fundamental statement in number theory that asserts that there are Infinite set, infinitely many prime number, prime numbers. It was first proven by Euclid in his work ''Euclid's Elements, Elements''. There are several proof ...

, there are an infinite number of prime numbers. Subsets of the prime numbers may be generated with various formulas for primes. The first 1000 primes are listed below, followed by lists of notable types of prime numbers in alphabetical order, giving their respective first terms. 1 is neither prime nor

composite Composite or compositing may refer to: Materials * Composite material, a material that is made from several different substances ** Metal matrix composite, composed of metal and other parts ** Cermet, a composite of ceramic and metallic material ...

The first 1000 prime numbers

The following table lists the first 1000 primes, with 20 columns of consecutive primes in each of the 50 rows. . The

Goldbach conjecture Goldbach's conjecture is one of the oldest and best-known unsolved problems in number theory and all of mathematics. It states that every even natural number greater than 2 is the sum of two prime numbers. The conjecture has been shown to ho ...

verification project reports that it has computed all primes smaller than 4×10. That means 95,676,260,903,887,607 primes (nearly 10), but they were not stored. There are known formulae to evaluate the

prime-counting function In mathematics, the prime-counting function is the function counting the number of prime numbers less than or equal to some real number . It is denoted by (unrelated to the number ). A symmetric variant seen sometimes is , which is equal ...

(the number of primes smaller than a given value) faster than computing the primes. This has been used to compute that there are 1,925,320,391,606,803,968,923 primes (roughly 2) smaller than 10. A different computation found that there are 18,435,599,767,349,200,867,866 primes (roughly 2) smaller than 10, if the

Riemann hypothesis In mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part . Many consider it to be the most important unsolved problem in pure ...

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

(including 0) in the definitions.

Balanced prime In number theory, a balanced prime is a prime number with equal-sized prime gaps above and below it, so that it is equal to the arithmetic mean of the nearest primes above and below. Or to put it algebraically, the nth prime number p_n is a balanced ...
s

Primes with equal-sized prime gaps after and before them, so that they are equal to the

arithmetic mean In mathematics and statistics, the arithmetic mean ( ), arithmetic average, or just the ''mean'' or ''average'' is the sum of a collection of numbers divided by the count of numbers in the collection. The collection is often a set of results fr ...

of the nearest primes after and before. * 5, 53, 157,

173 Year 173 ( CLXXIII) was a common year starting on Thursday of the Julian calendar. At the time, it was known as the Year of the Consulship of Severus and Pompeianus (or, less frequently, year 926 ''Ab urbe condita''). The denomination 173 for th ...

, 211,

257 __NOTOC__ Year 257 (Roman numerals, CCLVII) was a common year starting on Thursday of the Julian calendar. At the time, it was known as the Year of the Consulship of Valerianus and Gallienus (or, less frequently, year 1010 ''Ab urbe condita'') ...

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

Bell primes

Primes that are the number of

partitions of a set In mathematics, a partition of a set is a grouping of its elements into non-empty subsets, in such a way that every element is included in exactly one subset. Every equivalence relation on a set defines a partition of this set, and every part ...

with ''n'' members. 2, 5,

877 __NOTOC__ Year 877 ( DCCCLXXVII) was a common year starting on Tuesday of the Julian calendar. Events By place Europe * Summer – King Charles II ("the Bald") sets out for Italy, accompanied by his wife Richilde and a number ...

, 27644437, 35742549198872617291353508656626642567, 359334085968622831041960188598043661065388726959079837. The next term has 6,539 digits. ()

Chen primes

Where ''p'' is prime and ''p''+2 is either a prime or

semiprime In mathematics, a semiprime is a natural number that is the product of exactly two prime numbers. The two primes in the product may equal each other, so the semiprimes include the squares of prime numbers. Because there are infinitely many prime n ...

. 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 47, 53, 59, 67, 71, 83, 89, 101,

107 107 may refer to: *107 (number), the number *AD 107, a year in the 2nd century AD *107 BC, a year in the 2nd century BC *107 (New Jersey bus) *107 Camilla, a main-belt asteroid *Peugeot 107, a city car See also

*10/7 (disambiguation) *Bohrium, ...

, 109,

113 113 may refer to: *113 (number), a natural number *AD 113, a year *113 BC, a year *113 (band), a French hip hop group *113 (MBTA bus), Massachusetts Bay Transportation Authority bus route *113 (New Jersey bus), Ironbound Garage in Newark and run to ...

, 127, 131, 137,

139 139 may refer to: * 139 (number), an integer * AD 139, a year of the Julian calendar * 139 BC, a year of the pre-Julian Roman calendar * 139 (New Jersey bus) * 139 Juewa 139 Juewa ( ) is a very large and dark main belt asteroid. It is probabl ...

, 149,

157 Year 157 ( CLVII) was a common year starting on Friday of the Julian calendar. At the time, it was known as the Year of the Consulship of Civica and Aquillus (or, less frequently, year 910 ''Ab urbe condita''). The denomination 157 for this ye ...

, 167, 179,

181 Year 181 ( CLXXXI) was a common year starting on Sunday of the Julian calendar. At the time, it was known as the Year of the Consulship of Aurelius and Burrus (or, less frequently, year 934 ''Ab urbe condita''). The denomination 181 for this ye ...

, 191, 197,

199 Year 199 ( CXCIX) was a common year starting on Monday of the Julian calendar. At the time, it was sometimes known as year 952 ''Ab urbe condita''. The denomination 199 for this year has been used since the early medieval period, when the Anno ...

211 Year 211 ( CCXI) was a common year starting on Tuesday of the Julian calendar. At the time, in the Roman Empire it was known as the Year of the Consulship of Terentius and Bassus (or, less frequently, year 964 ''Ab urbe condita''). The denomin ...

, 227,

233 __NOTOC__ Year 233 ( CCXXXIII) was a common year starting on Tuesday of the Julian calendar. At the time, it was known as the Year of the Consulship of Claudius and Paternus (or, less frequently, year 986 ''Ab urbe condita''). The denomination ...

239 __NOTOC__ Year 239 ( CCXXXIX) was a common year starting on Tuesday of the Julian calendar. At the time, it was known as the Year of the Consulship of Gordianus and Aviola (or, less frequently, year 992 ''Ab urbe condita''). The denomination 23 ...

251 __NOTOC__ Year 251 (Roman numerals, CCLI) was a common year starting on Wednesday of the Julian calendar. At the time, in the Roman Empire, it was known as the Year of the Consulship of Traianus and Etruscus (or, less frequently, year 1004 ''A ...

263 __NOTOC__ Year 263 (Roman numerals, CCLXIII) was a common year starting on Thursday of the Julian calendar. At the time, it was known as the Year of the Consulship of Albinus and Dexter (or, less frequently, year 1016 ''Ab urbe condita''). The ...

269 Year 269 ( CCLXIX) was a common year starting on Friday of the Julian calendar. At the time, it was known as the Year of the Consulship of Claudius and Paternus (or, less frequently, year 1022 ''Ab urbe condita''). The denomination 269 for this ...

, 281, 293, 307, 311, 317,

337 __NOTOC__ Year 337 (Roman numerals, CCCXXXVII) was a common year starting on Saturday of the Julian calendar. At the time, it was known as the Year of the Consulship of Felicianus and Titianus (or, less frequently, year 1090 ''Ab urbe condita ...

, 347,

353 __NOTOC__ Year 353 (Roman numerals, CCCLIII) was a common year starting on Friday of the Julian calendar. At the time, it was known as the Year of the Consulship of Magnentius and Decentius (or, less frequently, year 1106 ''Ab urbe condita''). ...

359 __NOTOC__ Year 359 (Roman numerals, CCCLIX) was a common year starting on Friday of the Julian calendar. At the time, it was known as the Year of the Consulship of Eusebius and Hypatius (or, less frequently, year 1112 ''Ab urbe condita''). The ...

379 Year 379 ( CCCLXXIX) was a common year starting on Tuesday of the Julian calendar. At the time, it was known as the Year of the Consulship of Ausonius and Hermogenianus (or, less frequently, year 1132 ''Ab urbe condita''). The denomination 379 ...

389 Year 389 (Roman numerals, CCCLXXXIX) was a common year starting on Monday of the Julian calendar. At the time, it was known as the Year of the Consulship of Timasius and Promotus (or, less frequently, year 1142 ''Ab urbe condita''). The denomina ...

, 401, 409 ()

Circular primes

A circular prime number is a number that remains prime on any cyclic rotation of its digits (in base 10). 2, 3, 5, 7, 11, 13, 17, 31, 37, 71, 73, 79, 97,

, 131, 197,

, 311,

373 __NOTOC__ Year 373 ( CCCLXXIII) was a common year starting on Tuesday of the Julian calendar. At the time, it was known as the Year of the Consulship of Augustus and Valens (or, less frequently, year 1126 ''Ab urbe condita''). The denomination ...

719 __NOTOC__ Year 719 ( DCCXIX) was a common year starting on Sunday of the Julian calendar, the 719th year of the Common Era (CE) and Anno Domini (AD) designations, the 719th year of the 1st millennium, the 19th year of the 8th century, and the 1 ...

, 733,

919 __NOTOC__ Year 919 ( CMXIX) was a common year starting on Friday of the Julian calendar. Events By Place Byzantine Empire * March 25 – Romanos Lekapenos, admiral (''droungarios'') of the Byzantine navy, seizes the Boukoleon Pal ...

971 Year 971 ( CMLXXI) was a common year starting on Sunday of the Julian calendar. Events By place Byzantine Empire * Battle of Dorostolon: A Byzantine expeditionary army (possibly 30–40,000 men) attacks the Bulgarian frontier, perso ...

991 Year 991 (Roman numerals, CMXCI) was a common year starting on Thursday of the Julian calendar. Events * March 1: In Rouen, Pope John XV ratifies the first Peace and Truce of God, Truce of God, between Æthelred the Unready and Richard I o ...

, 1193,

1931 Events January * January 2 – South Dakota native Ernest Lawrence invents the cyclotron, used to accelerate particles to study nuclear physics. * January 4 – German pilot Elly Beinhorn begins her flight to Africa. * January 22 – Sir I ...

, 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 The On-Line Encyclopedia of Integer Sequences (OEIS) is an online database of integer sequences. It was created and maintained by Neil Sloane while researching at AT&T Labs. He transferred the intellectual property and hosting of the OEIS to th ...

really calls this sequence circular primes, but not the above sequence): 2, 3, 5, 7, 11, 13, 17, 37, 79,

, 197,

, 1193, 3779, 11939, 19937, 193939, 199933, 1111111111111111111, 11111111111111111111111 () All

repunit In recreational mathematics, a repunit is a number like 11, 111, or 1111 that contains only the digit 1 — a more specific type of repdigit. The term stands for "repeated unit" and was coined in 1966 by Albert H. Beiler in his book ''Recr ...

primes are circular.

