This is a list of articles about
prime number
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 ...
s. A prime number (or ''prime'') is a
natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called ''Cardinal n ...
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 divisible by ...
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 infinitely many prime numbers. It was first proved by Euclid in his work ''Elements''. There are several proofs of the theorem.
Euclid's proof
Euclid offered ...
, there are an infinite number of prime numbers. Subsets of the prime numbers may be generated with various
formulas for primes
In number theory, a formula for primes is a formula generating the prime numbers, exactly and without exception. No such formula which is efficiently computable is known. A number of constraints are known, showing what such a "formula" can and ca ...
. 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 materials
...
.
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 verification project reports that it has computed all primes below 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 (the number of primes below 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) below 10. A different computation found that there are 18,435,599,767,349,200,867,866 primes (roughly 2) below 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 ...
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
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called ''Cardinal n ...
(including 0) in the definitions.
Balanced primes
Primes with equal-sized
prime gaps
A prime gap is the difference between two successive prime numbers. The ''n''-th prime gap, denoted ''g'n'' or ''g''(''p'n'') is the difference between the (''n'' + 1)-th and the
''n''-th prime numbers, i.e.
:g_n = p_ - p_n.\
W ...
above and below them, so that they are equal to the
arithmetic mean
In mathematics and statistics, the arithmetic mean ( ) or arithmetic average, or just the ''mean'' or the ''average'' (when the context is clear), is the sum of a collection of numbers divided by the count of numbers in the collection. The colle ...
of the nearest primes above and below.
*
5,
53, 157,
173, 211,
257, 263, 373, 563, 593, 607, 653, 733, 947, 977, 1103, 1123, 1187, 1223, 1367, 1511, 1747, 1753, 1907, 2287, 2417, 2677, 2903, 2963, 3307, 3313, 3637, 3733, 4013, 4409, 4457, 4597, 4657, 4691, 4993, 5107, 5113, 5303, 5387, 5393 ().
Bell primes
Primes that are the number of
partitions of a set
Partition may refer to:
Computing Hardware
* Disk partitioning, the division of a hard disk drive
* Memory partition, a subdivision of a computer's memory, usually for use by a single job
Software
* Partition (database), the division of a ...
with ''n'' members.
2,
5,
877, 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 nu ...
.
2,
3,
5,
7,
11,
13,
17,
19,
23,
29,
31,
37,
41,
47,
53,
59,
67,
71,
83,
89,
101,
107,
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 ...
,
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 may refer to:
* 139 (number), an integer
* AD 139, a year of the Julian calendar
* 13 ...
,
149,
157,
167,
179,
181,
191,
197
Year 197 ( CXCVII) was a common year starting on Saturday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Magius and Rufinus (or, less frequently, year 950 '' Ab urbe con ...
,
199,
211,
227,
233,
239,
251,
257,
263
__NOTOC__
Year 263 ( CCLXIII) was a common year starting on Thursday (link will display the full calendar) 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 ' ...
,
269,
281,
293,
307
__NOTOC__
Year 307 ( CCCVII) was a common year starting on Wednesday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Severus and Maximinus (or, less frequently, year 1060 ...
,
311 311 may refer to:
* 311 (number), a natural number
* AD 311, a year of the Julian calendar, in the fourth century AD
* 311 BC, a year of the pre-Julian Roman calendar
* 311 (band), an American band
** ''311'' (album), band 311's self-titled album
...
,
317,
337,
347
Year 347 (Roman numerals, CCCXLVII) was a common year starting on Thursday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Rufinus and Eusebius (or, less frequently, year 1 ...
,
353,
359
__NOTOC__
Year 359 ( CCCLIX) was a common year starting on Friday (link will display the full calendar) 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 ' ...
,
379
Year 379 ( CCCLXXIX) was a common year starting on Tuesday (link will display the full calendar) 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 '' A ...
,
389,
401,
409 ()
Circular prime
A circular prime is a prime number with the property that the number generated at each intermediate step when cyclically permuting its (base 10) digits will be prime. For example, 1193 is a circular prime, since 1931, 9311 and 3119 all are also p ...
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,
13,
17,
31,
37,
71,
73,
79,
97,
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 ...
,
131,
197
Year 197 ( CXCVII) was a common year starting on Saturday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Magius and Rufinus (or, less frequently, year 950 '' Ab urbe con ...
,
199,
311 311 may refer to:
* 311 (number), a natural number
* AD 311, a year of the Julian calendar, in the fourth century AD
* 311 BC, a year of the pre-Julian Roman calendar
* 311 (band), an American band
** ''311'' (album), band 311's self-titled album
...
,
337,
373,
719,
733
__NOTOC__
Year 733 ( DCCXXXIII) was a common year starting on Thursday (link will display the full calendar) of the Julian calendar. The denomination 733 for this year has been used since the early medieval period, when the Anno Domini calend ...
,
919
__NOTOC__
Year 919 ( CMXIX) was a common year starting on Friday (link will display the full calendar) of the Julian calendar.
Events
By Place
Byzantine Empire
* March 25 – Romanos Lekapenos, admiral (''droungarios'') of the ...
,
971
Year 971 ( CMLXXI) was a common year starting on Sunday (link will display the full calendar) of the Julian calendar.
Events
By place
Byzantine Empire
* Battle of Dorostolon: A Byzantine expeditionary army (possibly 30–40,000 men) ...
,
991,
1193
Year 1193 ( MCXCIII) was a common year starting on Friday (link will display the full calendar) of the Julian calendar.
Events
By place
Levant
* March 4 – Saladin (the Lion) dies of a fever at Damascus. The lands of the Ayyu ...
,
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 the ...
really calls this sequence circular primes, but not the above sequence):
2,
3,
5,
7,
11,
13,
17,
37,
79,
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 ...
,
197
Year 197 ( CXCVII) was a common year starting on Saturday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Magius and Rufinus (or, less frequently, year 950 '' Ab urbe con ...
,
199,
337,
1193
Year 1193 ( MCXCIII) was a common year starting on Friday (link will display the full calendar) of the Julian calendar.
Events
By place
Levant
* March 4 – Saladin (the Lion) dies of a fever at Damascus. The lands of the Ayyu ...
,
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 ''Recreat ...
primes are circular.
Cluster primes
A cluster prime is a prime ''p'' such that every even
natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called ''Cardinal n ...
k ≤ p − 3 is the difference of two primes not exceeding ''p''.
3,
5,
7,
11,
13,
17,
19,
23, ...
All odd primes between 3 and 89, inclusive, are cluster primes. The first 10 primes that are ''not'' cluster primes are:
2,
97,
127,
149,
191,
211,
223,
227,
229,
251.
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,
107), (
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 ...
), (
127,
131), (
163
Year 163 ( CLXIII) was a common year starting on Friday (link will display the full calendar) 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 con ...
,
167), (
193,
197
Year 197 ( CXCVII) was a common year starting on Saturday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Magius and Rufinus (or, less frequently, year 950 '' Ab urbe con ...
), (
223,
227), (
229,
233), (
277
__NOTOC__
Year 277 ( CCLXXVII) was a common year starting on Monday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Probus and Paulinus (or, less frequently, year 1030 ''A ...
,
281) (, )
Cuban prime
A cuban prime is a prime number that is also a solution to one of two different specific equations involving differences between third powers of two integers ''x'' and ''y''.
First series
This is the first of these equations:
:p = \frac,\ x = ...
s
Of the form
where ''x'' = ''y'' + 1.
7,
19,
37,
61,
127,
271
__NOTOC__
Year 271 ( CCLXXI) was a common year starting on Sunday (link will display the full calendar) 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 '' ...
,
331
__NOTOC__
Year 331 (Roman numerals, CCCXXXI) was a common year starting on Friday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Bassus and Ablabius (or, less frequent ...
,
397,
547,
631
__NOTOC__
Year 631 ( DCXXXI) was a common year starting on Tuesday (link will display the full calendar) of the Julian calendar. The denomination 631 for this year has been used since the early medieval period, when the Anno Domini calendar er ...
,
919
__NOTOC__
Year 919 ( CMXIX) was a common year starting on Friday (link will display the full calendar) of the Julian calendar.
Events
By Place
Byzantine Empire
* March 25 – Romanos Lekapenos, admiral (''droungarios'') of the ...
,
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
where ''x'' = ''y'' + 2.
13,
109,
193,
433,
769,
1201
Year 1201 ( MCCI) was a common year starting on Monday (link will display the full calendar) of the Julian calendar.
Events
By place
Byzantine Empire
* July 31 – John Komnenos the Fat, a Byzantine aristocrat, attempts to usurp ...
,
1453,
2029,
3469,
3889,
4801,
10093,
12289,
13873,
18253,
20173,
21169,
22189,
28813,
37633,
43201,
47629,
60493,
63949,
65713,
69313,
73009,
76801,
84673,
106033,
108301,
112909,
115249 ()
Cullen primes
Of the form ''n''×2 + 1.
3, 393050634124102232869567034555427371542904833 ()
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 (normally or upside down), and surface (actual display or reflectio ...
s
Primes that remain prime when read upside down or mirrored in a
seven-segment display
A seven-segment display is a form of electronic display device for displaying decimal numerals that is an alternative to the more complex dot matrix displays.
Seven-segment displays are widely used in digital clocks, electronic meters, basic ...
.
2,
5,
11,
101,
181,
1181
Year 1181 ( MCLXXXI) was a common year starting on Thursday (link will display the full calendar) of the Julian calendar.
Events
By place
Europe
* King Philip II (Augustus) annuls all loans made by Jews to Christians, and takes a per ...
,
1811,
18181,
108881,
110881,
118081,
120121,
121021,
121151,
150151,
151051,
151121,
180181,
180811,
181081 ()
Eisenstein primes without imaginary part
Eisenstein integers that are
irreducible
In philosophy, systems theory, science, and art, emergence occurs when an entity is observed to have properties its parts do not have on their own, properties or behaviors that emerge only when the parts interact in a wider whole.
Emergence ...
and real numbers (primes of the form 3''n'' − 1).
2,
5,
11,
17,
23,
29,
41,
47,
53,
59,
71,
83,
89,
101,
107,
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 ...
,
131,
137,
149,
167,
173,
179,
191,
197
Year 197 ( CXCVII) was a common year starting on Saturday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Magius and Rufinus (or, less frequently, year 950 '' Ab urbe con ...
,
227,
233,
239,
251,
257,
263
__NOTOC__
Year 263 ( CCLXIII) was a common year starting on Thursday (link will display the full calendar) 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 ' ...
,
269,
281,
293,
311 311 may refer to:
* 311 (number), a natural number
* AD 311, a year of the Julian calendar, in the fourth century AD
* 311 BC, a year of the pre-Julian Roman calendar
* 311 (band), an American band
** ''311'' (album), band 311's self-titled album
...
,
317,
347
Year 347 (Roman numerals, CCCXLVII) was a common year starting on Thursday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Rufinus and Eusebius (or, less frequently, year 1 ...
,
353,
359
__NOTOC__
Year 359 ( CCCLIX) was a common year starting on Friday (link will display the full calendar) 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 ' ...
,
383,
389,
401 ()
Emirps
Primes that become a different prime when their decimal digits are reversed. The name "emirp" is obtained by reversing the word "prime".
13,
17,
31,
37,
71,
73,
79,
97,
107,
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 ...
