# Twin primes

From Number

## Contents

## 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 primes

- Chen prime is a prime number such that is either a prime number or a semiprime. The name arises because of Chen's theorem on primes and semiprimes with fixed separation, which essentially asserts that for any fixed separation, there are infinitely many pairs of a prime and a semiprime having that separation.

### 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. |