Cluster prime In number theory, a cluster prime is a prime number such that every even positive integer ''k'' ≤ p − 3 can be written as the difference between two prime numbers not exceeding (). For example, the number 23 is a cluster prime becau ...
s

A cluster prime is a prime ''p'' such that every even

''k'' ≤ ''p'' − 3 is the difference of two primes not exceeding ''p''. 3, 5, 7, 11, 13, 17, 19, 23, ... () All primes between 3 and 89, inclusive, are cluster primes. The first 10 primes that are ''not'' cluster primes are: 2, 97, 127, 149, 191,

223 __NOTOC__ Year 223 ( CCXXIII) was a common year starting on Wednesday of the Julian calendar. At the time, it was known as the Year of the Consulship of Maximus and Aelianus (or, less frequently, year 976 ''Ab urbe condita''). The denomination ...

, 227,

229 __NOTOC__ Year 229 ( CCXXIX) was a common year starting on Thursday of the Julian calendar. At the time, it was known as the Year of the Consulship of Severus and Cassius (or, less frequently, year 982 ''Ab urbe condita''). The denomination 22 ...

Cousin primes

Where (''p'', ''p'' + 4) are both prime. ( 3, 7), ( 7, 11), ( 13, 17), ( 19, 23), ( 37, 41), ( 43, 47), ( 67, 71), ( 79, 83), ( 97, 101), ( 103,

*10/7 (disambiguation) *Bohrium, ...

), ( 109,

), ( 127, 131), (

163 Year 163 ( CLXIII) was a common year starting on Friday of the Julian calendar. At the time, it was known as the Year of the Consulship of Laelianus and Pastor (or, less frequently, year 916 ''Ab urbe condita''). The denomination 163 for this y ...

, 167), (

193 Year 193 ( CXCIII) was a common year starting on Monday of the Julian calendar. At the time, it was known as the Year of the Consulship of Sosius and Ericius (or, less frequently, year 946 ''Ab urbe condita''). The denomination 193 for this ye ...

, 197), (

, 227), (

), ( 277, 281) (, )

Cuban primes

Of the form

\tfrac

where ''x'' = ''y'' + 1. 7, 19, 37, 61, 127,

271 __NOTOC__ Year 271 ( CCLXXI) was a common year starting on Sunday of the Julian calendar. At the time, it was known as the Year of the Consulship of Aurelianus and Bassus (or, less frequently, year 1024 ''Ab urbe condita''). The denomination 2 ...

331 __NOTOC__ Year 331 ( CCCXXXI) was a common year starting on Friday of the Julian calendar. At the time, it was known as the Year of the Consulship of Bassus and Ablabius (or, less frequently, year 1084 ''Ab urbe condita''). The denomination ...

, 397, 547, 631,

1657 Events January–March * January 8 – Miles Sindercombe and his group of disaffected Levellers are betrayed in their attempt to assassinate Oliver Cromwell by blowing up the Palace of Whitehall in London and are arrested. * Ja ...

1801 Events January–March *January 1 ** The legislative union of Great Britain and Ireland is completed under the Act of Union 1800, bringing about the United Kingdom of Great Britain and Ireland, and the abolition of the Parliament of Ir ...

1951 Events January * January 4 – Korean War: Third Battle of Seoul – Chinese and North Korean forces capture Seoul for the second time (having lost the Second Battle of Seoul in September 1950). * January 9 – The Government of the Uni ...

, 2269, 2437, 2791, 3169, 3571, 4219, 4447, 5167, 5419, 6211, 7057, 7351, 8269, 9241, 10267, 11719, 12097, 13267, 13669, 16651, 19441, 19927, 22447, 23497, 24571, 25117, 26227, 27361, 33391, 35317 () Of the form

\tfrac

where ''x'' = ''y'' + 2. 13, 109,

, 433, 769,

1201 Year 1201 ( MCCI) was a common year starting on Monday of the Julian calendar. Events By place Byzantine Empire * July 31 – John Komnenos the Fat, a Byzantine aristocrat, attempts to usurp the imperial throne; he is proclaim ...

1453 Year 1453 ( MCDLIII) was a common year starting on Monday of the Julian calendar, the 1453rd year of the Common Era (CE) and ''Anno Domini'' (AD) designations, the 453rd year of the 2nd millennium, the 53rd year of the 15th century, and the 4 ...

2029 Predicted and scheduled events * January 1 – Books, films and other works published in 1933 will enter the public domain. * March – The 2029 Special Olympics World Winter Games are scheduled to be held in Switzerland. * April 13 &n ...

, 3469, 3889, 4801, 10093, 12289, 13873, 18253, 20173, 21169, 22189, 28813, 37633, 43201, 47629, 60493, 63949, 65713, 69313, 73009, 76801, 84673, 106033, 108301, 112909, 115249 ()

Cullen primes

Of the form ''n''×2 + 1. 3, 393050634124102232869567034555427371542904833 ()

Delicate primes

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 ()

Dihedral primes

Primes that remain prime when read upside down or mirrored in a

seven-segment display A seven-segment display is a display device for Arabic numerals, less complex than a device that can show more characters such as dot matrix displays. Seven-segment displays are widely used in digital clocks, electronic meters, basic calculators, ...

. 2, 5, 11, 101,

, 1181,

1811 Events January–March * January 8 – An unsuccessful slave revolt is led by Charles Deslondes, in St. Charles and St. James Parishes, Louisiana. * January 17 – Mexican War of Independence – Battle of Calderón ...

, 18181, 108881, 110881, 118081, 120121, 121021, 121151, 150151, 151051, 151121, 180181, 180811, 181081 ()

Eisenstein primes without imaginary part

Eisenstein integer In mathematics, the Eisenstein integers (named after Gotthold Eisenstein), occasionally also known as Eulerian integers (after Leonhard Euler), are the complex numbers of the form : z = a + b\omega , where and are integers and : \omega = \frac ...

s that are irreducible and real numbers (primes of the form 3''n'' − 1). 2, 5, 11, 17, 23, 29, 41, 47, 53, 59, 71, 83, 89, 101,

*10/7 (disambiguation) *Bohrium, ...

, 131, 137, 149, 167,

, 179, 191, 197, 227,

, 281, 293, 311, 317, 347,

383 383 most commonly refers to: * 383 (number), the natural number following 382 and preceding 384. * AD 383, a year * 383 BC, a year 383 may also refer to: Astronomy * 383 Janina, a Themistian asteroid * Abell 383, a galaxy cluster * HOPS 383, a c ...

, 401 ()

Emirps

Primes that become a different prime when their decimal digits are reversed. The name "emirp" is the reverse of the word "prime". 13, 17, 31, 37, 71, 73, 79, 97,

*10/7 (disambiguation) *Bohrium, ...

, 149,

, 167, 179,

, 311,

, 347,

701 __NOTOC__ Year 701 ( DCCI) was a common year starting on Saturday of the Julian calendar, the 701st year of the Common Era (CE) and Anno Domini (AD) designations, the 701st year of the 1st millennium, the 1st year of the 8th century, and the ...

, 709, 733, 739, 743, 751, 761, 769, 907,

937 Year 937 ( CMXXXVII) was a common year starting on Sunday of the Julian calendar. Events By place Europe * A Hungarian army invades Burgundy, and burns the city of Tournus. Then they go southwards to Italy, pillaging the environs of ...

, 941,

953 Year 953 ( CMLIII) was a common year starting on Saturday of the Julian calendar. Events By place Byzantine Empire * Battle of Marash: Emir Sayf al-Dawla marches north into the Byzantine Empire and ravages the countryside of Malatya ...

967 Year 967 ( CMLXVII) was a common year starting on Tuesday of the Julian calendar. Events By place Europe * Spring – Emperor Otto I (the Great) calls for a council at Rome, to present the new government under Pope John XIII. He ...

983 Year 983 ( CMLXXXIII) was a common year starting on Monday of the Julian calendar. Events By place Europe * Summer – Diet of Verona: Emperor Otto II (the Red) declares war against the Byzantine Empire and the Emirate of Sicily ...

()

Euclid primes

Of the form ''p''# + 1 (a subset of

primorial prime In mathematics, a primorial prime is a prime number of the form ''pn''# ± 1, where ''pn''# is the primorial of ''pn'' (i.e. the product of the first ''n'' primes). Primality tests show that: : ''pn''# − 1 is prime for ...

s). 3, 7, 31,

, 2311, 200560490131 ( includes 2 =

empty product In mathematics, an empty product, or nullary product or vacuous product, is the result of multiplication, multiplying no factors. It is by convention equal to the multiplicative identity (assuming there is an identity for the multiplication operat ...

of first 0 primes plus 1, but 2 is excluded in this list.)

Euler irregular primes

A prime

p

that divides

Euler number Leonhard Euler ( ; ; ; 15 April 170718 September 1783) was a Swiss polymath who was active as a mathematician, physicist, astronomer, logician, geographer, and engineer. He founded the studies of graph theory and topology and made influential ...

E_

for some

0\leq 2n\leq p-3

. 19, 31, 43, 47, 61, 67, 71, 79, 101, 137,

, 149,

241 Year 241 ( CCXLI) was a common year starting on Friday of the Julian calendar. At the time, it was known as the Year of the Consulship of Gordianus and Pompeianus by the Romans (or, less frequently, year 994 ''Ab urbe condita''). The denominat ...

, 277, 307, 311,

349 __NOTOC__ Year 349 (Roman numerals, CCCXLIX) was a common year starting on Sunday of the Julian calendar. At the time, it was known as the Year of the Consulship of Limenius and Catullinus (or, less frequently, year 1102 ''Ab urbe condita''). T ...

, 419, 433, 461, 463,

491 __NOTOC__ Year 491 (Roman numerals, CDXCI) was a common year starting on Tuesday of the Julian calendar. At the time, it was known as the Year of the Consulship of Olybrius (consul 491), Olybrius without colleague (or, less frequently, year 12 ...

509 __NOTOC__ Year 509 (DIX) was a common year starting on Thursday of the Julian calendar. At the time, it was known as the Year of the Consulship of Inportunus without colleague (or, less frequently, year 1262 ''Ab urbe condita''). The denomina ...