,
149,
157,
167,
179,
199,
311 311 may refer to:
* 311 (number), a natural number
* AD 311, a year of the Julian calendar, in the fourth century AD
* 311 BC, a year of the pre-Julian Roman calendar
* 311 (band), an American band
** ''311'' (album), band 311's self-titled album
...
,
337,
347
Year 347 (Roman numerals, CCCXLVII) was a common year starting on Thursday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Rufinus and Eusebius (or, less frequently, year 1 ...
,
359
__NOTOC__
Year 359 ( CCCLIX) was a common year starting on Friday (link will display the full calendar) 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 ' ...
,
389,
701
__NOTOC__
Year 701 ( DCCI) was a common year starting on Saturday (link will display the full calendar) of the Julian calendar. The denomination 701 for this year has been used since the early medieval period, when the Anno Domini calendar era ...
,
709,
733
__NOTOC__
Year 733 ( DCCXXXIII) was a common year starting on Thursday (link will display the full calendar) of the Julian calendar. The denomination 733 for this year has been used since the early medieval period, when the Anno Domini calend ...
,
739,
743
__NOTOC__
Year 743 ( DCCXLIII) was a common year starting on Tuesday (link will display the full calendar) of the Julian calendar. The denomination 743 for this year has been used since the early medieval period, when the Anno Domini calendar ...
,
751,
761
__NOTOC__
Year 761 ( DCCLXI) was a common year starting on Thursday (link will display the full calendar) of the Julian calendar. The denomination 761 for this year has been used since the early medieval period, when the Anno Domini calendar er ...
,
769,
907
__NOTOC__
Year 907 ( CMVII) was a common year starting on Thursday (link will display the full calendar) of the Julian calendar.
Events
By place
Byzantine Empire
* Rus'–Byzantine War: Varangian prince Oleg of Novgorod leads the ...
,
937
Year 937 ( CMXXXVII) was a common year starting on Sunday (link will display the full calendar) of the Julian calendar.
Events
By place
Europe
* A Hungarian army invades Burgundy, and burns the city of Tournus. Then they go southward ...
,
941,
953
Year 953 ( CMLIII) was a common year starting on Saturday (link will display the full calendar) of the Julian calendar.
Events
By place
Byzantine Empire
* Battle of Marash: Emir Sayf al-Dawla marches north into the Byzantine Empire an ...
,
967,
971
Year 971 ( CMLXXI) was a common year starting on Sunday (link will display the full calendar) of the Julian calendar.
Events
By place
Byzantine Empire
* Battle of Dorostolon: A Byzantine expeditionary army (possibly 30–40,000 men) ...
,
983
Year 983 ( CMLXXXIII) was a common year starting on Monday (link will display the full calendar) of the Julian calendar.
Events
By place
Europe
* Summer – Diet of Verona: Emperor Otto II (the Red) declares war against the Byza ...
,
991 ()
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 ''n ...
s).
3,
7,
31,
211,
2311, 200560490131 (
[ includes 2 = ]empty product
In mathematics, an empty product, or nullary product or vacuous product, is the result of multiplying no factors. It is by convention equal to the multiplicative identity (assuming there is an identity for the multiplication operation in question ...
of first 0 primes plus 1, but 2 is excluded in this list.)
Euler irregular primes
A prime
that divides
Euler number
In mathematics, the Euler numbers are a sequence ''En'' of integers defined by the Taylor series expansion
:\frac = \frac = \sum_^\infty \frac \cdot t^n,
where \cosh (t) is the hyperbolic cosine function. The Euler numbers are related to a ...
for some
.
19,
31,
43,
47,
61,
67,
71,
79,
101,
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 may refer to:
* 139 (number), an integer
* AD 139, a year of the Julian calendar
* 13 ...
,
149,
193,
223,
241,
251,
263
__NOTOC__
Year 263 ( CCLXIII) was a common year starting on Thursday (link will display the full calendar) 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 ' ...
,
277
__NOTOC__
Year 277 ( CCLXXVII) was a common year starting on Monday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Probus and Paulinus (or, less frequently, year 1030 ''A ...
,
307
__NOTOC__
Year 307 ( CCCVII) was a common year starting on Wednesday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Severus and Maximinus (or, less frequently, year 1060 ...
,
311 311 may refer to:
* 311 (number), a natural number
* AD 311, a year of the Julian calendar, in the fourth century AD
* 311 BC, a year of the pre-Julian Roman calendar
* 311 (band), an American band
** ''311'' (album), band 311's self-titled album
...
,
349,
353,
359
__NOTOC__
Year 359 ( CCCLIX) was a common year starting on Friday (link will display the full calendar) 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 ' ...
,
373,
379
Year 379 ( CCCLXXIX) was a common year starting on Tuesday (link will display the full calendar) 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 '' A ...
,
419,
433,
461,
463
Year 463 ( CDLXIII) was a common year starting on Tuesday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Basilius and Vivianus (or, less frequently, year 1216 '' Ab urbe ...
,
491,
509,
541
__NOTOC__
Year 541 (Roman numerals, DXLI) was a common year starting on Tuesday of the Julian calendar. In the Roman Empire, it was known as the Year of the Consulship of Anicius Faustus Albinus Basilius, Basilius without colleague (or, less ...
,
563,
571,
577
__NOTOC__
Year 577 ( DLXXVII) was a common year starting on Friday (link will display the full calendar) of the Julian calendar. The denomination 577 for this year has been used since early medieval times, when the Anno Domini calendar era be ...
,
587
__NOTOC__
Year 587 ( DLXXXVII) was a common year starting on Wednesday (link will display the full calendar) of the Julian calendar. The denomination 587 for this year has been used since the early medieval period, when the Anno Domini calenda ...
()
Euler (''p'', ''p'' − 3) irregular primes
Primes
such that
is an Euler irregular pair.
149,
241,
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! +&n ...
s
Of the form ''n''
! − 1 or ''n''
! + 1.
2,
3,
5,
7,
23,
719,
5039,
39916801,
479001599, 87178291199, 10888869450418352160768000001, 265252859812191058636308479999999, 263130836933693530167218012159999999, 8683317618811886495518194401279999999 ()
Fermat prime
In mathematics, a Fermat number, named after Pierre de Fermat, who first 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, 257, 65537, 429496 ...
s
Of the form 2 + 1.
3,
5,
17,
257,
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). Therefore, a regular polygon with 65537 sides is constructible with compass and unmarked straightedge. Johann ...
()
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, who first 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, 257, 65537, 4294967 ...
Of the form ''a'' + 1 for fixed integer ''a''.
''a'' = 2:
3,
5,
17,
257,
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). Therefore, a regular polygon with 65537 sides is constructible with compass and unmarked straightedge. Johann ...
()
''a'' = 4:
5,
17,
257,
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). Therefore, a regular polygon with 65537 sides is constructible with compass and unmarked straightedge. Johann ...
''a'' = 6:
7,
37,
1297
''a'' = 8: (does not exist)
''a'' = 10:
11,
101
''a'' = 12:
13
''a'' = 14:
197
Year 197 ( CXCVII) was a common year starting on Saturday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Magius and Rufinus (or, less frequently, year 950 '' Ab urbe con ...
''a'' = 16:
17,
257,
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). Therefore, a regular polygon with 65537 sides is constructible with compass and unmarked straightedge. Johann ...
''a'' = 18:
19
''a'' = 20:
401, 160001
''a'' = 22:
23
''a'' = 24:
577
__NOTOC__
Year 577 ( DLXXVII) was a common year starting on Friday (link will display the full calendar) of the Julian calendar. The denomination 577 for this year has been used since early medieval times, when the Anno Domini calendar era be ...
, 331777
these are the only known generalized Fermat primes for ''a'' ≤ 24.
Fibonacci prime
A Fibonacci prime is a Fibonacci number that is prime, a type of integer sequence prime.
The first Fibonacci primes are :
: 2, 3, 5, 13, 89, 233, 1597, 28657, 514229, 433494437, 2971215073, ....
Known Fibonacci primes
It is not known whet ...
s
Primes in the
Fibonacci sequence
In mathematics, the Fibonacci numbers, commonly denoted , form a integer sequence, sequence, the Fibonacci sequence, in which each number is the sum of the two preceding ones. The sequence commonly starts from 0 and 1, although some authors start ...
''F'' = 0, ''F'' = 1,
''F'' = ''F'' + ''F''.
2,
3,
5,
13,
89,
233,
1597, 28657, 514229, 433494437, 2971215073, 99194853094755497, 1066340417491710595814572169, 19134702400093278081449423917 ()
Fortunate primes
Fortunate number
A Fortunate number, named after Reo Fortune, is the smallest integer ''m'' > 1 such that, for a given positive integer ''n'', ''p'n''# + ''m'' is a prime number, where the primorial ''p'n''# is the product of the first ''n'' prime numbers.
...
s that are prime (it has been conjectured they all are).
3,
5,
7,
13,
17,
19,
23,
37,
47,
59,
61,
67,
71,
79,
89,
101,
103,
107,
109,
127,
151
Year 151 (CLI) was a common year starting on Thursday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Condianus and Valerius (or, less frequently, year 904 ''Ab urbe cond ...
,
157,
163
Year 163 ( CLXIII) was a common year starting on Friday (link will display the full calendar) 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 con ...
,
167,
191,
197
Year 197 ( CXCVII) was a common year starting on Saturday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Magius and Rufinus (or, less frequently, year 950 '' Ab urbe con ...
,
199,
223,
229,
233,
239,
271
__NOTOC__
Year 271 ( CCLXXI) was a common year starting on Sunday (link will display the full calendar) 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 '' ...
,
277
__NOTOC__
Year 277 ( CCLXXVII) was a common year starting on Monday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Probus and Paulinus (or, less frequently, year 1030 ''A ...
,
283
Year 283 ( CCLXXXIII) was a common year starting on Monday (link will display the full calendar) 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 cond ...
,
293,
307
__NOTOC__
Year 307 ( CCCVII) was a common year starting on Wednesday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Severus and Maximinus (or, less frequently, year 1060 ...
,
311 311 may refer to:
* 311 (number), a natural number
* AD 311, a year of the Julian calendar, in the fourth century AD
* 311 BC, a year of the pre-Julian Roman calendar
* 311 (band), an American band
** ''311'' (album), band 311's self-titled album
...
,
313,
331
__NOTOC__
Year 331 (Roman numerals, CCCXXXI) was a common year starting on Friday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Bassus and Ablabius (or, less frequent ...
,
353,
373,
379
Year 379 ( CCCLXXIX) was a common year starting on Tuesday (link will display the full calendar) 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 '' A ...
,
383,
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 pri ...
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,
107,
127,
131,
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 may refer to:
* 139 (number), an integer
* AD 139, a year of the Julian calendar
* 13 ...
,
151
Year 151 (CLI) was a common year starting on Thursday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Condianus and Valerius (or, less frequently, year 904 ''Ab urbe cond ...
,
163
Year 163 ( CLXIII) was a common year starting on Friday (link will display the full calendar) 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 con ...
,
167,
179,
191,
199,
211,
223,
227,
239,
251,
263
__NOTOC__
Year 263 ( CCLXIII) was a common year starting on Thursday (link will display the full calendar) 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 ' ...
