Twin primes are pairs of primes of the form (
,
). The term "twin prime"
was coined by Paul Stäckel (1862-1919; Tietze 1965, p. 19). The first few
twin primes are 
for 
, 6, 12, 18, 30, 42, 60, 72, 102, 108,
138, 150, 180, 192, 198, 228, 240, 270, 282, ... (OEIS A014574).
Explicitly, these are (3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), ...
(OEIS A001359 and A006512).
All twin primes except (3, 5) are of the form 
.
It is conjectured that there are an infinite number of twin primes (this is one form of the twin prime conjecture), but proving this remains one of the most elusive open problems in number theory. An important result for twin primes is Brun's theorem, which states that the series of reciprocals of the odd twin primes,
 |
(1)
|
is a convergent series whose sum is Brun's constant, which expresses the scarcity
of twin primes, even if there are infinitely many of them (Ribenboim 1996, p. 201).
By contrast, the series of all prime
number reciprocals is a divergent
series tending to infinity, as follows from the Mertens
second theorem by letting 
.
The proved theorem on bounded gaps between primes gives infinitely many pairs of consecutive primes separated by at most 246 (Polymath 2014). It does not prove that infinitely many pairs are separated by exactly 2, so the twin prime conjecture remains open.
The following table gives the first few 
for the twin primes (
, 
),
cousin primes (
, 
),
sexy primes (
, 
),
etc.
| pair | OEIS | first member |
( ,  ) | A001359 | 3, 5, 11, 17, 29, 41, 59, 71, ... |
( ,  ) | A023200 | 3, 7, 13, 19, 37, 43, 67, 79, ... |
( ,  ) | A023201 | 5, 7, 11, 13, 17, 23, 31, 37, ... |
( ,  ) | A023202 | 3, 5, 11, 23, 29, 53, 59, 71, ... |
( ,  ) | A023203 | 3, 7, 13, 19, 31, 37, 43, 61, ... |
( ,  ) | A046133 | 5, 7, 11, 17, 19, 29, 31, 41, ... |
Let 
be the number of twin primes 
and 
such that 
. It is not known if there are infinitely many such primes (Wells 1986, p. 41; Shanks 1993), but it
seems almost certain to be true (Hardy and Wright 1979, p. 5).
Chen's theorem shows that there are infinitely many primes 
such that 
is either a prime or a semiprime (Chen 1973). Brun proved that there exists a
computable integer 
such that if 
, then
 |
(2)
|
(Ribenboim 1996, p. 261). It has been shown that
![pi_2(x)<=cproduct_(p>2)[1-1/((p-1)^2)]x/((lnx)^2)[1+O((lnlnx)/(lnx))],](/images/equations/TwinPrimes/NumberedEquation3.svg) |
(3)
|
written more concisely as
![pi_2(x)<=cPi_2x/((lnx)^2)[1+O((lnlnx)/(lnx))],](/images/equations/TwinPrimes/NumberedEquation4.svg) |
(4)
|
where 
is known as the twin primes constant and 
is another constant.
The constant 
has been reduced to 
(Fouvry and Iwaniec 1983), 
(Fouvry 1984), 7 (Bombieri et al. 1986),
6.9075 (Fouvry and Grupp 1986), 6.8354 (Wu 1990), and 6.8325 (Haugland 1999). The
latter calculation involved evaluation of 7-fold integrals
and fitting of three different parameters.
Hardy and Littlewood (1923) conjectured that 
(Ribenboim 1996, p. 262), and that 
satisfies the asymptotic
relation
 |
(5)
|
This result is sometimes called the strong twin prime conjecture and is a special case of the k-tuple conjecture. The prime number theorem already implies that
 |
(6)
|
where 
is the 
th
prime and 
is the prime
difference function.
Using the prime number theorem, the strong twin prime conjecture is equivalently expressed as
![pi_2(x)∼2Pi_2([pi(x)]^2)/x,](/images/equations/TwinPrimes/NumberedEquation7.svg) |
(7)
|
where 
is the prime counting function.
Wolf (2008) studied the analog of the Skewes number for twin primes, comparing his twin-prime count with
 |
(8)
|
With this lower integration limit and normalization, he reported 90,355 sign changes through 
(Wolf 2008, Table 1). This is an attributed result of computation
using his counting convention. Wolf (2008) proposed
 |
(9)
|
where 
counts these sign changes through 
. This growth approximation is conjectural. If the number of
sign changes were unbounded, there would be infinitely many twin primes.
The largest known twin-prime pair as of Aug. 2026 is
 |
(10)
|
whose members each have 388,342 decimal digits. Tom Greer found the pair as part of PrimeGrid in Sep. 2016 (PrimeGrid 2016, PrimePages 2026).
In 1995, Nicely discovered a flaw in the Intel® Pentium™ microprocessor by computing the reciprocals of 
and 
, which should have been accurate to 19 decimal
places but were incorrect from the tenth decimal place on (Cipra 1995, Nicely 1995,
Cipra 1996).
If 
, the integers 
and 
form a pair of twin primes iff
![4[(n-1)!+1]+n=0 (mod n(n+2)).](/images/equations/TwinPrimes/NumberedEquation11.svg) |
(11)
|
A positive integer 
is the product 
of a pair of twin primes 
iff
 |
(12)
|
where 
is the totient function and 
is the divisor function
(Ribenboim 1996, p. 259). For a positive integer 
, S. M. Ruiz's criterion (Ruiz
and Ariff 2004, pp. 357, 359) states that 
are twin primes iff
 |
(13)
|
for a real number 
, where 
is the floor function.
The values of 
were found by Brent (1976) up to 
. T. Nicely calculated them up to 
in his calculation of Brun's
constant. Fry et al. (2001) and Sebah (2002) independently obtained 
using distributed computation.
Later computations extended the values through 
. The following table gives known
values of 
(OEIS A007508; Ribenboim 1996, p. 263;
Nicely 1999; Sebah 2002).
 |  |
 | 35 |
 | 205 |
 | 1224 |
 |  |
 |  |
 |  |
 |  |
 |  |
 |  |
 |  |
 |  |
 |  |
 |  |
 |  |
 |  |
 |  |
 |  |
It is conjectured that every sufficiently large positive even number is a sum of two primes that each belong to a twin-prime pair, with equal summands allowed. The first few exceptions are 2, 4, 94, 96, 98, 400, 402, 404, 514, 516, 518, ... (OEIS A007534; Wells 1986, p. 132).
." Math. Comput. 30, 379, 1976.Chen,
J. R. "On the Representation of a Large Even Integer as the Sum of a Prime
and the Product of at Most Two Primes. I." Sci. Sinica 16, 157-176,
1973.Cipra, B. "How Number Theory Got the Best of the Pentium Chip."
Science 267, 175, 1995.Cipra, B. "Divide and Conquer."
of the Twin Primes and Brun's Constant." Virginia
J. Sci. 46, 195-204, 1995. 
and 
." J. Int. Seq. 6, 1-6, 2003.Parady,
B. K.; Smith, J. F.; and Zarantonello, S. E. "Largest Known Twin
Primes." Math. Comput. 55, 381-382, 1990.Polymath,
D. H. J. "Variants of the Selberg Sieve, and Bounded Intervals Containing
Many Primes." Res. Math. Sci. 1, Article 12, 2014.