, 541, 563, 571, 577, 587 ()

Euler (''p'', ''p'' − 3) irregular primes

Primes

p

such that

(p, p-3)

is an Euler irregular pair. 149,

, 2946901 ()

Factorial prime A factorial prime is a prime number that is one less or one more than a factorial (all factorials greater than 1 are even). The first 10 factorial primes (for ''n'' = 1, 2, 3, 4, 6, 7, 11, 12, 14) are : : 2 (0! + 1 or 1! + 1) ...
s

Of the form ''n'' ! − 1 or ''n'' ! + 1. 2, 3, 5, 7, 23,

, 5039, 39916801, 479001599, 87178291199, 10888869450418352160768000001, 265252859812191058636308479999999, 263130836933693530167218012159999999, 8683317618811886495518194401279999999 ()

Fermat prime In mathematics, a Fermat number, named after Pierre de Fermat (1601–1665), the first known to have studied them, is a positive integer of the form:F_ = 2^ + 1, where ''n'' is a non-negative integer. The first few Fermat numbers are: 3, 5, ...
s

Of the form 2 + 1. 3, 5, 17,

65537 65537 is the integer after 65536 and before 65538. In mathematics 65537 is the largest known prime number of the form 2^ +1 (n = 4), and is most likely the last one. Therefore, a regular polygon with 65537 sides is constructible with compass ...

() these are the only known Fermat primes, and conjecturally the only Fermat primes. The probability of the existence of another Fermat prime is less than one in a billion.

Generalized
Fermat primes In mathematics, a Fermat number, named after Pierre de Fermat (1601–1665), the first known to have studied them, is a positive integer of the form:F_ = 2^ + 1, where ''n'' is a non-negative integer. The first few Fermat numbers are: 3, 5, 17 ...

Of the form ''a'' + 1 for fixed integer ''a''. ''a'' = 2: 3, 5, 17,

() ''a'' = 4: 5, 17,

''a'' = 6: 7, 37, 1297 ''a'' = 8: (none exist) ''a'' = 10: 11, 101 ''a'' = 12: 13 ''a'' = 14: 197 ''a'' = 16: 17,

''a'' = 18: 19 ''a'' = 20: 401, 160001 ''a'' = 22: 23 ''a'' = 24: 577, 331777

Fibonacci primes

Primes in the

Fibonacci sequence In mathematics, the Fibonacci sequence is a Integer sequence, sequence in which each element is the sum of the two elements that precede it. Numbers that are part of the Fibonacci sequence are known as Fibonacci numbers, commonly denoted . Many w ...

''F'' = 0, ''F'' = 1, ''F'' = ''F'' + ''F''. 2, 3, 5, 13, 89,

1597 Events January–March * January 4 – Japan's Chancellor of the Realm, Toyotomi Hideyoshi, sends 26 European Christians, arrested on December 8, 1596, on a forced march from Kyoto to Nagasaki. * January 24 – Battle of Turnhout: M ...

, 28657, 514229, 433494437, 2971215073, 99194853094755497, 1066340417491710595814572169, 19134702400093278081449423917 ()

Fortunate primes

Fortunate numbers that are prime (it has been conjectured they all are). 3, 5, 7, 13, 17, 19, 23, 37, 47, 59, 61, 67, 71, 79, 89, 101, 103,

*10/7 (disambiguation) *Bohrium, ...

, 109, 127, 151,

, 167, 191, 197,

, 277,

283 Year 283 ( CCLXXXIII) was a common year starting on Monday of the Julian calendar. At the time, it was known as the Year of the Consulship of Carus and Carinus (or, less frequently, year 1036 ''Ab urbe condita''). The denomination 283 for this y ...

, 293, 307, 311, 313,

, 397 ()

Gaussian prime In number theory, a Gaussian integer is a complex number whose real and imaginary parts are both integers. The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain, usually written as \mathbf /ma ...
s

Prime element In mathematics, specifically in abstract algebra, a prime element of a commutative ring is an object satisfying certain properties similar to the prime numbers in the integers and to irreducible polynomials. Care should be taken to distinguish ...

s of the Gaussian integers; equivalently, primes of the form 4''n'' + 3. 3, 7, 11, 19, 23, 31, 43, 47, 59, 67, 71, 79, 83, 103,

*10/7 (disambiguation) *Bohrium, ...

, 127, 131,

, 151,

, 167, 179, 191,

, 227,

, 307, 311,

, 347,

, 367,

, 419, 431, 439, 443, 463,

467 __NOTOC__ Year 467 (Roman numerals, CDLXVII) was a common year starting on Sunday of the Julian calendar. At the time, it was known as the Year of the Consulship of Pusaeus and Iohannes (consul 467), Iohannes (or, less frequently, year 1220 '' ...

, 479, 487,

, 499,

503 __NOTOC__ Year 503 ( DIII) was a common year starting on Wednesday of the Julian calendar. At the time, it was known as the Year of the Consulship of Volusianus and Dexicrates (or, less frequently, year 1256 ''Ab urbe condita''). The denominati ...

()

Good prime A good prime is a prime number whose square is greater than the product of any two primes at the same number of positions before and after it in the sequence of primes. That is, good prime satisfies the inequality :p_n^2 > p_ \cdot p_ for all 1 ...
s

Primes ''p'' for which ''p'' > ''p'' ''p'' for all 1 ≤ ''i'' ≤ ''n''−1, where ''p'' is the ''n''th prime. 5, 11, 17, 29, 37, 41, 53, 59, 67, 71, 97, 101, 127, 149, 179, 191,

, 227,

, 307 ()

Happy prime In number theory, a happy number is a number which eventually reaches 1 when the number is replaced by the sum of the square of each digit. For instance, 13 is a happy number because 1^2+3^2=10, and 1^2+0^2=1. On the other hand, 4 is not a happy ...
s

Happy numbers that are prime. 7, 13, 19, 23, 31, 79, 97, 103, 109,

, 167,

, 293, 313,

, 367,

, 397, 409, 487, 563,

617 __NOTOC__ Year 617 ( DCXVII) was a common year starting on Saturday of the Julian calendar. The denomination 617 for this year has been used since the early medieval period, when the Anno Domini calendar era became the prevalent method in Euro ...

, 653,

673 __NOTOC__ Year 673 ( DCLXXIII) was a common year starting on Saturday of the Julian calendar. The denomination 673 for this year has been used since the early medieval period, when the Anno Domini calendar era became the prevalent method in Eur ...

683 __NOTOC__ Year 683 (Roman numerals, DCLXXXIII) was a common year starting on Thursday of the Julian calendar. The denomination 683 for this year has been used since the early medieval period, when the Anno Domini calendar era became the prevale ...

, 709, 739, 761, 863,

881 __NOTOC__ Year 881 ( DCCCLXXXI) was a common year starting on Sunday of the Julian calendar. Events By place Europe * February 12 – King Charles the Fat, the third son of the late Louis the German, is crowned as Holy Roman Emper ...

, 907,

1009 Year 1009 ( MIX) was a common year starting on Saturday of the Julian calendar. It was the 1009th year of the Common Era (CE) and Anno Domini (AD) designations, the 9th year of the 2nd millennium, the 9th year of the 11th century, and the 10th ...

1033 Year 1033 (Roman numerals, MXXXIII) was a common year starting on Monday (the wikilink will display the full calendar) of the Julian calendar. Events By place Asia * December 5 – A 1033 Jordan Rift Valley earthquake, major earth ...

1039 Year 1039 ( MXXXIX) was a common year starting on Monday of the Julian calendar. Events By place Europe * June 4 – Conrad II, Holy Roman Emperor, ("the Elder") dies of gout in Utrecht after a 12-year reign. He is succeeded by h ...

, 1093 ()

Harmonic primes

Primes ''p'' for which there are no solutions to ''H'' ≡ 0 (mod ''p'') and ''H'' ≡ −''ω'' (mod ''p'') for 1 ≤ ''k'' ≤ ''p''−2, where ''H'' denotes the ''k''-th

harmonic number In mathematics, the -th harmonic number is the sum of the reciprocals of the first natural numbers: H_n= 1+\frac+\frac+\cdots+\frac =\sum_^n \frac. Starting from , the sequence of harmonic numbers begins: 1, \frac, \frac, \frac, \frac, \dot ...

and ''ω'' denotes the Wolstenholme quotient. 5, 13, 17, 23, 41, 67, 73, 79,

*10/7 (disambiguation) *Bohrium, ...

, 149,

, 179, 191,

, 277, 281, 293, 307, 311, 317,

()

Higgs prime A Higgs prime, named after Denis Higgs, is a prime number with a totient (one less than the prime) that evenly divides the square of the product of the smaller Higgs primes. (This can be generalized to cubes, fourth powers, etc.) To put it algebrai ...
s for squares

Primes ''p'' for which ''p'' − 1 divides the square of the product of all earlier terms. 2, 3, 5, 7, 11, 13, 19, 23, 29, 31, 37, 43, 47, 53, 59, 61, 67, 71, 79, 101,

*10/7 (disambiguation) *Bohrium, ...

, 127, 131,

, 149, 151,

, 191, 197,

, 277,

, 311, 317,

, 347,

()

Highly cototient primes

Primes that are a cototient more often than any integer below it except 1. 2, 23, 47, 59, 83, 89,

, 167,

, 419,

, 659,

839 __NOTOC__ Year 839 ( DCCCXXXIX) was a common year starting on Wednesday of the Julian calendar. Events By place Europe * Prince Sicard of Benevento is assassinated by a conspiracy among the nobility. He is succeeded by Radelchis I, c ...

, 1049, 1259,

1889 Events January * January 1 ** The total solar eclipse of January 1, 1889 is seen over parts of California and Nevada. ** Paiute spiritual leader Wovoka experiences a vision, leading to the start of the Ghost Dance movement in the Dakotas ...

()

Home primes

For , write the prime factorization of in base 10 and concatenate the factors; iterate until a prime is reached. 2, 3, 211, 5, 23, 7, 3331113965338635107, 311, 773, 11, 223, 13, 13367, 1129, 31636373, 17, 233, 19, 3318308475676071413, 37, 211, 23, 331319, 773, 3251, 13367, 227, 29, 547, 31, 241271, 311, 31397, 1129, 71129, 37, 373, 313, 3314192745739, 41, 379, 43, 22815088913, 3411949, 223, 47, 6161791591356884791277 ()

Irregular prime In number theory, a regular prime is a special kind of prime number, defined by Ernst Kummer in 1850 to prove certain cases of Fermat's Last Theorem. Regular primes may be defined via the divisibility of either class number (number theory), clas ...
s

Odd primes ''p'' that divide the class number of the ''p''-th

cyclotomic field In algebraic number theory, a cyclotomic field is a number field obtained by adjoining a complex root of unity to \Q, the field of rational numbers. Cyclotomic fields played a crucial role in the development of modern algebra and number theory ...

