There are two related conjectures, each called the twin prime conjecture. The first version states that there are an infinite number of pairs of twin primes (Guy 1994, p. 19). It is not known if there are an infinite number of such primes (Wells 1986, p. 41; Shanks 1993, p. 30), but it seems almost certain to be true. While Hardy and Wright (1979, p. 5) note that "the evidence, when examined in detail, appears to justify the conjecture," and Shanks (1993, p. 219) states even more strongly, "the evidence is overwhelming," Hardy and Wright also note that the proof or disproof of conjectures of this type "is at present beyond the resources of mathematics."
Arenstorf (2004) published a purported proof of the conjecture (Weisstein 2004). Unfortunately, a serious error was found in the proof. As a result, the paper was retracted and the twin prime conjecture remains fully open.
The conjecture that there are infinitely many Sophie Germain primes, i.e., primes 
such that 
is also prime, is very closely related (Shanks 1993, p. 30).
A second twin prime conjecture states that adding a correction proportional to 
to a computation of Brun's
constant ending with 
will give an estimate with error less than 
. An extended form
of this conjecture, sometimes called the strong twin prime conjecture (Shanks 1993,
p. 30) or first Hardy-Littlewood
conjecture, states that the number 
of twin primes 
less than or equal to 
is asymptotically equal to
 |
where 
is the so-called twin primes constant (Hardy
and Littlewood 1923). The value of 
is plotted above for 
, with 
indicated in blue and taking 
.
This conjecture is a special case of the more general k-tuple conjecture (also known as the first Hardy-Littlewood
conjecture), which corresponds to the set 
.