,
271
__NOTOC__
Year 271 ( CCLXXI) was a common year starting on Sunday (link will display the full calendar) 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 '' ...
,
283
Year 283 ( CCLXXXIII) was a common year starting on Monday (link will display the full calendar) 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 cond ...
,
307
__NOTOC__
Year 307 ( CCCVII) was a common year starting on Wednesday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Severus and Maximinus (or, less frequently, year 1060 ...
,
311 311 may refer to:
* 311 (number), a natural number
* AD 311, a year of the Julian calendar, in the fourth century AD
* 311 BC, a year of the pre-Julian Roman calendar
* 311 (band), an American band
** ''311'' (album), band 311's self-titled album
...
,
331
__NOTOC__
Year 331 (Roman numerals, CCCXXXI) was a common year starting on Friday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Bassus and Ablabius (or, less frequent ...
,
347
Year 347 (Roman numerals, CCCXLVII) was a common year starting on Thursday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Rufinus and Eusebius (or, less frequently, year 1 ...
,
359
__NOTOC__
Year 359 ( CCCLIX) was a common year starting on Friday (link will display the full calendar) 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 ' ...
,
367
__NOTOC__
Year 367 ( CCCLXVII) was a common year starting on Monday of the Julian calendar. At the time, it was known as the Year of the Consulship of Lupicinus and Iovanus (or, less frequently, year 1120 ''Ab urbe condita''). The denomination ...
,
379
Year 379 ( CCCLXXIX) was a common year starting on Tuesday (link will display the full calendar) 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 '' A ...
,
383,
419,
431,
439,
443
__NOTOC__
Year 443 ( CDXLIII) was a common year starting on Friday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Maximus and Paterius (or, less frequently, year 1196 ' ...
,
463
Year 463 ( CDLXIII) was a common year starting on Tuesday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Basilius and Vivianus (or, less frequently, year 1216 '' Ab urbe ...
,
467,
479,
487,
491,
499
__NOTOC__
Year 499 (Roman numerals, CDXCIX) was a common year starting on Friday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Iohannes without colleague (or, less frequ ...
,
503 ()
Good primes
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,
223,
227,
251,
257,
269,
307
__NOTOC__
Year 307 ( CCCVII) was a common year starting on Wednesday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Severus and Maximinus (or, less frequently, year 1060 ...
()
Happy prime
In number theory, a happy number is a number which eventually reaches 1 when 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 number because ...
s
Happy numbers that are prime.
7,
13,
19,
23,
31,
79,
97,
103,
109,
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 may refer to:
* 139 (number), an integer
* AD 139, a year of the Julian calendar
* 13 ...
,
167,
193,
239,
263
__NOTOC__
Year 263 ( CCLXIII) was a common year starting on Thursday (link will display the full calendar) 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 ' ...
,
293,
313,
331
__NOTOC__
Year 331 (Roman numerals, CCCXXXI) was a common year starting on Friday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Bassus and Ablabius (or, less frequent ...
,
367
__NOTOC__
Year 367 ( CCCLXVII) was a common year starting on Monday of the Julian calendar. At the time, it was known as the Year of the Consulship of Lupicinus and Iovanus (or, less frequently, year 1120 ''Ab urbe condita''). The denomination ...
,
379
Year 379 ( CCCLXXIX) was a common year starting on Tuesday (link will display the full calendar) 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 '' A ...
,
383,
397,
409,
487,
563,
617
__NOTOC__
Year 617 ( DCXVII) was a common year starting on Saturday (link will display the full calendar) of the Julian calendar. The denomination 617 for this year has been used since the early medieval period, when the Anno Domini calendar e ...
,
653
__NOTOC__
Year 653 ( DCLIII) was a common year starting on Tuesday (link will display the full calendar) of the Julian calendar. The denomination 653 for this year has been used since the early medieval period, when the Anno Domini calendar era ...
,
673
__NOTOC__
Year 673 ( DCLXXIII) was a common year starting on Saturday (link will display the full calendar) of the Julian calendar. The denomination 673 for this year has been used since the early medieval period, when the Anno Domini calendar ...
,
683
__NOTOC__
Year 683 (Roman numerals, DCLXXXIII) was a common year starting on Thursday (link will display the full calendar) of the Julian calendar. The denomination 683 for this year has been used since the early medieval period, when the Anno ...
,
709,
739,
761
__NOTOC__
Year 761 ( DCCLXI) was a common year starting on Thursday (link will display the full calendar) of the Julian calendar. The denomination 761 for this year has been used since the early medieval period, when the Anno Domini calendar er ...
,
863
__NOTOC__
Year 863 ( DCCCLXIII) was a common year starting on Friday (link will display the full calendar) of the Julian calendar.
Events
By place Byzantine Empire
* September 3 – Battle of Lalakaon: A Byzantine army confronts ...
,
881,
907
__NOTOC__
Year 907 ( CMVII) was a common year starting on Thursday (link will display the full calendar) of the Julian calendar.
Events
By place
Byzantine Empire
* Rus'–Byzantine War: Varangian prince Oleg of Novgorod leads the ...
,
937
Year 937 ( CMXXXVII) was a common year starting on Sunday (link will display the full calendar) of the Julian calendar.
Events
By place
Europe
* A Hungarian army invades Burgundy, and burns the city of Tournus. Then they go southward ...
,
1009
Year 1009 ( MIX) was a common year starting on Saturday of the Julian calendar.
Events
By place
Europe
* February 14 or March 9 – The first known mention is made of the name of Lithuania, in connection with the murder of Bruno ...
,
1033,
1039,
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,
107,
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 ...
,
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 may refer to:
* 139 (number), an integer
* AD 139, a year of the Julian calendar
* 13 ...
,
149,
157,
179,
191,
193,
223,
239,
241,
251,
263
__NOTOC__
Year 263 ( CCLXIII) was a common year starting on Thursday (link will display the full calendar) 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 ' ...
,
277
__NOTOC__
Year 277 ( CCLXXVII) was a common year starting on Monday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Probus and Paulinus (or, less frequently, year 1030 ''A ...
,
281,
293,
307
__NOTOC__
Year 307 ( CCCVII) was a common year starting on Wednesday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Severus and Maximinus (or, less frequently, year 1060 ...
,
311 311 may refer to:
* 311 (number), a natural number
* AD 311, a year of the Julian calendar, in the fourth century AD
* 311 BC, a year of the pre-Julian Roman calendar
* 311 (band), an American band
** ''311'' (album), band 311's self-titled album
...
,
317,
331
__NOTOC__
Year 331 (Roman numerals, CCCXXXI) was a common year starting on Friday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Bassus and Ablabius (or, less frequent ...
,
337,
349 ()
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,
107,
127,
131,
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 may refer to:
* 139 (number), an integer
* AD 139, a year of the Julian calendar
* 13 ...
,
149,
151
Year 151 (CLI) was a common year starting on Thursday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Condianus and Valerius (or, less frequently, year 904 ''Ab urbe cond ...
,
157,
173,
181,
191,
197
Year 197 ( CXCVII) was a common year starting on Saturday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Magius and Rufinus (or, less frequently, year 950 '' Ab urbe con ...
,
199,
211,
223,
229,
263
__NOTOC__
Year 263 ( CCLXIII) was a common year starting on Thursday (link will display the full calendar) 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 ' ...
,
269,
277
__NOTOC__
Year 277 ( CCLXXVII) was a common year starting on Monday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Probus and Paulinus (or, less frequently, year 1030 ''A ...
,
283
Year 283 ( CCLXXXIII) was a common year starting on Monday (link will display the full calendar) 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 cond ...
,
311 311 may refer to:
* 311 (number), a natural number
* AD 311, a year of the Julian calendar, in the fourth century AD
* 311 BC, a year of the pre-Julian Roman calendar
* 311 (band), an American band
** ''311'' (album), band 311's self-titled album
...
,
317,
331
__NOTOC__
Year 331 (Roman numerals, CCCXXXI) was a common year starting on Friday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Bassus and Ablabius (or, less frequent ...
,
347
Year 347 (Roman numerals, CCCXLVII) was a common year starting on Thursday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Rufinus and Eusebius (or, less frequently, year 1 ...
,
349 ()
Highly cototient primes
Primes that are a
cototient
In number theory, Euler's totient function counts the positive integers up to a given integer that are relatively prime to . It is written using the Greek letter phi as \varphi(n) or \phi(n), and may also be called Euler's phi function. In ot ...
more often than any integer below it except 1.
2,
23,
47,
59,
83,
89,
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 ...
,
167,
269,
389,
419,
509,
659
__NOTOC__
Year 659 ( DCLIX) was a common year starting on Tuesday (link will display the full calendar) of the Julian calendar. The denomination 659 for this year has been used since the early medieval period, when the Anno Domini calendar era ...
,
839
__NOTOC__
Year 839 ( DCCCXXXIX) was a common year starting on Wednesday (link will display the full calendar) of the Julian calendar.
Events
By place Europe
* Prince Sicard of Benevento is assassinated by a conspiracy among the nobil ...
,
1049
Year 1049 ( MXLIX) was a common year starting on Sunday (link will display the full calendar) of the Julian calendar.
Events
By place
Byzantine Empire
* Spring – Pecheneg Revolt: Emperor Constantine IX decides to transfer 15,0 ...
,
1259
Year 1259 ( MCCLIX) was a common year starting on Wednesday (link will display the full calendar) of the Julian calendar.
Events
By place Europe
* September – Battle of Pelagonia: The Empire of Nicaea defeats the Principality of Ac ...
,
1889 ()
Home prime In number theory, the home prime HP(''n'') of an integer ''n'' greater than 1 is the prime number obtained by repeatedly factoring the increasing concatenation of prime factors including repetitions. The ''m''th intermediate stage in the process o ...
s
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 numbers or of Bernoulli num ...
s
Odd primes ''p'' that divide the
class number of the ''p''-th
cyclotomic field
In number theory, a cyclotomic field is a number field obtained by adjoining a complex root of unity to , the field of rational numbers.
Cyclotomic fields played a crucial role in the development of modern algebra and number theory because of ...
.
37,
59,
67,
101,
103,
131,
149,
157,
233,
257,
263
__NOTOC__
Year 263 ( CCLXIII) was a common year starting on Thursday (link will display the full calendar) 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 ' ...
,
271
__NOTOC__
Year 271 ( CCLXXI) was a common year starting on Sunday (link will display the full calendar) 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 '' ...
,
283
Year 283 ( CCLXXXIII) was a common year starting on Monday (link will display the full calendar) 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 cond ...
,
293,
307
__NOTOC__
Year 307 ( CCCVII) was a common year starting on Wednesday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Severus and Maximinus (or, less frequently, year 1060 ...
,
311 311 may refer to:
* 311 (number), a natural number
* AD 311, a year of the Julian calendar, in the fourth century AD
* 311 BC, a year of the pre-Julian Roman calendar
* 311 (band), an American band
** ''311'' (album), band 311's self-titled album
...
,
347
Year 347 (Roman numerals, CCCXLVII) was a common year starting on Thursday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Rufinus and Eusebius (or, less frequently, year 1 ...
,
353,
379
Year 379 ( CCCLXXIX) was a common year starting on Tuesday (link will display the full calendar) 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 '' A ...
,
389,
401,
409,
421,
433,
461,
463
Year 463 ( CDLXIII) was a common year starting on Tuesday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Basilius and Vivianus (or, less frequently, year 1216 '' Ab urbe ...