. 37, 59, 67, 101, 103, 131, 149,

, 293, 307, 311, 347,

, 401, 409,

421 __NOTOC__ Year 421 ( CDXXI) was a common year starting on Saturday of the Julian calendar. At the time, it was known as the Year of the Consulship of Agricola and Eustathius (or, less frequently, year 1174 ''Ab urbe condita''). The denominat ...

, 433, 461, 463,

523 __NOTOC__ Year 523 ( DXXIII) was a common year starting on Sunday of the Julian calendar. At the time, it was known as the Year of the Consulship of Maximus without colleague (or, less frequently, year 1276 ''Ab urbe condita''). The denominati ...

, 541, 547, 557, 577, 587,

593 __NOTOC__ Year 593 (Roman numerals, DXCIII) was a common year starting on Thursday of the Julian calendar. The denomination 593 for this year has been used since the early medieval period, when the Anno Domini calendar era became the prevalent ...

, 607, 613 ()

(''p'', ''p'' − 3) irregular primes

(See

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 3. Wolstenholme primes ...

)

(''p'', ''p'' − 5) irregular primes

Primes ''p'' such that (''p'', ''p''−5) is an irregular pair. 37

(''p'', ''p'' − 9) irregular primes

Primes ''p'' such that (''p'', ''p'' − 9) is an irregular pair. 67,

()

Isolated prime A twin prime is a prime number that is either 2 less or 2 more than another prime number—for example, either member of the twin prime pair or In other words, a twin prime is a prime that has a prime gap of two. Sometimes the term ''twin prime'' ...
s

Primes ''p'' such that neither ''p'' − 2 nor ''p'' + 2 is prime. 2, 23, 37, 47, 53, 67, 79, 83, 89, 97,

, 127, 131,

, 167,

, 277, 293, 307, 317,

, 367,

, 397, 401, 409, 439, 443,

449 Year 449 ( CDXLIX) was a common year starting on Saturday of the Julian calendar. At the time, it was known as the Year of the Consulship of Astyrius and Romanus (or, less frequently, year 1202 ''Ab urbe condita''). The denomination 449 for t ...

, 457,

, 479, 487,

, 499,

, 541, 547, 557, 563, 577, 587,

, 607, 613, 631, 647, 653,

, 677,

691 __NOTOC__ Year 691 ( DCXCI) was a common year starting on Sunday of the Julian calendar. The denomination 691 for this year has been used since the early medieval period, when the Anno Domini calendar era became the prevalent method in Europe f ...

, 709,

, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797,

853 __NOTOC__ Year 853 ( DCCCLIII) was a common year starting on Sunday of the Julian calendar. Events By place Byzantine Empire * May 22 – A Byzantine fleet (85 ships and 5,000 men) sacks and destroys the port city of Damietta, lo ...

, 863,

887 __NOTOC__ Year 887 ( DCCCLXXXVII) was a common year starting on Sunday of the Julian calendar. Events By place Europe * November 17 – East Frankish magnates revolt against the inept emperor Charles III (the Fat) in an assembly ...

, 907,

911 911, 9/11 or Nine Eleven may refer to: Dates * AD 911 * 911 BC * September 11 ** The 2001 September 11 attacks on the United States by al-Qaeda, commonly referred to as 9/11 ** 11 de Septiembre, Chilean coup d'état in 1973 that ousted the ...

, 929,

, 941,

947 Year 947 (Roman numerals, CMXLVII) was a common year starting on Friday of the Julian calendar. Events By place Europe * Summer – A Principality of Hungary, Hungarian army led by Grand Prince Taksony of Hungary, Taksony campaign ...

977 Year 977 ( CMLXXVII) was a common year starting on Monday of the Julian calendar. Events By place Europe * May – Boris II, dethroned emperor (''tsar'') of Bulgaria, and his brother Roman manage to escape from captivity in Const ...

997 Year 997 ( CMXCVII) was a common year starting on Friday of the Julian calendar. Events By place Japan * 1 February: Empress Teishi gives birth to Princess Shushi - she is the first child of the emperor, but because of the power stru ...

()

Leyland primes

Of the form ''x'' + ''y'', with 1 < ''x'' < ''y''. 17,

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

Long primes

Primes ''p'' for which, in a given base ''b'',

\frac

gives a

cyclic number A cyclic number is an integer for which cyclic permutations of the digits are successive integer multiples of the number. The most widely known is the six-digit number 142857, whose first six integer multiples are :142857 × 1 = 142857 : ...

. They are also called full reptend primes. Primes ''p'' for base 10: 7, 17, 19, 23, 29, 47, 59, 61, 97, 109,

, 131, 149, 167, 179,

, 313,

, 367,

, 419, 433, 461, 487,

, 499,

, 541, 571, 577,

()

Lucas prime The Lucas sequence is an integer sequence named after the mathematician François Édouard Anatole Lucas (1842–1891), who studied both that sequence and the closely related Fibonacci sequence. Individual numbers in the Lucas sequence are ...
s

Primes in the Lucas number sequence ''L'' = 2, ''L'' = 1, ''L'' = ''L'' + ''L''. 2, 3, 7, 11, 29, 47,

521 Year 521 (Roman numerals, DXXI) was a common year starting on Friday of the Julian calendar. At the time, it was known as the Year of the Consulship of Justinian I, Sabbatius and Valerius (or, less frequently, year 1274 ''Ab urbe condita''). ...

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

Lucky primes

Lucky numbers that are prime. 3, 7, 13, 31, 37, 43, 67, 73, 79, 127, 151,

, 307,

, 367, 409,

, 433, 463, 487, 541, 577, 601, 613, 619, 631, 643,

, 727, 739, 769, 787, 823,

883 __NOTOC__ Year 883 ( DCCCLXXXIII) was a common year starting on Tuesday of the Julian calendar. Events By place Europe * Spring – Viking raiders ravage Flanders, and sack the abbey at Saint-Quentin. King Carloman II blocks thei ...

()

Mersenne prime In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form for some integer . They are named after Marin Mersenne, a French Minim friar, who studied them in the early 1 ...
s

Of the form 2 − 1. 3, 7, 31, 127, 8191, 131071, 524287, 2147483647, 2305843009213693951, 618970019642690137449562111, 162259276829213363391578010288127, 170141183460469231731687303715884105727 () , there are 52 known Mersenne primes. The 13th, 14th, and 52nd have respectively 157, 183, and 41,024,320 digits. This includes the largest known prime 2^136,279,841−1, which is the 52nd Mersenne prime.

Mersenne divisors

Primes ''p'' that divide 2 − 1, for some prime number n. 3, 7, 23, 31, 47, 89, 127, 167, 223, 233, 263, 359, 383, 431, 439, 479, 503, 719, 839, 863, 887, 983, 1103, 1319, 1367, 1399, 1433, 1439, 1487, 1823, 1913, 2039, 2063, 2089, 2207, 2351, 2383, 2447, 2687, 2767, 2879, 2903, 2999, 3023, 3119, 3167, 3343 () All Mersenne primes are, by definition, members of this sequence.

Mersenne prime exponents

Primes ''p'' such that 2 − 1 is prime. 2, 3, 5, 7, 13, 17, 19, 31, 61, 89,

*10/7 (disambiguation) *Bohrium, ...

, 127,

, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583, 25964951, 30402457, 32582657, 37156667, 42643801, 43112609, 57885161 () , four more are known to be in the sequence, but it is not known whether they are the next:
74207281, 77232917, 82589933, 136279841

Double Mersenne primes

A subset of Mersenne primes of the form 2 − 1 for prime ''p''. 7, 127, 2147483647, 170141183460469231731687303715884105727 (primes in )

Generalized repunit primes

Of the form (''a'' − 1) / (''a'' − 1) for fixed integer ''a''. For ''a'' = 2, these are the Mersenne primes, while for ''a'' = 10 they are the repunit primes. For other small ''a'', they are given below: ''a'' = 3: 13, 1093, 797161, 3754733257489862401973357979128773, 6957596529882152968992225251835887181478451547013 () ''a'' = 4: 5 (the only prime for ''a'' = 4) ''a'' = 5: 31, 19531, 12207031, 305175781, 177635683940025046467781066894531, 14693679385278593849609206715278070972733319459651094018859396328480215743184089660644531 () ''a'' = 6: 7, 43, 55987, 7369130657357778596659, 3546245297457217493590449191748546458005595187661976371 () ''a'' = 7: 2801, 16148168401, 85053461164796801949539541639542805770666392330682673302530819774105141531698707146930307290253537320447270457 ''a'' = 8: 73 (the only prime for ''a'' = 8) ''a'' = 9: none exist

Other generalizations and variations

Many generalizations of Mersenne primes have been defined. This include the following: * Primes of the form , including the Mersenne primes and the cuban primes as special cases * Williams primes, of the form

Mills primes

Of the form ⌊θ⌋, where θ is Mills' constant. This form is prime for all positive integers ''n''. 2, 11, 1361, 2521008887, 16022236204009818131831320183 ()

Minimal primes

Primes for which there is no shorter sub-sequence of the decimal digits that form a prime. There are exactly 26 minimal primes: 2, 3, 5, 7, 11, 19, 41, 61, 89, 409,

, 499,

, 6469, 6949,

9001 9001 may mean: * AD 9001, the first year of the 10th millennium * 9001 BC, the last year of the 10th millennium BC The 10th millennium BC spanned the years 10,000 BC to 9001 BC (c. 12 ka to c. 11 ka). It marks the beginning of th ...

