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 number obtained by adding the reciprocals of the odd twin primes,
 |
(1)
|
converges to a definite number ("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 reciprocals
diverges to infinity, as follows from the Mertens
second theorem by letting 
.
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 an infinite number of
such primes (Wells 1986, p. 41; Shanks 1993),
but it seems almost certain to be true (Hardy and Wright 1979, p. 5).
J. R. Chen has shown there exists an infinite number of primes 
such that 
has at most two factors (Le Lionnais 1979, p. 49).
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 
is asymptotically equal to
 |
(5)
|
This result is sometimes called the strong twin prime conjecture and is a special case of the k-tuple conjecture. A necessary (but not sufficient) condition for the twin prime conjecture to hold is that the prime gaps constant, defined by
 |
(6)
|
where 
is the 
th
prime and 
is the prime difference function, satisfies
.
Wolf notes that the formula
![pi_2(x)∼2Pi_2([pi(x)]^2)/x,](/images/equations/TwinPrimes/NumberedEquation7.svg) |
(7)
|
(which has asymptotic growth 
) agrees with numerical data much better than
does 
, although not as well as
.
Extending the search done by Brent in 1974 or 1975, Wolf has searched for the analog of the Skewes number for twins, i.e., an 
such that 
changes sign. Wolf checked numbers up to
and found more than 
sign changes. From this data, Wolf conjectured that the
number of sign changes 
for 
of 
is given by
 |
(8)
|
Proof of this conjecture would also imply the existence an infinite number of twin primes.
The largest known twin primes as of Sep. 2016 correspond to
 |
(9)
|
each having 
decimal digits and found by PrimeGrid on Dec. 25, 2011 (http://primes.utm.edu/top20/page.php?id=1#records).
In 1995, Nicely discovered a flaw in the Intel® PentiumTM 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, 1996; Nicely 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/NumberedEquation10.svg) |
(10)
|
where 
is a pair of twin primes iff
 |
(11)
|
(Ribenboim 1996, p. 259). S. M. Ruiz has found the unexpected result that 
are twin primes iff
 |
(12)
|
for 
, 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.
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 even number is a sum of a pair of twin primes except a finite number of exceptions whose first few terms are 2, 4, 94, 96, 98, 400, 402, 404, 514, 516, 518, ... (OEIS A007534; Wells 1986, p. 132).
." Math. Comput. 30, 379, 1976.Caldwell,
C. 
of the Twin Primes and Brun's Constant." Virginia
J. Sci. 46, 195-204, 1996. 
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.Ribenboim,
P. "Twin Primes." §4.3 in