A

**twin prime** is a

prime
number that differs from another prime number by

two. Except for the pair (2, 3), this is the smallest
possible difference between two primes. Some examples of twin prime
pairs are (3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43),
and (skipping quite a few), (821, 823). Sometimes the term

*twin
prime* is used for a pair of twin primes; an alternative name
for this is

**prime twin**.

The question of whether there exist infinitely many twin primes has
been one of the great open questions in

number theory for many years.

This is the content of the

twin
prime conjecture. A strong form of the twin prime conjecture,
the

Hardy–Littlewood
conjecture, postulates a distribution law for twin primes akin
to the

prime number
theorem.

Using his celebrated

sieve methods,

Viggo Brun shows that the number of twin
primes less than

`x` is

O(

`x`/(log

`x`)

^{2}). This result implies that the

sum of the

reciprocals of all twin primes
converges (see

Brun's constant and

Brun's theorem). This is in contrast
to the sum of the reciprocals of all primes, which diverges.He also
shows that every

even number can be
represented in infinitely many ways as a difference of two numbers
both having at most 9 prime factors.

Chen
Jingrun's well known theorem states that for any

`m`
even, there are infinitely many primes that differ by

`m`
from a number having at most two prime factors.(Before Brun
attacked the twin prime problem,

Jean
Merlin (1876–1914) had also attempted to solve this problem
using the sieve method.)

Every twin prime pair except (3, 5) is of the form (6

*n* −
1, 6

*n* + 1) for some

natural
number *n*, and with the exception of

`n` = 1,

`n` must end in 0, 2, 3, 5, 7, or 8.

It has been proved that the pair

*m*,

*m* + 2 is a
twin prime

if and only if
- 4((m-1)! + 1) \equiv -m \pmod {m(m+2)}.

If

*m* − 4 or

*m* + 6 is also prime then the 3 primes
are called a

prime triplet.

## Largest known twin prime

On January 15, 2007 two

distributed computing projects,

Twin Prime Search and

PrimeGrid found the largest known twin primes,
2003663613 · 2

^{195000} ± 1. The numbers have 58711

decimal digits.

Their
discoverer was Eric Vautier of France.
On August 6, 2009 those same two projects announced that a new
record twin prime had been found. It is 65516468355 ·
2

^{333333} ± 1. The numbers have 100355 decimal
digits.

An empirical analysis of all prime pairs up to 4.35 ·
10

^{15} shows that the number of such pairs less than

`x` is

`x`·f(

`x`)/(log

`x`)

^{2} where f(

`x`) is about 1.7 for
small

`x` and decreases to about 1.3 as

`x` tends
to infinity.The limiting value of f(

`x`) is conjectured to
equal twice the

twin prime
constant (not to be confused with

Brun's constant)

- 2 \prod_{\textstyle{p\;{\rm prime}\atop p \ge 3}} \left(1 -
\frac{1}{(p-1)^2}\right) = 1.3203236\ldots;

this conjecture would imply the twin prime conjecture, but remains
unresolved.

## The first 35 twin prime pairs

There are 35 twin prime pairs below 1000, given in the following
list:

- (3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), (59,
61), (71, 73), (101, 103), (107, 109), (137, 139), (149, 151),
(179, 181), (191, 193), (197, 199), (227, 229), (239, 241), (269,
271), (281, 283), (311, 313), (347, 349), (419, 421), (431, 433),
(461, 463), (521, 523), (569, 571), (599, 601), (617, 619), (641,
643), (659, 661), (809, 811), (821, 823), (827, 829), (857, 859),
(881, 883).

Every third odd number greater than seven is divisible by 3, so 5
is the only prime which is part of two pairs. The lower member of a
pair is by definition a

Chen prime.

Polignac's conjecture from
1849 states that for every even natural number

*k*, there
are infinitely many prime pairs

*p* and

*p′* such
that

*p − p′* =

*k*. The case

*k* = 2 is the
twin prime conjecture. The case

*k* = 4 corresponds to

cousin primes and the case

*k* =
6 to

sexy primes. The conjecture has not
been proved or disproved for any value of

*k*.

## See also

## References

## External links