, 9049, 9649, 9949, 60649, 666649, 946669, 60000049, 66000049, 66600049 ()

Newman–Shanks–Williams primes

Newman–Shanks–Williams numbers that are prime. 7, 41,

, 9369319, 63018038201, 489133282872437279, 19175002942688032928599 ()

Non-generous primes

Primes ''p'' for which the least positive primitive root is not a primitive root of ''p''². Three such primes are known; it is not known whether there are more. 2, 40487, 6692367337 ()

Palindromic primes

Primes that remain the same when their decimal digits are read backwards. 2, 3, 5, 7, 11, 101, 131, 151,

, 191, 313,

, 727, 757, 787, 797,

, 929, 10301, 10501, 10601, 11311, 11411, 12421, 12721, 12821, 13331, 13831, 13931, 14341, 14741 ()

Palindromic wing primes

Primes of the form

\frac \pm b \times 10^

with

0 \le a \pm b < 10

. This means all digits except the middle digit are equal. 101, 131, 151,

, 191, 313,

, 727, 757, 787, 797,

, 929, 11311, 11411, 33533, 77377, 77477, 77977, 1114111, 1117111, 3331333, 3337333, 7772777, 7774777, 7778777, 111181111, 111191111, 777767777, 77777677777, 99999199999 ()

Partition primes

Partition function values that are prime. 2, 3, 5, 7, 11, 101, 17977, 10619863, 6620830889, 80630964769, 228204732751, 1171432692373, 1398341745571, 10963707205259, 15285151248481, 10657331232548839, 790738119649411319, 18987964267331664557 ()

Pell primes

Primes in the Pell number sequence ''P'' = 0, ''P'' = 1, ''P'' = 2''P'' + ''P''. 2, 5, 29, 5741, 33461, 44560482149, 1746860020068409, 68480406462161287469, 13558774610046711780701, 4125636888562548868221559797461449 ()

Permutable prime A permutable prime, also known as anagrammatic prime, is a prime number which, in a given radix, base, can have its digits' positions switched through any permutation and still be a prime number. H. E. Richert, who is supposedly the first to stu ...
s

Any permutation of the decimal digits is a prime. 2, 3, 5, 7, 11, 13, 17, 31, 37, 71, 73, 79, 97,

, 131,

, 311,

, 733,

, 1111111111111111111, 11111111111111111111111 ()

Perrin primes

Primes in the Perrin number sequence ''P''(0) = 3, ''P''(1) = 0, ''P''(2) = 2, ''P''(''n'') = ''P''(''n''−2) + ''P''(''n''−3). 2, 3, 5, 7, 17, 29, 277, 367,

, 14197, 43721, 1442968193, 792606555396977, 187278659180417234321, 66241160488780141071579864797 ()

Pierpont prime In number theory, a Pierpont prime is a prime number of the form 2^u\cdot 3^v + 1\, for some nonnegative integers and . That is, they are the prime numbers for which is 3-smooth. They are named after the mathematician James Pierpont, who us ...
s

Of the form 23 + 1 for some

integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...

s ''u'',''v'' ≥ 0. These are also class 1- primes. 2, 3, 5, 7, 13, 17, 19, 37, 73, 97, 109,

, 433, 487, 577, 769, 1153, 1297, 1459, 2593, 2917, 3457, 3889, 10369, 12289, 17497, 18433, 39367, 52489,

, 139969, 147457 ()

Pillai prime In number theory, a Pillai prime is a prime number ''p'' for which there is an integer ''n'' > 0 such that the factorial of ''n'' is one less than a multiple of the prime, but the prime is not one more than a multiple of ''n''. To put it algebraical ...
s

Primes ''p'' for which there exist ''n'' > 0 such that ''p'' divides ''n''! + 1 and ''n'' does not divide ''p'' − 1. 23, 29, 59, 61, 67, 71, 79, 83, 109, 137,

, 149,

, 227,

, 277, 293, 307, 311, 317,

, 397, 401, 419, 431,

, 461, 463,

, 479, 499 ()

Primes of the form ''n''⁴ + 1

Of the form ''n''⁴ + 1. 2, 17,

, 1297,

, 160001, 331777, 614657, 1336337, 4477457, 5308417, 8503057, 9834497, 29986577, 40960001, 45212177, 59969537, 65610001, 126247697, 193877777, 303595777, 384160001, 406586897, 562448657, 655360001 ()

Primeval primes

Primes for which there are more prime permutations of some or all the decimal digits than for any smaller number. 2, 13, 37,

*10/7 (disambiguation) *Bohrium, ...

, 137,

1013 Year 1013 ( MXIII) was a common year starting on Thursday of the Julian calendar. Events By place Europe * King Henry II of Germany signs a peace treaty at Merseburg with Duke Bolesław I the Brave) of Poland. As part of the treaty, Bo ...

, 1237, 1367, 10079 ()

Primorial prime In mathematics, a primorial prime is a prime number of the form ''pn''# ± 1, where ''pn''# is the primorial of ''pn'' (i.e. the product of the first ''n'' primes). Primality tests show that: : ''pn''# − 1 is prime for ...
s

Of the form ''p''# ± 1. 3, 5, 7, 29, 31,

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

Proth primes

Of the form ''k''×2 + 1, with odd ''k'' and ''k'' < 2. 3, 5, 13, 17, 41, 97,

, 577,

641 __NOTOC__ Year 641 ( DCXLI) was a common year starting on Monday of the Julian calendar. The denomination 641 for this year has been used since the early medieval period, when the Anno Domini calendar era became the prevalent method in Europe ...

, 769, 929, 1153, 1217,

1409 Year 1409 ( MCDIX) was a common year starting on Tuesday of the Julian calendar. Events January–March * January 1 – The Welsh surrender Harlech Castle to the English. * January 18 – The Decree of Kutná Hora strengthe ...

1601 This Epoch (reference date)#Computing, epoch is the beginning of the 400-year Gregorian leap-year cycle within which digital files first existed; the last year of any such cycle is the only leap year whose year number is divisible by 100. Jan ...

, 2113, 2689, 2753, 3137, 3329, 3457, 4481, 4993, 6529, 7297, 7681, 7937, 9473, 9601, 9857 ()

Pythagorean prime A Pythagorean prime is a prime number of the Pythagorean primes are exactly the odd prime numbers that are the sum of two squares; this characterization is Fermat's theorem on sums of two squares. Equivalently, by the Pythagorean theorem, they ...
s

Of the form 4''n'' + 1. 5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101, 109,

, 137, 149,

, 197,

, 277, 281, 293, 313, 317,

, 397, 401, 409,

, 433,

()

Prime quadruplet In number theory, a prime quadruplet (sometimes called a prime quadruple) is a set of four prime numbers of the form This represents the closest possible grouping of four primes larger than 3, and is the only prime constellation of length 4. P ...
s

Where (''p'', ''p''+2, ''p''+6, ''p''+8) are all prime. ( 5, 7, 11, 13), (11, 13, 17, 19), ( 101, 103,

*10/7 (disambiguation) *Bohrium, ...

, 109), ( 191,

, 197,

), (

821 __NOTOC__ Year 821 (Roman numerals, DCCCXXI) was a common year starting on Tuesday of the Julian calendar. Events By place Byzantine Empire * Byzantine general Thomas the Slav leads a revolt, and secures control over most of the Byzan ...

, 823,

827 __FORCETOC__ Year 827 ( DCCCXXVII) was a common year starting on Tuesday of the Julian calendar. Events By place Byzantine Empire * June 14 – Euphemius, exiled Byzantine admiral, asks for the help of North African Arabs, to ret ...

, 829), ( 1481,

1483 Year 1483 ( MCDLXXXIII) was a common year starting on Wednesday of the Julian calendar. Events January–December * January 1 – The Jews are expelled from Andalusia. * February 11 – The ''General Council of the Inquisiti ...

, 1487, 1489), (

1871 Events January–March * January 3 – Franco-Prussian War: Battle of Bapaume – Prussians win a strategic victory. * January 18 – Proclamation of the German Empire: The member states of the North German Confederation and the sout ...

1873 Events January * January 1 ** Japan adopts the Gregorian calendar. ** The California Penal Code goes into effect. * January 17 – American Indian Wars: Modoc War: First Battle of the Stronghold – Modoc Indians defeat the Unit ...

1877 Events January * January 1 – Queen Victoria is proclaimed Empress of India by the Royal Titles Act 1876, introduced by Benjamin Disraeli, the Prime Minister of the United Kingdom . * January 8 – Great Sioux War of 1876: Batt ...

1879 Events January * January 1 ** The Specie Resumption Act takes effect. The United States Note is valued the same as gold, for the first time since the American Civil War. ** Brahms' Violin Concerto is premiered in Leipzig with Joseph Joachim ...

), (

2081 In contemporary history, the third millennium is the current millennium in the ''Anno Domini'' or Common Era, under the Gregorian calendar. It began on 1 January 2001 (Roman numerals, MMI) and will end on 31 December 3000 (Roman numerals, MMM ...

2083 In contemporary history, the third millennium is the current millennium in the ''Anno Domini'' or Common Era, under the Gregorian calendar. It began on 1 January 2001 ( MMI) and will end on 31 December 3000 ( MMM), spanning the 21st to 30th ...

2087 In contemporary history, the third millennium is the current millennium in the ''Anno Domini'' or Common Era, under the Gregorian calendar. It began on 1 January 2001 ( MMI) and will end on 31 December 3000 ( MMM), spanning the 21st to 30t ...

2089 In contemporary history, the third millennium is the current millennium in the ''Anno Domini'' or Common Era, under the Gregorian calendar. It began on 1 January 2001 ( MMI) and will end on 31 December 3000 ( MMM), spanning the 21st to 30th ...

), ( 3251, 3253, 3257, 3259), ( 3461, 3463, 3467, 3469), (5651, 5653, 5657, 5659), (9431, 9433, 9437, 9439) (, , , )

Quartan primes

Of the form ''x'' + ''y'', where ''x'',''y'' > 0. 2, 17, 97,

()

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 In 1919, Ramanujan published a new proof of Bertrand's postulate which, as ...
s

Integers ''R'' that are the smallest to give at least ''n'' primes from ''x''/2 to ''x'' for all ''x'' ≥ ''R'' (all such integers are primes). 2, 11, 17, 29, 41, 47, 59, 67, 71, 97, 101,

*10/7 (disambiguation) *Bohrium, ...

, 127, 149, 151, 167, 179,

, 227,

, 281, 307, 311, 347,

, 367,

, 401, 409, 419, 431, 433, 439, 461, 487,

()

Regular prime In number theory, a regular prime is a special kind of prime number, defined by Ernst Kummer in 1850 to prove certain cases of Fermat's Last Theorem. Regular primes may be defined via the divisibility of either class numbers or of Bernoulli num ...
s

Primes ''p'' that do not divide the class number of the ''p''-th

. 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 43, 47, 53, 61, 71, 73, 79, 83, 89, 97,

*10/7 (disambiguation) *Bohrium, ...

, 109,

, 127, 137,

, 151,

, 167,

, 179,

, 191,

, 197,

, 227,

, 277, 281 ()

Repunit primes

Primes containing only the decimal digit 1. 11, 1111111111111111111 (19 digits), 11111111111111111111111 (23 digits) () The next have 317, 1031, 49081, 86453, 109297, and 270343 digits, respectively ().

Residue classes of primes

Of the form ''an'' + ''d'' for fixed integers ''a'' and ''d''. Also called primes congruent to ''d''

modulo In computing and mathematics, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another, the latter being called the '' modulus'' of the operation. Given two positive numbers and , mo ...

''a''. The primes of the form 2''n''+1 are the odd primes, including all primes other than 2. Some sequences have alternate names: 4''n''+1 are Pythagorean primes, 4''n''+3 are the integer Gaussian primes, and 6''n''+5 are the Eisenstein primes (with 2 omitted). The classes 10''n''+''d'' (''d'' = 1, 3, 7, 9) are primes ending in the decimal digit ''d''. 2''n''+1: 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53 ()
4''n''+1: 5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101, 109,

, 137 ()
4''n''+3: 3, 7, 11, 19, 23, 31, 43, 47, 59, 67, 71, 79, 83, 103,

*10/7 (disambiguation) *Bohrium, ...

