The twin prime conjecture refers to two related conjectures. The first version states that there are infinitely many pairs of twin primes
(Guy 1994, p. 19). It is not known if there are infinitely many 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 As a result, the paper was retracted.
The theorem on bounded gaps between primes does not resolve the conjecture. Zhang (2014) proved that the limit
inferior
is finite, and Polymath (2014) established the unconditional bound . This gives infinitely many pairs of consecutive
primes separated by at most 246, but it does not show that gap 2 occurs infinitely
often. The twin prime conjecture is the stronger assertion .
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 .
is finite, and Polymath (2014) established the unconditional bound 
. This gives infinitely many pairs of consecutive
primes separated by at most 246, but it does not show that gap 2 occurs infinitely
often. The twin prime conjecture is the stronger assertion 
.
such that 
is also prime, is very closely related (Shanks 1993, p. 30).
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
is the so-called twin primes constant (Hardy
and Littlewood 1923). The value of 
is plotted above for 
, with 
indicated in blue and taking 
.
.
