Definition
The term twin primes is used for a pair of odd prime numbers that differ by two. In other words, primes
are termed twin primes. Either member of a pair of twin primes may be referred to as a twin prime.
The twin prime conjecture states that there are infinitely many twin primes.
Basic facts
- If
and
are both prime, they both must be odd numbers.
- For
, if
are twin primes, then
(hence
) and
(hence
).
- In particular, with the exception of the prime
, any member of a pair of twin primes cannot be a member of another pair of twin primes, i.e., the lesser member in one pair of twin primes cannot be the greater member in another pair.
Occurrence
Lesser of twin primes
The
lesser of twin primes is the sequence (in increasing order) of all the primes

for which

is prime. This list goes:
3,
5,
11,
17,
29,
41,
59,
71,
101,
[SHOW MORE]107, 137, 149, 179, 191, 197, 227, 239, 269, 281, 311, 347, 419, 431, 461, 521, 569, 599, 617, 641, 659, 809, 821, 827, 857, 881, 1019, 1031, 1049, 1061, 1091, 1151, 1229, 1277, 1289, 1301, 1319, 1427, 1451, 1481, 1487, 1607
View list on OEIS
Greater of twin primes
The
greater of twin primes is the sequence (in increasing order) of all the primes

for which

is prime. This list goes:
5,
7,
13,
19,
31,
43,
61,
73,
103,
[SHOW MORE]109, 139, 151, 181, 193, 199, 229, 241, 271, 283, 313, 349, 421, 433, 463, 523, 571, 601, 619, 643, 661, 811, 823, 829, 859, 883, 1021, 1033, 1051, 1063, 1093, 1153, 1231, 1279, 1291, 1303, 1321, 1429, 1453, 1483, 1489, 1609
View list on OEIS
Proportion of primes
|
Number of primes |
Number of twin prime pairs |
Proportion of primes that are members of twin prime pairs
|
10 |
4 |
2 |
|
100 |
25 |
8 |
|
1000 |
168 |
35 |
|
10000 |
1229 |
205 |
|
Relation with other properties
Related properties for pairs of primes
Property |
Meaning |
Comment
|
Cousin primes |
two primes that differ by |
Note that for , if both and are prime, is not prime. Hence, the prime gap in this case is .
|
Sexy primes |
two primes that differ by (with no prime in between) |
Since this is a pair of successive primes, the prime gap is .
|
Sophie Germain prime |
a prime such that is also prime |
the corresponding prime is a safe prime
|
safe prime |
a prime such that is also prime |
the corresponding prime is a Sophie Germain prime
|
Related properties for primes
Related properties for more than two primes
Related facts/conjectures
Broad concern |
Name of fact/conjecture |
Statement |
Status
|
Infinitude |
twin primes conjecture |
there are infinitely many twin primes |
open
|
Largeness, i.e., sum of reciprocals |
Brun's theorem |
the sum of reciprocals of all twin prime pairs (i.e., we add the reciprocal of each member of each twin prime pair) is finite |
proved. This sum is Brun's constant.
|
Density |
first Hardy-Littlewood conjecture |
In the particular case of twin primes, the claim is that the number of twin prime pairs is , where is a specified constant.
|