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

()
6''n''+5: 5, 11, 17, 23, 29, 41, 47, 53, 59, 71, 83, 89, 101, 107, 113 ()
8''n''+1: 17, 41, 73, 89, 97, 113, 137,

, 281, 313,

()
8''n''+3: 3, 11, 19, 43, 59, 67, 83, 107, 131, 139,

, 179,

, 227,

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

, 197,

()
8''n''+7: 7, 23, 31, 47, 71, 79, 103, 127, 151, 167, 191,

()
10''n''+1: 11, 31, 41, 61, 71, 101, 131, 151, 181, 191, 211, 241, 251,

, 281 ()
10''n''+3: 3, 13, 23, 43, 53, 73, 83, 103, 113, 163, 173, 193, 223, 233, 263 ()
10''n''+7: 7, 17, 37, 47, 67, 97, 107, 127, 137, 157, 167, 197, 227, 257, 277 ()
10''n''+9: 19, 29, 59, 79, 89, 109, 139, 149, 179, 199, 229, 239, 269,

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

Safe prime In number theory, a prime number ''p'' is a if 2''p'' + 1 is also prime. The number 2''p'' + 1 associated with a Sophie Germain prime is called a . For example, 11 is a Sophie Germain prime and 2 × 11 +&nbs ...
s

Where ''p'' and (''p''−1) / 2 are both prime. 5, 7, 11, 23, 47, 59, 83,

*10/7 (disambiguation) *Bohrium, ...

, 167, 179, 227,

, 347,

, 479,

, 563, 587,

, 863,

1019 Year 1019 (Roman numerals, MXIX) was a common year starting on Thursday of the Julian calendar. Events By place Europe * Sviatopolk I of Kiev, Sviatopolk I dies, and is succeeded by his brother Yaroslav the Wise, Yaroslav I (the Wise ...

, 1187,

1283 Year 1283 ( MCCLXXXIII) was a common year starting on Friday of the Julian calendar. Events By place Europe * June 1 – Treaty of Rheinfelden: The 11-year-old Rudolf II is forced to relinquish his claim on the Duchies of Austr ...

, 1307, 1319, 1367, 1439, 1487,

1523 Year 1523 (Roman numerals, MDXXIII) was a common year starting on Thursday of the Julian calendar. Events January–March * January 20 – Christian II is forced to abdicate as King of Denmark and Norway after the nobles of the ...

1619 Events January– March * January 12 – James I of England's Banqueting House, Whitehall in London is destroyed by fire."Fires, Great", in ''The Insurance Cyclopeadia: Being an Historical Treasury of Events and Circumstances Co ...

1823 Events January–March * January 22 – By secret treaty signed at the Congress of Verona, the Quintuple Alliance gives France a mandate to invade Spain for the purpose of restoring Ferdinand VII (who has been captured by armed revoluti ...

1907 Events January * January 14 – 1907 Kingston earthquake: A 6.5 Moment magnitude scale, Mw earthquake in Kingston, Jamaica, kills between 800 and 1,000. February * February 9 – The "Mud March (suffragists), Mud March", the ...

()

Self primes in base 10

Primes that cannot be generated by any integer added to the sum of its decimal digits. 3, 5, 7, 31, 53, 97,

, 277, 367,

, 457, 479, 547, 569, 613, 659, 727,

, 929, 1021, 1087, 1109,

1223 Year 1223 (MCCXXIII) was a common year starting on Sunday of the Julian calendar. Events January–March * January 4 – Germanus II of Constantinople, Germanus II takes office as Patriarch of Constantinople, leader of the Eastern O ...

, 1289,

1447 Year 1447 (Roman numerals, MCDXLVII) was a common year starting on Sunday of the Julian calendar. Events January–March * January 4 – Barnaba Adorno becomes the new Doge of Genoa, Doge of the Republic of Genoa when his cousin :i ...

, 1559,

1627 Events January–March * January 26 – The Dutch ship '' 't Gulden Zeepaert'', skippered by François Thijssen, makes the first recorded sighting of the coast of South Australia. * February 15 – The administrative rura ...

1693 Events January–March * January 11 – The Mount Etna volcano erupts in Italy, causing a devastating earthquake that kills 60,000 people in Sicily and Malta Malta, officially the Republic of Malta, is an island country in Sout ...

1783 Events January–March * January 20 – At Versailles, Kingdom of Great Britain, Great Britain signs preliminary peace treaties with the Kingdom of France and the Kingdom of Spain. * January 23 – The Confederation Congress ...

()

Sexy prime In number theory, sexy primes are prime numbers that differ from each other by . For example, the numbers and are a pair of sexy primes, because both are prime and 11 - 5 = 6. The term "sexy prime" is a pun stemming from the Latin word for six ...
s

Where (''p'', ''p'' + 6) are both prime. ( 5, 11), ( 7, 13), (11, 17), (13, 19), (17, 23), (23, 29), ( 31, 37), (37, 43), ( 41, 47), (47, 53), (53, 59), ( 61, 67), (67, 73), (73, 79), ( 83, 89), ( 97, 103), ( 101,

*10/7 (disambiguation) *Bohrium, ...

), (103, 109), (107,

), ( 131, 137), ( 151,

), (157,

), ( 167,

), (173, 179), ( 191, 197), (

) (, )

Smarandache–Wellin primes

Primes that are the concatenation of the first ''n'' primes written in decimal. 2, 23, 2357 () The fourth Smarandache-Wellin prime is the 355-digit concatenation of the first 128 primes that end with 719.

Solinas primes

Of the form 2 − ''c''·2 − ''c''·2 − ... − ''c''. * 3, 5, 7, 11, 13 () * ''2 − 5'', the largest prime that fits into 32 bits of memory. * ''2 − 59'', the largest prime that fits into 64 bits of memory.

Sophie Germain prime In number theory, a prime number ''p'' is a if 2''p'' + 1 is also prime. The number 2''p'' + 1 associated with a Sophie Germain prime is called a . For example, 11 is a Sophie Germain prime and 2 × 11 +&nbs ...
s

Where ''p'' and 2''p'' + 1 are both prime. A Sophie Germain prime has a corresponding

safe prime In number theory, a prime number ''p'' is a if 2''p'' + 1 is also prime. The number 2''p'' + 1 associated with a Sophie Germain prime is called a . For example, 11 is a Sophie Germain prime and 2 × 11 +&nbs ...

. 2, 3, 5, 11, 23, 29, 41, 53, 83, 89,

, 131,

, 179, 191,

, 281, 293,

, 419, 431, 443,

, 653, 659,

, 743, 761, 809,

()

Stern primes

Primes that are not the sum of a smaller prime and twice the square of a nonzero integer. 2, 3, 17, 137, 227,

, 1187,

1493 Year 1493 ( MCDXCIII) was a common year starting on Tuesday of the Julian calendar. Events January–December * January 19 – Treaty of Barcelona: Charles VIII of France returns Cerdagne and Roussillon to Ferdinand of Aragon. ...

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

Super-prime Super-prime numbers, also known as higher-order primes or prime-indexed primes (PIPs), are the subsequence of prime numbers that occupy prime-numbered positions within the sequence of all prime numbers. In other words, if prime numbers are matched ...
s

Primes with prime-numbered indexes in the sequence of prime numbers (the 2nd, 3rd, 5th, ... prime). 3, 5, 11, 17, 31, 41, 59, 67, 83, 109, 127,

, 179, 191,

, 277,

, 367, 401, 431, 461,

, 547, 563, 587,

599 __NOTOC__ Year 599 (Roman numerals, DXCIX) was a common year starting on Thursday of the Julian calendar. The denomination 599 for this year has been used since the early medieval period, when the Anno Domini calendar era became the prevalent m ...

, 709, 739, 773, 797, 859,

()

Supersingular primes

There are exactly fifteen supersingular primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59, 71 ()

Thabit prime In number theory, a Thabit number, Thâbit ibn Qurra number, or 321 number is an integer of the form 3 \cdot 2^n - 1 for a non-negative integer ''n''. The first few Thabit numbers are: : 2, 5, 11, 23, 47, 95, 191, 383, 767, 1535, 3071, 614 ...
s

Of the form 3×2 − 1. 2, 5, 11, 23, 47, 191,

, 6143, 786431, 51539607551, 824633720831, 26388279066623, 108086391056891903, 55340232221128654847, 226673591177742970257407 () The primes of the form 3×2 + 1 are related. 7, 13, 97,

, 769, 12289, 786433, 3221225473, 206158430209, 6597069766657 ()

Prime triplet In number theory, a prime triplet is a set of three prime numbers in which the smallest and largest of the three differ by 6. In particular, the sets must have the form or . With the exceptions of and , this is the closest possible grouping of ...
s

Where (''p'', ''p''+2, ''p''+6) or (''p'', ''p''+4, ''p''+6) are all prime. ( 5, 7, 11), (7, 11, 13), (11, 13, 17), (13, 17, 19), (17, 19, 23), ( 37, 41, 43), (41, 43, 47), ( 67, 71, 73), ( 97, 101, 103), (101, 103,

*10/7 (disambiguation) *Bohrium, ...

), (103, 107, 109), (107, 109,

), ( 191,

, 197), (193, 197,

), (

, 227,

), (227, 229,

), ( 277, 281,

), ( 307, 311, 313), (311, 313, 317), ( 347,

) (, , )

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 ("left") digit is successively removed, then all resulting numbers are prime. For example, 9137, since 9137, 137, 37 and 7 are ...

Left-truncatable

Primes that remain prime when the leading decimal digit is successively removed. 2, 3, 5, 7, 13, 17, 23, 37, 43, 47, 53, 67, 73, 83, 97,

, 137, 167,

, 197,

, 313, 317,

, 347,

, 367,

, 397, 443,

, 547, 613,

, 643, 647, 653,

()

Right-truncatable

Primes that remain prime when the least significant decimal digit is successively removed. 2, 3, 5, 7, 23, 29, 31, 37, 53, 59, 71, 73, 79,

, 293, 311, 313, 317,

, 733, 739, 797, 2333, 2339, 2393, 2399, 2939, 3119, 3137, 3733, 3739, 3793, 3797 ()

Two-sided

Primes that are both left-truncatable and right-truncatable. There are exactly fifteen two-sided primes: 2, 3, 5, 7, 23, 37, 53, 73, 313, 317,

, 797, 3137, 3797, 739397 ()

Twin prime A twin prime is a prime number that is either 2 less or 2 more than another prime number—for example, either member of the twin prime pair or In other words, a twin prime is a prime that has a prime gap of two. Sometimes the term ''twin prime' ...
s

Where (''p'', ''p''+2) are both prime. ( 3, 5), (5, 7), ( 11, 13), ( 17, 19), ( 29, 31), ( 41, 43), ( 59, 61), ( 71, 73), ( 101, 103), (

*10/7 (disambiguation) *Bohrium, ...