,
467,
491,
523,
541
__NOTOC__
Year 541 (Roman numerals, DXLI) was a common year starting on Tuesday of the Julian calendar. In the Roman Empire, it was known as the Year of the Consulship of Anicius Faustus Albinus Basilius, Basilius without colleague (or, less ...
,
547,
557,
577
__NOTOC__
Year 577 ( DLXXVII) was a common year starting on Friday (link will display the full calendar) of the Julian calendar. The denomination 577 for this year has been used since early medieval times, when the Anno Domini calendar era be ...
,
587
__NOTOC__
Year 587 ( DLXXXVII) was a common year starting on Wednesday (link will display the full calendar) of the Julian calendar. The denomination 587 for this year has been used since the early medieval period, when the Anno Domini calenda ...
,
593,
607,
613
__NOTOC__
Year 613 ( DCXIII) was a common year starting on Monday (link will display the full calendar) of the Julian calendar. The denomination 613 for this year has been used since the early medieval period, when the Anno Domini calendar era ...
()
(''p'', ''p'' − 3) irregular primes
(See
Wolstenholme prime)
(''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,
877 ()
Isolated primes
Primes ''p'' such that neither ''p'' − 2 nor ''p'' + 2 is prime.
2,
23,
37,
47,
53,
67,
79,
83,
89,
97,
113 113 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 ...
,
127,
131,
157,
163
Year 163 ( CLXIII) was a common year starting on Friday (link will display the full calendar) 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 con ...
,
167,
173,
211,
223,
233,
251,
257,
263
__NOTOC__
Year 263 ( CCLXIII) was a common year starting on Thursday (link will display the full calendar) 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 ' ...
,
277
__NOTOC__
Year 277 ( CCLXXVII) was a common year starting on Monday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Probus and Paulinus (or, less frequently, year 1030 ''A ...
,
293,
307
__NOTOC__
Year 307 ( CCCVII) was a common year starting on Wednesday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Severus and Maximinus (or, less frequently, year 1060 ...
,
317,
331
__NOTOC__
Year 331 (Roman numerals, CCCXXXI) was a common year starting on Friday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Bassus and Ablabius (or, less frequent ...
,
337,
353,
359
__NOTOC__
Year 359 ( CCCLIX) was a common year starting on Friday (link will display the full calendar) 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 ' ...
,
367
__NOTOC__
Year 367 ( CCCLXVII) was a common year starting on Monday of the Julian calendar. At the time, it was known as the Year of the Consulship of Lupicinus and Iovanus (or, less frequently, year 1120 ''Ab urbe condita''). The denomination ...
,
373,
379
Year 379 ( CCCLXXIX) was a common year starting on Tuesday (link will display the full calendar) 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 '' A ...
,
383,
389,
397,
401,
409,
439,
443
__NOTOC__
Year 443 ( CDXLIII) was a common year starting on Friday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Maximus and Paterius (or, less frequently, year 1196 ' ...
,
449,
457,
467,
479,
487,
491,
499
__NOTOC__
Year 499 (Roman numerals, CDXCIX) was a common year starting on Friday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Iohannes without colleague (or, less frequ ...
,
503,
509,
541
__NOTOC__
Year 541 (Roman numerals, DXLI) was a common year starting on Tuesday of the Julian calendar. In the Roman Empire, it was known as the Year of the Consulship of Anicius Faustus Albinus Basilius, Basilius without colleague (or, less ...
,
547,
557,
563,
577
__NOTOC__
Year 577 ( DLXXVII) was a common year starting on Friday (link will display the full calendar) of the Julian calendar. The denomination 577 for this year has been used since early medieval times, when the Anno Domini calendar era be ...
,
587
__NOTOC__
Year 587 ( DLXXXVII) was a common year starting on Wednesday (link will display the full calendar) of the Julian calendar. The denomination 587 for this year has been used since the early medieval period, when the Anno Domini calenda ...
,
593,
607,
613
__NOTOC__
Year 613 ( DCXIII) was a common year starting on Monday (link will display the full calendar) of the Julian calendar. The denomination 613 for this year has been used since the early medieval period, when the Anno Domini calendar era ...
,
631
__NOTOC__
Year 631 ( DCXXXI) was a common year starting on Tuesday (link will display the full calendar) of the Julian calendar. The denomination 631 for this year has been used since the early medieval period, when the Anno Domini calendar er ...
,
647
__NOTOC__
Year 647 ( DCXLVII) was a common year starting on Monday (link will display the full calendar) of the Julian calendar. The denomination 647 for this year has been used since the early medieval period, when the Anno Domini calendar era ...
,
653
__NOTOC__
Year 653 ( DCLIII) was a common year starting on Tuesday (link will display the full calendar) of the Julian calendar. The denomination 653 for this year has been used since the early medieval period, when the Anno Domini calendar era ...
,
673
__NOTOC__
Year 673 ( DCLXXIII) was a common year starting on Saturday (link will display the full calendar) of the Julian calendar. The denomination 673 for this year has been used since the early medieval period, when the Anno Domini calendar ...
,
677
__NOTOC__
Year 677 ( DCLXXVII) was a common year starting on Thursday (link will display the full calendar) of the Julian calendar. The denomination 677 for this year has been used since the early medieval period, when the Anno Domini calendar e ...
,
683
__NOTOC__
Year 683 (Roman numerals, DCLXXXIII) was a common year starting on Thursday (link will display the full calendar) of the Julian calendar. The denomination 683 for this year has been used since the early medieval period, when the Anno ...
,
691
__NOTOC__
Year 691 ( DCXCI) was a common year starting on Sunday (link will display the full calendar) of the Julian calendar. The denomination 691 for this year has been used since the early medieval period, when the Anno Domini calendar era b ...
,
701
__NOTOC__
Year 701 ( DCCI) was a common year starting on Saturday (link will display the full calendar) of the Julian calendar. The denomination 701 for this year has been used since the early medieval period, when the Anno Domini calendar era ...
,
709,
719,
727,
733
__NOTOC__
Year 733 ( DCCXXXIII) was a common year starting on Thursday (link will display the full calendar) of the Julian calendar. The denomination 733 for this year has been used since the early medieval period, when the Anno Domini calend ...
,
739,
743
__NOTOC__
Year 743 ( DCCXLIII) was a common year starting on Tuesday (link will display the full calendar) of the Julian calendar. The denomination 743 for this year has been used since the early medieval period, when the Anno Domini calendar ...
,
751,
757
757 may refer to:
* Boeing 757: a narrow-body airliner
* AD 757: a year
* 757 BC: a year
* 757 (number): a number
* Area code 757
Image:Area code 757.png, The area colored red indicates the southeast corner of Virginia served by area code 757
po ...
,
761
__NOTOC__
Year 761 ( DCCLXI) was a common year starting on Thursday (link will display the full calendar) of the Julian calendar. The denomination 761 for this year has been used since the early medieval period, when the Anno Domini calendar er ...
,
769,
773
__NOTOC__
Year 773 ( DCCLXXIII) was a common year starting on Friday (link will display the full calendar) of the Julian calendar. The denomination 773 for this year has been used since the early medieval period, when the Anno Domini calendar e ...
,
787
787 may refer to:
* Boeing 787 Dreamliner, a jet airliner
* AD 787, a year
* 787 BC, a year
* Mazda 787/787B, a Japanese rotary-engine race car which won the 1991 Le Mans Race
* Porsche 787, a race car from the 1960s
* 787 series, a train model o ...
,
797,
839
__NOTOC__
Year 839 ( DCCCXXXIX) was a common year starting on Wednesday (link will display the full calendar) of the Julian calendar.
Events
By place Europe
* Prince Sicard of Benevento is assassinated by a conspiracy among the nobil ...
,
853
__NOTOC__
Year 853 ( DCCCLIII) was a common year starting on Sunday (link will display the full calendar) of the Julian calendar.
Events
By place Byzantine Empire
* May 22 – A Byzantine fleet (85 ships and 5,000 men) sacks and d ...
,
863
__NOTOC__
Year 863 ( DCCCLXIII) was a common year starting on Friday (link will display the full calendar) of the Julian calendar.
Events
By place Byzantine Empire
* September 3 – Battle of Lalakaon: A Byzantine army confronts ...
,
877,
887,
907
__NOTOC__
Year 907 ( CMVII) was a common year starting on Thursday (link will display the full calendar) of the Julian calendar.
Events
By place
Byzantine Empire
* Rus'–Byzantine War: Varangian prince Oleg of Novgorod leads the ...
,
911
911 or 9/11 may refer to:
Dates
* AD 911
* 911 BC
* September 11
** 9/11, the September 11 attacks of 2001
** 11 de Septiembre, Chilean coup d'état in 1973 that outed the democratically elected Salvador Allende
* November 9
Numbers
* 91 ...
,
919
__NOTOC__
Year 919 ( CMXIX) was a common year starting on Friday (link will display the full calendar) of the Julian calendar.
Events
By Place
Byzantine Empire
* March 25 – Romanos Lekapenos, admiral (''droungarios'') of the ...
,
929
Year 929 ( CMXXIX) was a common year starting on Thursday (link will display the full calendar) of the Julian calendar.
Events
By place
Europe
* January 16 – Emir Abd-al-Rahman III of Córdoba proclaims himself caliph and create ...
,
937
Year 937 ( CMXXXVII) was a common year starting on Sunday (link will display the full calendar) of the Julian calendar.
Events
By place
Europe
* A Hungarian army invades Burgundy, and burns the city of Tournus. Then they go southward ...
,
941,
947
Year 947 ( CMXLVII) was a common year starting on Friday (link will display the full calendar) of the Julian calendar.
Events
By place
Europe
* Summer – A Hungarian army led by Grand Prince Taksony campaigns in Italy, heading ...
,
953
Year 953 ( CMLIII) was a common year starting on Saturday (link will display the full calendar) of the Julian calendar.
Events
By place
Byzantine Empire
* Battle of Marash: Emir Sayf al-Dawla marches north into the Byzantine Empire an ...
,
967,
971
Year 971 ( CMLXXI) was a common year starting on Sunday (link will display the full calendar) of the Julian calendar.
Events
By place
Byzantine Empire
* Battle of Dorostolon: A Byzantine expeditionary army (possibly 30–40,000 men) ...
,
977,
983
Year 983 ( CMLXXXIII) was a common year starting on Monday (link will display the full calendar) of the Julian calendar.
Events
By place
Europe
* Summer – Diet of Verona: Emperor Otto II (the Red) declares war against the Byza ...
,
991,
997
Year 997 (Roman numerals, CMXCVII) was a common year starting on Friday (link will display the full calendar) of the Julian calendar.
Events
By place
Japan
* 1 February: Empress Teishi gives birth to Princess Shushi - she is the first ...
()
Leyland primes
Of the form ''x'' + ''y'', with 1 < ''x'' < ''y''.
17,
593, 32993, 2097593, 8589935681, 59604644783353249, 523347633027360537213687137, 43143988327398957279342419750374600193 ()
Long primes
Primes ''p'' for which, in a given base ''b'',
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
:14 ...
. They are also called full reptend primes. Primes ''p'' for base 10:
7,
17,
19,
23,
29,
47,
59,
61,
97,
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 ...
