# Twin primes

## 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]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]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 more than two primes

Property Meaning Comment
Prime quadruplet a collection of four primes  there can be no further primes in between
Prime constellation a sequence of consecutive primes  for which the difference between the first and last prime is the least possible based on considerations of modular arithmetic relative to smaller primes we are usually interested in prime constellation having a particular constellation pattern.
Bitwin chain combines the idea of twin primes, Cunningham chain of the first kind, and Cunningham chain of the second kind

## 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.