, 109), ( 137,

), ( 149, 151), ( 179,

), ( 191,

), ( 197,

), ( 227,

), (

), ( 281,

), ( 311, 313), ( 347,

), ( 419,

), ( 431, 433), ( 461, 463) (, )

Unique primes

The list of primes ''p'' for which the period length of the decimal expansion of 1/''p'' is unique (no other prime gives the same period). 3, 11, 37, 101, 9091, 9901, 333667, 909091, 99990001, 999999000001, 9999999900000001, 909090909090909091, 1111111111111111111, 11111111111111111111111, 900900900900990990990991 ()

Wagstaff primes

Of the form (2 + 1) / 3. 3, 11, 43,

, 2731, 43691, 174763, 2796203, 715827883, 2932031007403, 768614336404564651, 201487636602438195784363, 845100400152152934331135470251, 56713727820156410577229101238628035243 () Values of ''n'': 3, 5, 7, 11, 13, 17, 19, 23, 31, 43, 61, 79, 101, 127, 167, 191,

, 313, 347,

1709 In the Swedish calendar it was a common year starting on Friday, one day ahead of the Julian and ten days behind the Gregorian calendar. Events January–March * January 1 – Battle of St. John's: The French capture St. John' ...

, 2617, 3539, 5807, 10501, 10691, 11279, 12391, 14479, 42737, 83339, 95369, 117239, 127031, 138937, 141079, 267017, 269987, 374321 ()

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 Let p be a prime number. When each term in the sequence of Fibona ...
s

A prime ''p'' > 5, if ''p'' divides the

Fibonacci number In mathematics, the Fibonacci sequence is a Integer sequence, sequence in which each element is the sum of the two elements that precede it. Numbers that are part of the Fibonacci sequence are known as Fibonacci numbers, commonly denoted . Many w ...

F_

, where the

Legendre symbol In number theory, the Legendre symbol is a multiplicative function with values 1, −1, 0 that is a quadratic character modulo of an odd prime number ''p'': its value at a (nonzero) quadratic residue mod ''p'' is 1 and at a non-quadratic re ...

\left(\frac\right)

is defined as :

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

, no Wall-Sun-Sun primes are known.

Wieferich prime In number theory, a Wieferich prime is a prime number ''p'' such that ''p''2 divides , therefore connecting these primes with Fermat's little theorem, which states that every odd prime ''p'' divides . Wieferich primes were first described by A ...
s

Primes ''p'' such that for fixed integer ''a'' > 1. 2^{''p'' − 1} ≡ 1 (mod ''p''²): 1093, 3511 ()
3^{''p'' − 1} ≡ 1 (mod ''p''²): 11, 1006003 ()
4^{''p'' − 1} ≡ 1 (mod ''p''²): 1093, 3511
5^{''p'' − 1} ≡ 1 (mod ''p''²): 2, 20771, 40487, 53471161, 1645333507, 6692367337, 188748146801 ()
6^{''p'' − 1} ≡ 1 (mod ''p''²): 66161, 534851, 3152573 ()
7^{''p'' − 1} ≡ 1 (mod ''p''²): 5, 491531 ()
8^{''p'' − 1} ≡ 1 (mod ''p''²): 3, 1093, 3511
9^{''p'' − 1} ≡ 1 (mod ''p''²): 2, 11, 1006003
10^{''p'' − 1} ≡ 1 (mod ''p''²): 3, 487, 56598313 ()
11^{''p'' − 1} ≡ 1 (mod ''p''²): 71
12^{''p'' − 1} ≡ 1 (mod ''p''²): 2693, 123653 ()
13^{''p'' − 1} ≡ 1 (mod ''p''²): 2, 863, 1747591 ()
14^{''p'' − 1} ≡ 1 (mod ''p''²): 29,

, 7596952219 ()
15^{''p'' − 1} ≡ 1 (mod ''p''²): 29131, 119327070011 ()
16^{''p'' − 1} ≡ 1 (mod ''p''²): 1093, 3511
17^{''p'' − 1} ≡ 1 (mod ''p''²): 2, 3, 46021, 48947 ()
18^{''p'' − 1} ≡ 1 (mod ''p''²): 5, 7, 37,

, 33923, 1284043 ()
19^{''p'' − 1} ≡ 1 (mod ''p''²): 3, 7, 13, 43, 137, 63061489 ()
20^{''p'' − 1} ≡ 1 (mod ''p''²): 281, 46457, 9377747, 122959073 ()
21^{''p'' − 1} ≡ 1 (mod ''p''²): 2
22^{''p'' − 1} ≡ 1 (mod ''p''²): 13,

, 1595813, 492366587, 9809862296159 ()
23^{''p'' − 1} ≡ 1 (mod ''p''²): 13, 2481757, 13703077, 15546404183, 2549536629329 ()
24^{''p'' − 1} ≡ 1 (mod ''p''²): 5, 25633
25^{''p'' − 1} ≡ 1 (mod ''p''²): 2, 20771, 40487, 53471161, 1645333507, 6692367337, 188748146801 , these are all known Wieferich primes with ''a'' ≤ 25.

Wilson prime In number theory, a Wilson prime is a prime number p such that p^2 divides (p-1)!+1, where "!" denotes the factorial function; compare this with Wilson's theorem, which states that every prime p divides (p-1)!+1. Both are named for 18th-century ...
s

Primes ''p'' for which ''p'' divides (''p''−1)! + 1. 5, 13, 563 () , these are the only known Wilson primes.

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 3. Wolstenholme primes ...
s

Primes ''p'' for which the

binomial coefficient In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the t ...

\equiv 1 \pmod.

16843, 2124679 () , these are the only known Wolstenholme primes.

Woodall primes

Of the form ''n''×2 − 1. 7, 23,

, 32212254719, 2833419889721787128217599, 195845982777569926302400511, 4776913109852041418248056622882488319 ()

References

External links

All prime numbers from 31 to 6,469,693,189 for free download.
Lists of Primes
at the Prime Pages.
The Nth Prime Page
Nth prime through n=10^12, pi(x) through x=3*10^13, Random primes in same range.

(primes less than 2,000,000,000) *
Selected prime related sequences
in On-Line Encyclopedia of Integer Sequences, OEIS. * Fischer, R
Thema: Fermatquotient B^(P−1)

1 (mod P^2)
(Lists Wieferich primes in all bases up to 1052) * {{Prime number classes * *

Prime numbers A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...

Prime A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...

The first 1000 prime numbers

Lists of primes by type

Balanced prime In number theory, a balanced prime is a prime number with equal-sized prime gaps above and below it, so that it is equal to the arithmetic mean of the nearest primes above and below. Or to put it algebraically, the nth prime number p_n is a balanced ...s

Bell primes

Circular primes

Cluster prime In number theory, a cluster prime is a prime number such that every even positive integer ''k'' ≤ p − 3 can be written as the difference between two prime numbers not exceeding (). For example, the number 23 is a cluster prime becau ...s

Cuban primes

Delicate primes

Dihedral primes

Eisenstein primes without imaginary part

Factorial prime A factorial prime is a prime number that is one less or one more than a factorial (all factorials greater than 1 are even). The first 10 factorial primes (for ''n'' = 1, 2, 3, 4, 6, 7, 11, 12, 14) are : : 2 (0! + 1 or 1! + 1) ...s

Fermat prime In mathematics, a Fermat number, named after Pierre de Fermat (1601–1665), the first known to have studied them, is a positive integer of the form:F_ = 2^ + 1, where ''n'' is a non-negative integer. The first few Fermat numbers are: 3, 5, ...s

Generalized Fermat primes In mathematics, a Fermat number, named after Pierre de Fermat (1601–1665), the first known to have studied them, is a positive integer of the form:F_ = 2^ + 1, where ''n'' is a non-negative integer. The first few Fermat numbers are: 3, 5, 17 ...

Fibonacci primes

Gaussian prime In number theory, a Gaussian integer is a complex number whose real and imaginary parts are both integers. The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain, usually written as \mathbf /ma ...s

Good prime A good prime is a prime number whose square is greater than the product of any two primes at the same number of positions before and after it in the sequence of primes. That is, good prime satisfies the inequality :p_n^2 > p_ \cdot p_ for all 1 ...s

Happy prime In number theory, a happy number is a number which eventually reaches 1 when the number is replaced by the sum of the square of each digit. For instance, 13 is a happy number because 1^2+3^2=10, and 1^2+0^2=1. On the other hand, 4 is not a happy ...s

Harmonic primes

Higgs prime A Higgs prime, named after Denis Higgs, is a prime number with a totient (one less than the prime) that evenly divides the square of the product of the smaller Higgs primes. (This can be generalized to cubes, fourth powers, etc.) To put it algebrai ...s for squares

Home primes

Irregular prime In number theory, a regular prime is a special kind of prime number, defined by Ernst Kummer in 1850 to prove certain cases of Fermat's Last Theorem. Regular primes may be defined via the divisibility of either class number (number theory), clas ...s

Isolated prime A twin prime is a prime number that is either 2 less or 2 more than another prime number—for example, either member of the twin prime pair or In other words, a twin prime is a prime that has a prime gap of two. Sometimes the term ''twin prime'' ...s

Lucas prime The Lucas sequence is an integer sequence named after the mathematician François Édouard Anatole Lucas (1842–1891), who studied both that sequence and the closely related Fibonacci sequence. Individual numbers in the Lucas sequence are ...s

Mersenne prime In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form for some integer . They are named after Marin Mersenne, a French Minim friar, who studied them in the early 1 ...s

Mersenne divisors

Mersenne prime exponents

Double Mersenne primes

Generalized repunit primes

Other generalizations and variations

Newman–Shanks–Williams primes

Non-generous primes

Palindromic wing primes

Permutable prime A permutable prime, also known as anagrammatic prime, is a prime number which, in a given radix, base, can have its digits' positions switched through any permutation and still be a prime number. H. E. Richert, who is supposedly the first to stu ...s

Pierpont prime In number theory, a Pierpont prime is a prime number of the form 2^u\cdot 3^v + 1\, for some nonnegative integers and . That is, they are the prime numbers for which is 3-smooth. They are named after the mathematician James Pierpont, who us ...s

Pillai prime In number theory, a Pillai prime is a prime number ''p'' for which there is an integer ''n'' > 0 such that the factorial of ''n'' is one less than a multiple of the prime, but the prime is not one more than a multiple of ''n''. To put it algebraical ...s