,
131,
149,
167,
179,
181,
193,
223,
229,
233,
257,
263
__NOTOC__
Year 263 ( CCLXIII) was a common year starting on Thursday (link will display the full calendar) 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 ' ...
,
269,
313,
337,
367
__NOTOC__
Year 367 ( CCCLXVII) was a common year starting on Monday of the Julian calendar. At the time, it was known as the Year of the Consulship of Lupicinus and Iovanus (or, less frequently, year 1120 ''Ab urbe condita''). The denomination ...
,
379
Year 379 ( CCCLXXIX) was a common year starting on Tuesday (link will display the full calendar) 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 '' A ...
,
383,
389,
419,
433,
461,
487,
491,
499
__NOTOC__
Year 499 (Roman numerals, CDXCIX) was a common year starting on Friday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Iohannes without colleague (or, less frequ ...
,
503,
509,
541
__NOTOC__
Year 541 (Roman numerals, DXLI) was a common year starting on Tuesday of the Julian calendar. In the Roman Empire, it was known as the Year of the Consulship of Anicius Faustus Albinus Basilius, Basilius without colleague (or, less ...
,
571,
577
__NOTOC__
Year 577 ( DLXXVII) was a common year starting on Friday (link will display the full calendar) of the Julian calendar. The denomination 577 for this year has been used since early medieval times, when the Anno Domini calendar era be ...
,
593 ()
Lucas prime
The Lucas numbers or Lucas series are an integer sequence named after the mathematician Édouard Lucas, François Édouard Anatole Lucas (1842–1891), who studied both that sequence and the closely related Fibonacci numbers. Lucas numbers a ...
s
Primes in the Lucas number sequence ''L'' = 2, ''L'' = 1,
''L'' = ''L'' + ''L''.
2,
3,
7,
11,
29,
47,
199,
521,
2207,
3571, 9349, 3010349, 54018521, 370248451, 6643838879, 119218851371, 5600748293801, 688846502588399, 32361122672259149 ()
Lucky primes
Lucky numbers that are prime.
3,
7,
13,
31,
37,
43,
67,
73,
79,
127,
151
Year 151 (CLI) was a common year starting on Thursday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Condianus and Valerius (or, less frequently, year 904 ''Ab urbe cond ...
,
163
Year 163 ( CLXIII) was a common year starting on Friday (link will display the full calendar) 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 con ...
,
193,
211,
223,
241,
283
Year 283 ( CCLXXXIII) was a common year starting on Monday (link will display the full calendar) 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 cond ...
,
307
__NOTOC__
Year 307 ( CCCVII) was a common year starting on Wednesday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Severus and Maximinus (or, less frequently, year 1060 ...
,
331
__NOTOC__
Year 331 (Roman numerals, CCCXXXI) was a common year starting on Friday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Bassus and Ablabius (or, less frequent ...
,
349,
367
__NOTOC__
Year 367 ( CCCLXVII) was a common year starting on Monday of the Julian calendar. At the time, it was known as the Year of the Consulship of Lupicinus and Iovanus (or, less frequently, year 1120 ''Ab urbe condita''). The denomination ...
,
409,
421,
433,
463
Year 463 ( CDLXIII) was a common year starting on Tuesday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Basilius and Vivianus (or, less frequently, year 1216 '' Ab urbe ...
,
487,
541
__NOTOC__
Year 541 (Roman numerals, DXLI) was a common year starting on Tuesday of the Julian calendar. In the Roman Empire, it was known as the Year of the Consulship of Anicius Faustus Albinus Basilius, Basilius without colleague (or, less ...
,
577
__NOTOC__
Year 577 ( DLXXVII) was a common year starting on Friday (link will display the full calendar) of the Julian calendar. The denomination 577 for this year has been used since early medieval times, when the Anno Domini calendar era be ...
,
601,
613
__NOTOC__
Year 613 ( DCXIII) was a common year starting on Monday (link will display the full calendar) of the Julian calendar. The denomination 613 for this year has been used since the early medieval period, when the Anno Domini calendar era ...
,
619
__NOTOC__
Year 619 ( DCXIX) was a common year starting on Monday (link will display the full calendar) of the Julian calendar. The denomination 619 for this year has been used since the early medieval period, when the Anno Domini calendar era b ...
,
631
__NOTOC__
Year 631 ( DCXXXI) was a common year starting on Tuesday (link will display the full calendar) of the Julian calendar. The denomination 631 for this year has been used since the early medieval period, when the Anno Domini calendar er ...
,
643
__NOTOC__
Year 643 (Roman numerals, DCXLIII) was a common year starting on Wednesday (link will display the full calendar) of the Julian calendar. The denomination 643 for this year has been used since the early medieval period, when the Anno D ...
,
673
__NOTOC__
Year 673 ( DCLXXIII) was a common year starting on Saturday (link will display the full calendar) of the Julian calendar. The denomination 673 for this year has been used since the early medieval period, when the Anno Domini calendar ...
,
727,
739,
769,
787
787 may refer to:
* Boeing 787 Dreamliner, a jet airliner
* AD 787, a year
* 787 BC, a year
* Mazda 787/787B, a Japanese rotary-engine race car which won the 1991 Le Mans Race
* Porsche 787, a race car from the 1960s
* 787 series, a train model o ...
,
823
__NOTOC__
Year 823 ( DCCCXXIII) was a common year starting on Thursday (link will display the full calendar) of the Julian calendar.
Events
By place
Byzantine Empire
* Emperor Michael II defeats the rebel forces under Thomas the Sla ...
,
883
__NOTOC__
Year 883 ( DCCCLXXXIII) was a common year starting on Tuesday (link will display the full calendar) of the Julian calendar.
Events
By place Europe
* Spring – Viking raiders ravage Flanders, and sack the abbey at Saint- ...
,
937
Year 937 ( CMXXXVII) was a common year starting on Sunday (link will display the full calendar) of the Julian calendar.
Events
By place
Europe
* A Hungarian army invades Burgundy, and burns the city of Tournus. Then they go southward ...
,
991,
997
Year 997 (Roman numerals, CMXCVII) was a common year starting on Friday (link will display the full calendar) of the Julian calendar.
Events
By place
Japan
* 1 February: Empress Teishi gives birth to Princess Shushi - she is the first ...
()
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 17t ...
s
Of the form 2 − 1.
3,
7,
31,
127,
8191,
131071,
524287,
2147483647, 2305843009213693951, 618970019642690137449562111, 162259276829213363391578010288127, 170141183460469231731687303715884105727 ()
, there are 51 known Mersenne primes. The 13th, 14th, and 51st have respectively 157, 183, and 24,862,048 digits.
, this class of prime numbers also contains the largest known prime: M
82589933, the 51st known 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,
107,
127,
521,
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 ()
, three more are known to be in the sequence, but it is not known whether they are the next:
74207281, 77232917, 82589933
Double Mersenne prime
In mathematics, a double Mersenne number is a Mersenne number of the form
:M_ = 2^-1
where ''p'' is prime.
Examples
The first four terms of the sequence of double Mersenne numbers areChris Caldwell''Mersenne Primes: History, Theorems and ...
s
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 prime
A cuban prime is a prime number that is also a solution to one of two different specific equations involving differences between third powers of two integers ''x'' and ''y''.
First series
This is the first of these equations:
:p = \frac,\ x = ...
s 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,
449,
499
__NOTOC__
Year 499 (Roman numerals, CDXCIX) was a common year starting on Friday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Iohannes without colleague (or, less frequ ...
,
881,
991, 6469, 6949,
9001, 9049, 9649, 9949, 60649, 666649, 946669, 60000049, 66000049, 66600049 ()
Newman–Shanks–Williams prime
In mathematics, a Newman–Shanks–Williams prime (NSW prime) is a prime number ''p'' which can be written in the form
:S_=\frac.
NSW primes were first described by Morris Newman, Daniel Shanks and Hugh C. Williams in 1981 during the study of ...
s
Newman–Shanks–Williams numbers that are prime.
7,
41,
239, 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
Year 151 (CLI) was a common year starting on Thursday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Condianus and Valerius (or, less frequently, year 904 ''Ab urbe cond ...
,
181,
191,
313,
353,
373,
383,
727,
757
757 may refer to:
* Boeing 757: a narrow-body airliner
* AD 757: a year
* 757 BC: a year
* 757 (number): a number
* Area code 757
Image:Area code 757.png, The area colored red indicates the southeast corner of Virginia served by area code 757
po ...
,
787
787 may refer to:
* Boeing 787 Dreamliner, a jet airliner
* AD 787, a year
* 787 BC, a year
* Mazda 787/787B, a Japanese rotary-engine race car which won the 1991 Le Mans Race
* Porsche 787, a race car from the 1960s
* 787 series, a train model o ...
,
797,
919
__NOTOC__
Year 919 ( CMXIX) was a common year starting on Friday (link will display the full calendar) of the Julian calendar.
Events
By Place
Byzantine Empire
* March 25 – Romanos Lekapenos, admiral (''droungarios'') of the ...
,
929
Year 929 ( CMXXIX) was a common year starting on Thursday (link will display the full calendar) of the Julian calendar.
Events
By place
Europe
* January 16 – Emir Abd-al-Rahman III of Córdoba proclaims himself caliph and create ...
, 10301, 10501, 10601, 11311, 11411, 12421, 12721, 12821, 13331, 13831, 13931, 14341, 14741 ()
Palindromic wing primes
Primes of the form
with
. This means all digits except the middle digit are equal.
101,
131,
151
Year 151 (CLI) was a common year starting on Thursday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Condianus and Valerius (or, less frequently, year 904 ''Ab urbe cond ...
,
181,
191,
313,
353,
373,
383,
727,
757
757 may refer to:
* Boeing 757: a narrow-body airliner
* AD 757: a year
* 757 BC: a year
* 757 (number): a number
* Area code 757
Image:Area code 757.png, The area colored red indicates the southeast corner of Virginia served by area code 757
po ...
,
787
787 may refer to:
* Boeing 787 Dreamliner, a jet airliner
* AD 787, a year
* 787 BC, a year
* Mazda 787/787B, a Japanese rotary-engine race car which won the 1991 Le Mans Race
* Porsche 787, a race car from the 1960s
* 787 series, a train model o ...
,
797,
919
__NOTOC__
Year 919 ( CMXIX) was a common year starting on Friday (link will display the full calendar) of the Julian calendar.
Events
By Place
Byzantine Empire
* March 25 – Romanos Lekapenos, admiral (''droungarios'') of the ...
,
929
Year 929 ( CMXXIX) was a common year starting on Thursday (link will display the full calendar) of the Julian calendar.
Events
By place
Europe
* January 16 – Emir Abd-al-Rahman III of Córdoba proclaims himself caliph and create ...
, 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 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 study the ...
s
Any permutation of the decimal digits is a prime.
2,
3,
5,
7,
11,
13,
17,
31,
37,
71,
73,
79,
97,
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 ...
,
131,
199,
311 311 may refer to:
* 311 (number), a natural number
* AD 311, a year of the Julian calendar, in the fourth century AD
* 311 BC, a year of the pre-Julian Roman calendar
* 311 (band), an American band
** ''311'' (album), band 311's self-titled album
...