Primes of the form ''n''4 + 1

Primeval primes

Primorial prime In mathematics, a primorial prime is a prime number of the form ''pn''# ± 1, where ''pn''# is the primorial of ''pn'' (i.e. the product of the first ''n'' primes). Primality tests show that: : ''pn''# − 1 is prime for ...s

Pythagorean prime A Pythagorean prime is a prime number of the Pythagorean primes are exactly the odd prime numbers that are the sum of two squares; this characterization is Fermat's theorem on sums of two squares. Equivalently, by the Pythagorean theorem, they ...s

Prime quadruplet In number theory, a prime quadruplet (sometimes called a prime quadruple) is a set of four prime numbers of the form This represents the closest possible grouping of four primes larger than 3, and is the only prime constellation of length 4. P ...s

Quartan primes

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 In 1919, Ramanujan published a new proof of Bertrand's postulate which, as ...s

Regular prime In number theory, a regular prime is a special kind of prime number, defined by Ernst Kummer in 1850 to prove certain cases of Fermat's Last Theorem. Regular primes may be defined via the divisibility of either class numbers or of Bernoulli num ...s

Safe prime In number theory, a prime number ''p'' is a if 2''p'' + 1 is also prime. The number 2''p'' + 1 associated with a Sophie Germain prime is called a . For example, 11 is a Sophie Germain prime and 2 × 11 +&nbs ...s

Self primes in base 10

Sexy prime In number theory, sexy primes are prime numbers that differ from each other by . For example, the numbers and are a pair of sexy primes, because both are prime and 11 - 5 = 6. The term "sexy prime" is a pun stemming from the Latin word for six ...s

Solinas primes

Sophie Germain prime In number theory, a prime number ''p'' is a if 2''p'' + 1 is also prime. The number 2''p'' + 1 associated with a Sophie Germain prime is called a . For example, 11 is a Sophie Germain prime and 2 × 11 +&nbs ...s

Stern primes

Super-prime Super-prime numbers, also known as higher-order primes or prime-indexed primes (PIPs), are the subsequence of prime numbers that occupy prime-numbered positions within the sequence of all prime numbers. In other words, if prime numbers are matched ...s

Thabit prime In number theory, a Thabit number, Thâbit ibn Qurra number, or 321 number is an integer of the form 3 \cdot 2^n - 1 for a non-negative integer ''n''. The first few Thabit numbers are: : 2, 5, 11, 23, 47, 95, 191, 383, 767, 1535, 3071, 614 ...s

Prime triplet In number theory, a prime triplet is a set of three prime numbers in which the smallest and largest of the three differ by 6. In particular, the sets must have the form or . With the exceptions of and , this is the closest possible grouping of ...s

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 ("left") digit is successively removed, then all resulting numbers are prime. For example, 9137, since 9137, 137, 37 and 7 are ...

Left-truncatable

Right-truncatable

Two-sided

Twin prime A twin prime is a prime number that is either 2 less or 2 more than another prime number—for example, either member of the twin prime pair or In other words, a twin prime is a prime that has a prime gap of two. Sometimes the term ''twin prime' ...s

Unique primes

Wagstaff primes

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 Let p be a prime number. When each term in the sequence of Fibona ...s

Wieferich prime In number theory, a Wieferich prime is a prime number ''p'' such that ''p''2 divides , therefore connecting these primes with Fermat's little theorem, which states that every odd prime ''p'' divides . Wieferich primes were first described by A ...s

Wilson prime In number theory, a Wilson prime is a prime number p such that p^2 divides (p-1)!+1, where "!" denotes the factorial function; compare this with Wilson's theorem, which states that every prime p divides (p-1)!+1. Both are named for 18th-century ...s

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 3. Wolstenholme primes ...s

See also

References

External links

Balanced prime In number theory, a balanced prime is a prime number with equal-sized prime gaps above and below it, so that it is equal to the arithmetic mean of the nearest primes above and below. Or to put it algebraically, the nth prime number p_n is a balanced ...
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Cluster prime In number theory, a cluster prime is a prime number such that every even positive integer ''k'' ≤ p − 3 can be written as the difference between two prime numbers not exceeding (). For example, the number 23 is a cluster prime becau ...
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Factorial prime A factorial prime is a prime number that is one less or one more than a factorial (all factorials greater than 1 are even). The first 10 factorial primes (for ''n'' = 1, 2, 3, 4, 6, 7, 11, 12, 14) are : : 2 (0! + 1 or 1! + 1) ...
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Fermat prime In mathematics, a Fermat number, named after Pierre de Fermat (1601–1665), the first known to have studied them, is a positive integer of the form:F_ = 2^ + 1, where ''n'' is a non-negative integer. The first few Fermat numbers are: 3, 5, ...
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Generalized
Fermat primes In mathematics, a Fermat number, named after Pierre de Fermat (1601–1665), the first known to have studied them, is a positive integer of the form:F_ = 2^ + 1, where ''n'' is a non-negative integer. The first few Fermat numbers are: 3, 5, 17 ...

Gaussian prime In number theory, a Gaussian integer is a complex number whose real and imaginary parts are both integers. The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain, usually written as \mathbf /ma ...
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Good prime A good prime is a prime number whose square is greater than the product of any two primes at the same number of positions before and after it in the sequence of primes. That is, good prime satisfies the inequality :p_n^2 > p_ \cdot p_ for all 1 ...
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Happy prime In number theory, a happy number is a number which eventually reaches 1 when the number is replaced by the sum of the square of each digit. For instance, 13 is a happy number because 1^2+3^2=10, and 1^2+0^2=1. On the other hand, 4 is not a happy ...
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Higgs prime A Higgs prime, named after Denis Higgs, is a prime number with a totient (one less than the prime) that evenly divides the square of the product of the smaller Higgs primes. (This can be generalized to cubes, fourth powers, etc.) To put it algebrai ...
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Irregular prime In number theory, a regular prime is a special kind of prime number, defined by Ernst Kummer in 1850 to prove certain cases of Fermat's Last Theorem. Regular primes may be defined via the divisibility of either class number (number theory), clas ...
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Isolated prime A twin prime is a prime number that is either 2 less or 2 more than another prime number—for example, either member of the twin prime pair or In other words, a twin prime is a prime that has a prime gap of two. Sometimes the term ''twin prime'' ...
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Lucas prime The Lucas sequence is an integer sequence named after the mathematician François Édouard Anatole Lucas (1842–1891), who studied both that sequence and the closely related Fibonacci sequence. Individual numbers in the Lucas sequence are ...
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Mersenne prime In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form for some integer . They are named after Marin Mersenne, a French Minim friar, who studied them in the early 1 ...
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Permutable prime A permutable prime, also known as anagrammatic prime, is a prime number which, in a given radix, base, can have its digits' positions switched through any permutation and still be a prime number. H. E. Richert, who is supposedly the first to stu ...
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Pierpont prime In number theory, a Pierpont prime is a prime number of the form 2^u\cdot 3^v + 1\, for some nonnegative integers and . That is, they are the prime numbers for which is 3-smooth. They are named after the mathematician James Pierpont, who us ...
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Pillai prime In number theory, a Pillai prime is a prime number ''p'' for which there is an integer ''n'' > 0 such that the factorial of ''n'' is one less than a multiple of the prime, but the prime is not one more than a multiple of ''n''. To put it algebraical ...
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Primes of the form ''n''⁴ + 1

Primorial prime In mathematics, a primorial prime is a prime number of the form ''pn''# ± 1, where ''pn''# is the primorial of ''pn'' (i.e. the product of the first ''n'' primes). Primality tests show that: : ''pn''# − 1 is prime for ...
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Pythagorean prime A Pythagorean prime is a prime number of the Pythagorean primes are exactly the odd prime numbers that are the sum of two squares; this characterization is Fermat's theorem on sums of two squares. Equivalently, by the Pythagorean theorem, they ...
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Prime quadruplet In number theory, a prime quadruplet (sometimes called a prime quadruple) is a set of four prime numbers of the form This represents the closest possible grouping of four primes larger than 3, and is the only prime constellation of length 4. P ...
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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 In 1919, Ramanujan published a new proof of Bertrand's postulate which, as ...
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Regular prime In number theory, a regular prime is a special kind of prime number, defined by Ernst Kummer in 1850 to prove certain cases of Fermat's Last Theorem. Regular primes may be defined via the divisibility of either class numbers or of Bernoulli num ...
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Safe prime In number theory, a prime number ''p'' is a if 2''p'' + 1 is also prime. The number 2''p'' + 1 associated with a Sophie Germain prime is called a . For example, 11 is a Sophie Germain prime and 2 × 11 +&nbs ...
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Sexy prime In number theory, sexy primes are prime numbers that differ from each other by . For example, the numbers and are a pair of sexy primes, because both are prime and 11 - 5 = 6. The term "sexy prime" is a pun stemming from the Latin word for six ...
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Sophie Germain prime In number theory, a prime number ''p'' is a if 2''p'' + 1 is also prime. The number 2''p'' + 1 associated with a Sophie Germain prime is called a . For example, 11 is a Sophie Germain prime and 2 × 11 +&nbs ...
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Super-prime Super-prime numbers, also known as higher-order primes or prime-indexed primes (PIPs), are the subsequence of prime numbers that occupy prime-numbered positions within the sequence of all prime numbers. In other words, if prime numbers are matched ...
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Thabit prime In number theory, a Thabit number, Thâbit ibn Qurra number, or 321 number is an integer of the form 3 \cdot 2^n - 1 for a non-negative integer ''n''. The first few Thabit numbers are: : 2, 5, 11, 23, 47, 95, 191, 383, 767, 1535, 3071, 614 ...
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Prime triplet In number theory, a prime triplet is a set of three prime numbers in which the smallest and largest of the three differ by 6. In particular, the sets must have the form or . With the exceptions of and , this is the closest possible grouping of ...
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Twin prime A twin prime is a prime number that is either 2 less or 2 more than another prime number—for example, either member of the twin prime pair or In other words, a twin prime is a prime that has a prime gap of two. Sometimes the term ''twin prime' ...
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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 Let p be a prime number. When each term in the sequence of Fibona ...
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Wieferich prime In number theory, a Wieferich prime is a prime number ''p'' such that ''p''2 divides , therefore connecting these primes with Fermat's little theorem, which states that every odd prime ''p'' divides . Wieferich primes were first described by A ...
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Wilson prime In number theory, a Wilson prime is a prime number p such that p^2 divides (p-1)!+1, where "!" denotes the factorial function; compare this with Wilson's theorem, which states that every prime p divides (p-1)!+1. Both are named for 18th-century ...
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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 3. Wolstenholme primes ...
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