,
337,
373,
733
__NOTOC__
Year 733 ( DCCXXXIII) was a common year starting on Thursday (link will display the full calendar) of the Julian calendar. The denomination 733 for this year has been used since the early medieval period, when the Anno Domini calend ...
,
919
__NOTOC__
Year 919 ( CMXIX) was a common year starting on Friday (link will display the full calendar) of the Julian calendar.
Events
By Place
Byzantine Empire
* March 25 – Romanos Lekapenos, admiral (''droungarios'') of the ...
,
991, 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
__NOTOC__
Year 277 ( CCLXXVII) was a common year starting on Monday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Probus and Paulinus (or, less frequently, year 1030 ''A ...
,
367
__NOTOC__
Year 367 ( CCCLXVII) was a common year starting on Monday of the Julian calendar. At the time, it was known as the Year of the Consulship of Lupicinus and Iovanus (or, less frequently, year 1120 ''Ab urbe condita''). The denomination ...
,
853
__NOTOC__
Year 853 ( DCCCLIII) was a common year starting on Sunday (link will display the full calendar) of the Julian calendar.
Events
By place Byzantine Empire
* May 22 – A Byzantine fleet (85 ships and 5,000 men) sacks and d ...
, 14197, 43721, 1442968193, 792606555396977, 187278659180417234321, 66241160488780141071579864797 ()
Pierpont primes
Of the form 23 + 1 for some integers ''u'',''v'' ≥ 0.
These are also Prime number#Classification of prime numbers, class 1- primes.
2,
3,
5,
7,
13,
17,
19,
37,
73,
97,
109,
163
Year 163 ( CLXIII) was a common year starting on Friday (link will display the full calendar) 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 con ...
,
193,
257,
433,
487,
577
__NOTOC__
Year 577 ( DLXXVII) was a common year starting on Friday (link will display the full calendar) of the Julian calendar. The denomination 577 for this year has been used since early medieval times, when the Anno Domini calendar era be ...
,
769, 1153 (number), 1153,
1297, 1459 (number), 1459, 2593 (number), 2593, 2917 (number), 2917, 3457 (number), 3457,
3889, 10369, 12289, 17497, 18433, 39367, 52489,
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). Therefore, a regular polygon with 65537 sides is constructible with compass and unmarked straightedge. Johann ...
, 139969, 147457 ()
Pillai primes
Primes ''p'' for which there exist ''n'' > 0 such that ''p'' divides ''n''! + 1 and ''n'' does not divide ''p'' − 1.
23,
29,
59,
61,
67,
71,
79,
83,
109,
137,
139 139 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 may refer to:
* 139 (number), an integer
* AD 139, a year of the Julian calendar
* 13 ...
,
149,
193,
227,
233,
239,
251,
257,
269,
271
__NOTOC__
Year 271 ( CCLXXI) was a common year starting on Sunday (link will display the full calendar) 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 '' ...
,
277
__NOTOC__
Year 277 ( CCLXXVII) was a common year starting on Monday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Probus and Paulinus (or, less frequently, year 1030 ''A ...
,
293,
307
__NOTOC__
Year 307 ( CCCVII) was a common year starting on Wednesday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Severus and Maximinus (or, less frequently, year 1060 ...
,
311 311 may refer to:
* 311 (number), a natural number
* AD 311, a year of the Julian calendar, in the fourth century AD
* 311 BC, a year of the pre-Julian Roman calendar
* 311 (band), an American band
** ''311'' (album), band 311's self-titled album
...
,
317,
359
__NOTOC__
Year 359 ( CCCLIX) was a common year starting on Friday (link will display the full calendar) 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 ' ...
,
379
Year 379 ( CCCLXXIX) was a common year starting on Tuesday (link will display the full calendar) 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 '' A ...
,
383,
389,
397,
401,
419,
431,
449,
461,
463
Year 463 ( CDLXIII) was a common year starting on Tuesday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Basilius and Vivianus (or, less frequently, year 1216 '' Ab urbe ...
,
467,
479,
499
__NOTOC__
Year 499 (Roman numerals, CDXCIX) was a common year starting on Friday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Iohannes without colleague (or, less frequ ...
()
Primes of the form ''n''4 + 1
Of the form ''n''
4 + 1.
2,
17,
257,
1297,
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). Therefore, a regular polygon with 65537 sides is constructible with compass and unmarked straightedge. Johann ...
, 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,
107,
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 ...
,
137, 1013 (number), 1013, 1237 (number), 1237, 1367 (number), 1367, 10079 ()
Primorial primes
Of the form ''p''# ± 1.
3,
5,
7,
29,
31,
211, 2309 (number), 2309,
2311, 30029, 200560490131, 304250263527209, 23768741896345550770650537601358309 (union of and
)
Proth number, Proth primes
Of the form ''k''×2 + 1, with odd ''k'' and ''k'' < 2.
3,
5,
13,
17,
41,
97,
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 ...
,
193,
241,
257,
353,
449,
577
__NOTOC__
Year 577 ( DLXXVII) was a common year starting on Friday (link will display the full calendar) of the Julian calendar. The denomination 577 for this year has been used since early medieval times, when the Anno Domini calendar era be ...
, 641 (number), 641,
673
__NOTOC__
Year 673 ( DCLXXIII) was a common year starting on Saturday (link will display the full calendar) of the Julian calendar. The denomination 673 for this year has been used since the early medieval period, when the Anno Domini calendar ...
,
769,
929
Year 929 ( CMXXIX) was a common year starting on Thursday (link will display the full calendar) of the Julian calendar.
Events
By place
Europe
* January 16 – Emir Abd-al-Rahman III of Córdoba proclaims himself caliph and create ...
, 1153 (number), 1153, 1217 (number), 1217, 1409 (number), 1409, 1601 (number), 1601, 2113 (number), 2113, 2689 (number), 2689, 2753 (number), 2753, 3137 (number), 3137, 3329 (number), 3329, 3457 (number), 3457, 4481, 4993, 6529, 7297, 7681, 7937, 9473, 9601, 9857 ()
Pythagorean primes
Of the form 4''n'' + 1.
5,
13,
17,
29,
37,
41,
53,
61,
73,
89,
97,
101,
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 ...
,
137,
149,
157,
173,
181,
193,
197
Year 197 ( CXCVII) was a common year starting on Saturday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Magius and Rufinus (or, less frequently, year 950 '' Ab urbe con ...
,
229,
233,
241,
257,
269,
277
__NOTOC__
Year 277 ( CCLXXVII) was a common year starting on Monday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Probus and Paulinus (or, less frequently, year 1030 ''A ...
,
281,
293,
313,
317,
337,
349,
353,
373,
389,
397,
401,
409,
421,
433,
449 ()
Prime quadruplet, Prime quadruplets
Where (''p'', ''p''+2, ''p''+6, ''p''+8) are all prime.
(
5,
7,
11,
13), (11, 13,
17,
19), (
101,
103,
107,
109), (
191,
193,
197
Year 197 ( CXCVII) was a common year starting on Saturday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Magius and Rufinus (or, less frequently, year 950 '' Ab urbe con ...
,
199), (821 (number), 821,
823
__NOTOC__
Year 823 ( DCCCXXIII) was a common year starting on Thursday (link will display the full calendar) of the Julian calendar.
Events
By place
Byzantine Empire
* Emperor Michael II defeats the rebel forces under Thomas the Sla ...
, 827 (number), 827, 829 (number), 829), (1481 (number), 1481, 1483 (number), 1483, 1487 (number), 1487, 1489 (number), 1489), (1871 (number), 1871, 1873 (number), 1873, 1877 (number), 1877, 1879 (number), 1879), (2081 (number), 2081, 2083 (number), 2083, 2087 (number), 2087, 2089 (number), 2089), (3251 (number), 3251, 3253 (number), 3253, 3257 (number), 3257, 3259 (number), 3259), (3461 (number), 3461, 3463 (number), 3463, 3467 (number), 3467,
3469), (5651, 5653, 5657, 5659), (9431, 9433, 9437, 9439) (, , , )
Quartan primes
Of the form ''x'' + ''y'', where ''x'',''y'' > 0.
2,
17,
97,
257,
337, 641 (number), 641,
881 ()
Ramanujan primes
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,
107,
127,
149,
151
Year 151 (CLI) was a common year starting on Thursday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Condianus and Valerius (or, less frequently, year 904 ''Ab urbe cond ...
,
167,
179,
181,
227,
229,
233,
239,
241,
263
__NOTOC__
Year 263 ( CCLXIII) was a common year starting on Thursday (link will display the full calendar) 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 ' ...
,
269,
281,
307
__NOTOC__
Year 307 ( CCCVII) was a common year starting on Wednesday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Severus and Maximinus (or, less frequently, year 1060 ...
,
311 311 may refer to:
* 311 (number), a natural number
* AD 311, a year of the Julian calendar, in the fourth century AD
* 311 BC, a year of the pre-Julian Roman calendar
* 311 (band), an American band
** ''311'' (album), band 311's self-titled album
...
,
347
Year 347 (Roman numerals, CCCXLVII) was a common year starting on Thursday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Rufinus and Eusebius (or, less frequently, year 1 ...
,
349,
367
__NOTOC__
Year 367 ( CCCLXVII) was a common year starting on Monday of the Julian calendar. At the time, it was known as the Year of the Consulship of Lupicinus and Iovanus (or, less frequently, year 1120 ''Ab urbe condita''). The denomination ...
,
373,
401,
409,
419,
431,
433,
439,
461,
487,
491 ()
Regular primes
Primes ''p'' that do not divide the
class number of the ''p''-th
cyclotomic field
In number theory, a cyclotomic field is a number field obtained by adjoining a complex root of unity to , the field of rational numbers.
Cyclotomic fields played a crucial role in the development of modern algebra and number theory because of ...
.
3,
5,
7,
11,
13,
17,
19,
23,
29,
31,
41,
43,
47,
53,
61,
71,
73,
79,
83,
89,
97,
107,
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 ...
,
127,
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 may refer to:
* 139 (number), an integer
* AD 139, a year of the Julian calendar
* 13 ...
,
151
Year 151 (CLI) was a common year starting on Thursday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Condianus and Valerius (or, less frequently, year 904 ''Ab urbe cond ...
,
163
Year 163 ( CLXIII) was a common year starting on Friday (link will display the full calendar) 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 con ...
,
167,
173,
179,
181,
191,
193,
197
Year 197 ( CXCVII) was a common year starting on Saturday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Magius and Rufinus (or, less frequently, year 950 '' Ab urbe con ...
,
199,
211,
223,
227,
229,
239,
241,
251,
269,
277
__NOTOC__
Year 277 ( CCLXXVII) was a common year starting on Monday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Probus and Paulinus (or, less frequently, year 1030 ''A ...
,
281 ()
Repunit#Decimal repunit primes, Repunit primes
Primes containing only the decimal digit 1.
11, 1111111111111111111 (19 digits), 11111111111111111111111 (23 digits) ()
The next have 317, 1031, 49081, 86453, 109297, 270343 digits ()
Dirichlet's theorem on arithmetic progressions, Residue classes of primes
Of the form ''an'' + ''d'' for fixed integers ''a'' and ''d''. Also called primes congruent to ''d'' Modular arithmetic, modulo ''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,
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 ...
,
137 ()
4''n''+3: 3, 7, 11, 19, 23, 31, 43, 47,
59,
67,
71,
79,
83,
103,
107 ()
6''n''+1: 7, 13, 19, 31, 37, 43, 61, 67, 73, 79, 97, 103, 109,
127,
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 may refer to:
* 139 (number), an integer
* AD 139, a year of the Julian calendar
* 13 ...
()
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,
193,
233,
241,
257,
281,
313,
337,
353 ()
8''n''+3: 3, 11, 19, 43, 59, 67, 83, 107,
131, 139,
163
Year 163 ( CLXIII) was a common year starting on Friday (link will display the full calendar) 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 con ...
,
179,
211,
227,
251 ()
8''n''+5: 5, 13, 29, 37, 53, 61, 101, 109,
149,
157,
173,
181,
197
Year 197 ( CXCVII) was a common year starting on Saturday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Magius and Rufinus (or, less frequently, year 950 '' Ab urbe con ...
,
229,
269 ()
8''n''+7: 7, 23, 31, 47, 71, 79, 103, 127,
151
Year 151 (CLI) was a common year starting on Thursday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Condianus and Valerius (or, less frequently, year 904 ''Ab urbe cond ...
,
167,
191,
199,
223,
239,
263
__NOTOC__
Year 263 ( CCLXIII) was a common year starting on Thursday (link will display the full calendar) 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 ' ...
()
10''n''+1: 11, 31, 41, 61, 71, 101, 131, 151, 181, 191, 211, 241, 251,
271
__NOTOC__
Year 271 ( CCLXXI) was a common year starting on Sunday (link will display the full calendar) 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 '' ...
, 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
__NOTOC__
Year 277 ( CCLXXVII) was a common year starting on Monday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Probus and Paulinus (or, less frequently, year 1030 ''A ...
()
10''n''+9: 19, 29, 59, 79, 89, 109, 139, 149, 179, 199, 229, 239, 269,
349,
359
__NOTOC__
Year 359 ( CCCLIX) was a common year starting on Friday (link will display the full calendar) 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 ' ...
()
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 primes
Where ''p'' and (''p''−1) / 2 are both prime.
5,
7,
11,
23,
47,
59,
83,
107,
167,
179,
227,
263
__NOTOC__
Year 263 ( CCLXIII) was a common year starting on Thursday (link will display the full calendar) 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 ' ...
,
347
Year 347 (Roman numerals, CCCXLVII) was a common year starting on Thursday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Rufinus and Eusebius (or, less frequently, year 1 ...
,
359
__NOTOC__
Year 359 ( CCCLIX) was a common year starting on Friday (link will display the full calendar) 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 ' ...
,
383,
467,
479,
503,
563,
587
__NOTOC__
Year 587 ( DLXXXVII) was a common year starting on Wednesday (link will display the full calendar) of the Julian calendar. The denomination 587 for this year has been used since the early medieval period, when the Anno Domini calenda ...
,
719,
839
__NOTOC__
Year 839 ( DCCCXXXIX) was a common year starting on Wednesday (link will display the full calendar) of the Julian calendar.
Events
By place Europe
* Prince Sicard of Benevento is assassinated by a conspiracy among the nobil ...
,
863
__NOTOC__
Year 863 ( DCCCLXIII) was a common year starting on Friday (link will display the full calendar) of the Julian calendar.
Events
By place Byzantine Empire
* September 3 – Battle of Lalakaon: A Byzantine army confronts ...
,
887,
983
Year 983 ( CMLXXXIII) was a common year starting on Monday (link will display the full calendar) of the Julian calendar.
Events
By place
Europe
* Summer – Diet of Verona: Emperor Otto II (the Red) declares war against the Byza ...
, 1019 (number), 1019, 1187 (number), 1187, 1283 (number), 1283, 1307 (number), 1307, 1319 (number), 1319, 1367 (number), 1367, 1439 (number), 1439, 1487 (number), 1487, 1523 (number), 1523, 1619 (number), 1619, 1823 (number), 1823, 1907 (number), 1907 ()
Self number, Self primes in base 10
Primes that cannot be generated by any integer added to the sum of its decimal digits.
3,
5,
7,
31,
53,
97,
211,
233,
277
__NOTOC__
Year 277 ( CCLXXVII) was a common year starting on Monday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Probus and Paulinus (or, less frequently, year 1030 ''A ...
,
367
__NOTOC__
Year 367 ( CCCLXVII) was a common year starting on Monday of the Julian calendar. At the time, it was known as the Year of the Consulship of Lupicinus and Iovanus (or, less frequently, year 1120 ''Ab urbe condita''). The denomination ...
,
389,
457,
479,
547, 569 (number), 569,
613
__NOTOC__
Year 613 ( DCXIII) was a common year starting on Monday (link will display the full calendar) of the Julian calendar. The denomination 613 for this year has been used since the early medieval period, when the Anno Domini calendar era ...
,
659
__NOTOC__
Year 659 ( DCLIX) was a common year starting on Tuesday (link will display the full calendar) of the Julian calendar. The denomination 659 for this year has been used since the early medieval period, when the Anno Domini calendar era ...
,
727,
839
__NOTOC__
Year 839 ( DCCCXXXIX) was a common year starting on Wednesday (link will display the full calendar) of the Julian calendar.
Events
By place Europe
* Prince Sicard of Benevento is assassinated by a conspiracy among the nobil ...
,
883
__NOTOC__
Year 883 ( DCCCLXXXIII) was a common year starting on Tuesday (link will display the full calendar) of the Julian calendar.
Events
By place Europe
* Spring – Viking raiders ravage Flanders, and sack the abbey at Saint- ...
,
929
Year 929 ( CMXXIX) was a common year starting on Thursday (link will display the full calendar) of the Julian calendar.
Events
By place
Europe
* January 16 – Emir Abd-al-Rahman III of Córdoba proclaims himself caliph and create ...
, 1021 (number), 1021, 1087 (number), 1087, 1109 (number), 1109, 1223 (number), 1223, 1289 (number), 1289, 1447 (number), 1447, 1559 (number), 1559, 1627 (number), 1627, 1693 (number), 1693, 1783 (number), 1783, 1873 (number), 1873 ()
Sexy primes
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,
107), (103,
109), (107,
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 ...
), (
131,
137), (
151
Year 151 (CLI) was a common year starting on Thursday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Condianus and Valerius (or, less frequently, year 904 ''Ab urbe cond ...
,
157), (157,
163
Year 163 ( CLXIII) was a common year starting on Friday (link will display the full calendar) 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 con ...
), (
167,
173), (173,
179), (
191,
197
Year 197 ( CXCVII) was a common year starting on Saturday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Magius and Rufinus (or, less frequently, year 950 '' Ab urbe con ...
), (
193,
199) (, )
Smarandache–Wellin number, Smarandache–Wellin primes
Primes that are the concatenation of the first ''n'' primes written in decimal.
2,
23, 2357 (number), 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 ± 2 ± 1, where 0 < ''b'' < ''a''.
3,
5,
7,
11,
13 ()
Sophie Germain primes
Where ''p'' and 2''p'' + 1 are both prime. A Sophie Germain prime has a corresponding #Safe primes, safe prime.
2,
3,
5,
11,
23,
29,
41,
53,
83,
89,
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 ...
,
131,
173,
179,
191,
233,
239,
251,
281,
293,
359
__NOTOC__
Year 359 ( CCCLIX) was a common year starting on Friday (link will display the full calendar) 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 ' ...
,
419,
431,
443
__NOTOC__
Year 443 ( CDXLIII) was a common year starting on Friday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Maximus and Paterius (or, less frequently, year 1196 ' ...
,
491,
509,
593, 641 (number), 641,
653
__NOTOC__
Year 653 ( DCLIII) was a common year starting on Tuesday (link will display the full calendar) of the Julian calendar. The denomination 653 for this year has been used since the early medieval period, when the Anno Domini calendar era ...
,
659
__NOTOC__
Year 659 ( DCLIX) was a common year starting on Tuesday (link will display the full calendar) of the Julian calendar. The denomination 659 for this year has been used since the early medieval period, when the Anno Domini calendar era ...
,
683
__NOTOC__
Year 683 (Roman numerals, DCLXXXIII) was a common year starting on Thursday (link will display the full calendar) of the Julian calendar. The denomination 683 for this year has been used since the early medieval period, when the Anno ...
,
719,
743
__NOTOC__
Year 743 ( DCCXLIII) was a common year starting on Tuesday (link will display the full calendar) of the Julian calendar. The denomination 743 for this year has been used since the early medieval period, when the Anno Domini calendar ...
,
761
__NOTOC__
Year 761 ( DCCLXI) was a common year starting on Thursday (link will display the full calendar) of the Julian calendar. The denomination 761 for this year has been used since the early medieval period, when the Anno Domini calendar er ...
, 809 (number), 809,
911
911 or 9/11 may refer to:
Dates
* AD 911
* 911 BC
* September 11
** 9/11, the September 11 attacks of 2001
** 11 de Septiembre, Chilean coup d'état in 1973 that outed the democratically elected Salvador Allende
* November 9
Numbers
* 91 ...
,
953
Year 953 ( CMLIII) was a common year starting on Saturday (link will display the full calendar) of the Julian calendar.
Events
By place
Byzantine Empire
* Battle of Marash: Emir Sayf al-Dawla marches north into the Byzantine Empire an ...
()
Stern primes
Primes that are not the sum of a smaller prime and twice the square of a nonzero integer.
2,
3,
17,
137,
227,
977, 1187 (number), 1187, 1493 (number), 1493 ()
, these are the only known Stern primes, and possibly the only existing.
Super-primes
Primes with a prime index in the sequence of prime numbers (the 2nd, 3rd, 5th, ... prime).
3,
5,
11,
17,
31,
41,
59,
67,
83,
109,
127,
157,
179,
191,
211,
241,
277
__NOTOC__
Year 277 ( CCLXXVII) was a common year starting on Monday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Probus and Paulinus (or, less frequently, year 1030 ''A ...
,
283
Year 283 ( CCLXXXIII) was a common year starting on Monday (link will display the full calendar) 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 cond ...
,
331
__NOTOC__
Year 331 (Roman numerals, CCCXXXI) was a common year starting on Friday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Bassus and Ablabius (or, less frequent ...
,
353,
367
__NOTOC__
Year 367 ( CCCLXVII) was a common year starting on Monday of the Julian calendar. At the time, it was known as the Year of the Consulship of Lupicinus and Iovanus (or, less frequently, year 1120 ''Ab urbe condita''). The denomination ...
,
401,
431,
461,
509,
547,
563,
587
__NOTOC__
Year 587 ( DLXXXVII) was a common year starting on Wednesday (link will display the full calendar) of the Julian calendar. The denomination 587 for this year has been used since the early medieval period, when the Anno Domini calenda ...
, 599 (number), 599,
617
__NOTOC__
Year 617 ( DCXVII) was a common year starting on Saturday (link will display the full calendar) of the Julian calendar. The denomination 617 for this year has been used since the early medieval period, when the Anno Domini calendar e ...
,
709,
739,
773
__NOTOC__
Year 773 ( DCCLXXIII) was a common year starting on Friday (link will display the full calendar) of the Julian calendar. The denomination 773 for this year has been used since the early medieval period, when the Anno Domini calendar e ...
,
797, 859 (number), 859,
877,
919
__NOTOC__
Year 919 ( CMXIX) was a common year starting on Friday (link will display the full calendar) of the Julian calendar.
Events
By Place
Byzantine Empire
* March 25 – Romanos Lekapenos, admiral (''droungarios'') of the ...
,
967,
991 ()
Supersingular prime (moonshine theory), Supersingular primes
There are exactly fifteen supersingular primes:
2,
3,
5,
7,
11,
13,
17,
19,
23,
29,
31,
41,
47,
59,
71 ()
Thabit primes
Of the form 3×2 − 1.
2,
5,
11,
23,
47,
191,
383, 6143, 786431, 51539607551, 824633720831, 26388279066623, 108086391056891903, 55340232221128654847, 226673591177742970257407 ()
The primes of the form 3×2 + 1 are related.
7,
13,
97,
193,
769, 12289, 786433, 3221225473, 206158430209, 6597069766657 ()
Prime triplet, Prime triplets
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,
107), (103, 107,
109), (107, 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 ...
), (
191,
193,
197
Year 197 ( CXCVII) was a common year starting on Saturday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Magius and Rufinus (or, less frequently, year 950 '' Ab urbe con ...
), (193, 197,
199), (
223,
227,
229), (227, 229,
233), (
277
__NOTOC__
Year 277 ( CCLXXVII) was a common year starting on Monday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Probus and Paulinus (or, less frequently, year 1030 ''A ...
,
281,
283
Year 283 ( CCLXXXIII) was a common year starting on Monday (link will display the full calendar) 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 cond ...
), (
307
__NOTOC__
Year 307 ( CCCVII) was a common year starting on Wednesday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Severus and Maximinus (or, less frequently, year 1060 ...
,
311 311 may refer to:
* 311 (number), a natural number
* AD 311, a year of the Julian calendar, in the fourth century AD
* 311 BC, a year of the pre-Julian Roman calendar
* 311 (band), an American band
** ''311'' (album), band 311's self-titled album
...
,
313), (311, 313,
317), (
347
Year 347 (Roman numerals, CCCXLVII) was a common year starting on Thursday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Rufinus and Eusebius (or, less frequently, year 1 ...
,
349,
353) (, , )
Truncatable prime
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,
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 ...
,
137,
167,
173,
197
Year 197 ( CXCVII) was a common year starting on Saturday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Magius and Rufinus (or, less frequently, year 950 '' Ab urbe con ...
,
223,
283
Year 283 ( CCLXXXIII) was a common year starting on Monday (link will display the full calendar) 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 cond ...
,
313,
317,
337,
347
Year 347 (Roman numerals, CCCXLVII) was a common year starting on Thursday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Rufinus and Eusebius (or, less frequently, year 1 ...
,
353,
367
__NOTOC__
Year 367 ( CCCLXVII) was a common year starting on Monday of the Julian calendar. At the time, it was known as the Year of the Consulship of Lupicinus and Iovanus (or, less frequently, year 1120 ''Ab urbe condita''). The denomination ...
,
373,
383,
397,
443
__NOTOC__
Year 443 ( CDXLIII) was a common year starting on Friday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Maximus and Paterius (or, less frequently, year 1196 ' ...
,
467,
523,
547,
613
__NOTOC__
Year 613 ( DCXIII) was a common year starting on Monday (link will display the full calendar) of the Julian calendar. The denomination 613 for this year has been used since the early medieval period, when the Anno Domini calendar era ...
,
617
__NOTOC__
Year 617 ( DCXVII) was a common year starting on Saturday (link will display the full calendar) of the Julian calendar. The denomination 617 for this year has been used since the early medieval period, when the Anno Domini calendar e ...
,
643
__NOTOC__
Year 643 (Roman numerals, DCXLIII) was a common year starting on Wednesday (link will display the full calendar) of the Julian calendar. The denomination 643 for this year has been used since the early medieval period, when the Anno D ...
,
647
__NOTOC__
Year 647 ( DCXLVII) was a common year starting on Monday (link will display the full calendar) of the Julian calendar. The denomination 647 for this year has been used since the early medieval period, when the Anno Domini calendar era ...
,
653
__NOTOC__
Year 653 ( DCLIII) was a common year starting on Tuesday (link will display the full calendar) of the Julian calendar. The denomination 653 for this year has been used since the early medieval period, when the Anno Domini calendar era ...
,
673
__NOTOC__
Year 673 ( DCLXXIII) was a common year starting on Saturday (link will display the full calendar) of the Julian calendar. The denomination 673 for this year has been used since the early medieval period, when the Anno Domini calendar ...
,
683
__NOTOC__
Year 683 (Roman numerals, DCLXXXIII) was a common year starting on Thursday (link will display the full calendar) of the Julian calendar. The denomination 683 for this year has been used since the early medieval period, when the Anno ...
()
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,
233,
239,
293,
311 311 may refer to:
* 311 (number), a natural number
* AD 311, a year of the Julian calendar, in the fourth century AD
* 311 BC, a year of the pre-Julian Roman calendar
* 311 (band), an American band
** ''311'' (album), band 311's self-titled album
...
,
313,
317,
373,
379
Year 379 ( CCCLXXIX) was a common year starting on Tuesday (link will display the full calendar) 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 '' A ...
,
593, 599 (number), 599,
719,
733
__NOTOC__
Year 733 ( DCCXXXIII) was a common year starting on Thursday (link will display the full calendar) of the Julian calendar. The denomination 733 for this year has been used since the early medieval period, when the Anno Domini calend ...
,
739,
797, 2333 (number), 2333, 2339 (number), 2339, 2393 (number), 2393, 2399 (number), 2399, 2939 (number), 2939,
3119, 3137 (number), 3137, 3733 (number), 3733, 3739 (number), 3739, 3793 (number), 3793, 3797 (number), 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,
373,
797, 3137 (number), 3137, 3797 (number), 3797, 739397 ()
Twin primes
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), (
107,
109), (
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 may refer to:
* 139 (number), an integer
* AD 139, a year of the Julian calendar
* 13 ...
), (
149,
151
Year 151 (CLI) was a common year starting on Thursday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Condianus and Valerius (or, less frequently, year 904 ''Ab urbe cond ...
), (
179,
181), (
191,
193), (
197
Year 197 ( CXCVII) was a common year starting on Saturday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Magius and Rufinus (or, less frequently, year 950 '' Ab urbe con ...
,
199), (
227,
229), (
239,
241), (
269,
271
__NOTOC__
Year 271 ( CCLXXI) was a common year starting on Sunday (link will display the full calendar) 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 '' ...
), (
281,
283
Year 283 ( CCLXXXIII) was a common year starting on Monday (link will display the full calendar) 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 cond ...
), (
311 311 may refer to:
* 311 (number), a natural number
* AD 311, a year of the Julian calendar, in the fourth century AD
* 311 BC, a year of the pre-Julian Roman calendar
* 311 (band), an American band
** ''311'' (album), band 311's self-titled album
...
,
313), (
347
Year 347 (Roman numerals, CCCXLVII) was a common year starting on Thursday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Rufinus and Eusebius (or, less frequently, year 1 ...
,
349), (
419,
421), (
431,
433), (
461,
463
Year 463 ( CDLXIII) was a common year starting on Tuesday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Basilius and Vivianus (or, less frequently, year 1216 '' Ab urbe ...
) (, )
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,
683
__NOTOC__
Year 683 (Roman numerals, DCLXXXIII) was a common year starting on Thursday (link will display the full calendar) of the Julian calendar. The denomination 683 for this year has been used since the early medieval period, when the Anno ...
, 2731 (number), 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,
199,
313,
347
Year 347 (Roman numerals, CCCXLVII) was a common year starting on Thursday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Rufinus and Eusebius (or, less frequently, year 1 ...
,
701
__NOTOC__
Year 701 ( DCCI) was a common year starting on Saturday (link will display the full calendar) of the Julian calendar. The denomination 701 for this year has been used since the early medieval period, when the Anno Domini calendar era ...
, 1709 (number), 1709, 2617 (number), 2617, 3539 (number), 3539, 5807 (number), 5807, 10501 (number), 10501, 10691, 11279, 12391, 14479, 42737, 83339, 95369, 117239, 127031, 138937, 141079, 267017, 269987, 374321 ()
Wall–Sun–Sun primes
A prime ''p'' > 5, if ''p'' divides the Fibonacci number
, where the Legendre symbol
is defined as
:
, no Wall-Sun-Sun primes are known.
Weakly prime numbers
Primes that having any one of their (base 10) digits changed to any other value will always result in a composite number.
294001, 505447, 584141, 604171, 971767, 1062599, 1282529, 1524181, 2017963, 2474431, 2690201, 3085553, 3326489, 4393139 ()
Wieferich primes
Primes ''p'' such that for fixed integer ''a'' > 1.
2
''p'' − 1 ≡ 1 (mod ''p''
2):
1093, 3511 (number), 3511 ()
3
''p'' − 1 ≡ 1 (mod ''p''
2):
11, 1006003 ()
4
''p'' − 1 ≡ 1 (mod ''p''
2):
1093, 3511 (number), 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 (number), 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 (number), 2693, 123653 ()
13
''p'' − 1 ≡ 1 (mod ''p''
2):
2,
863
__NOTOC__
Year 863 ( DCCCLXIII) was a common year starting on Friday (link will display the full calendar) of the Julian calendar.
Events
By place Byzantine Empire
* September 3 – Battle of Lalakaon: A Byzantine army confronts ...
, 1747591 ()
14
''p'' − 1 ≡ 1 (mod ''p''
2):
29,
353, 7596952219 ()
15
''p'' − 1 ≡ 1 (mod ''p''
2): 29131, 119327070011 ()
16
''p'' − 1 ≡ 1 (mod ''p''
2):
1093, 3511 (number), 3511
17
''p'' − 1 ≡ 1 (mod ''p''
2):
2,
3, 46021, 48947 ()
18
''p'' − 1 ≡ 1 (mod ''p''
2):
5,
7,
37,
331
__NOTOC__
Year 331 (Roman numerals, CCCXXXI) was a common year starting on Friday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Bassus and Ablabius (or, less frequent ...
, 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,
673
__NOTOC__
Year 673 ( DCLXXIII) was a common year starting on Saturday (link will display the full calendar) of the Julian calendar. The denomination 673 for this year has been used since the early medieval period, when the Anno Domini calendar ...
, 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 primes
Primes ''p'' for which ''p'' divides (''p''−1)! + 1.
5,
13,
563 ()
, these are the only known Wilson primes.
Wolstenholme primes
Primes ''p'' for which the binomial coefficient
16843 (number), 16843, 2124679 ()
, these are the only known Wolstenholme primes.
Woodall number, Woodall primes
Of the form ''n''×2 − 1.
7,
23,
383, 32212254719, 2833419889721787128217599, 195845982777569926302400511, 4776913109852041418248056622882488319 ()
See also
*
*
*
*
*
*
*
*
*
*
References
External links
Lists of Primesat the Prime Pages.
The Nth Prime PageNth prime through n=10^12, pi(x) through x=3*10^13, Random prime in same range.
Prime Numbers ListFull list for prime numbers below 10,000,000,000, partial list for up to 400 digits.
(primes less than 2,000,000,000)
*
Selected prime related sequencesin 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, *
Classes of prime numbers, *
Mathematics-related lists, Prime numbers
Number-related lists, Prime