# Cousin primes

## Contents

## Definition

Two odd primes that differ by are called **cousin primes**. In other words, **cousin primes** are a pair of primes .

The term **cousin prime** is typically used for either member of a pair of cousin primes.

## Basic facts

- For , if form a pair of cousin primes, then (and hence ) and (and hence ). In particular, is divisible by and cannot be prime. Hence, apart from the pair , every pair of cousin primes is a pair of
*consecutive*primes. - For , it is not possible for both and to be pairs of cousin primes. Hence, the only prime that occurs in two pairs of cousin primes is the prime .

## Particular cases

### The greater of the cousin primes

The ID of the sequence in the Online Encyclopedia of Integer Sequences is A046132

The list begins: 7, 11, 17, 23, 41, 47, 71, 83, 101, 107, 113, 131, 167, 197, 227, 233, 281, 311, 317, 353, 383, 401, 443, 461, 467, 491, 503, 617, 647, 677, 743, 761, 773, 827, 857, 863, 881, 887, 911, 941, 971, 1013, 1091, 1097, 1217, 1283, 1301, 1307, 1427, 1433, 1451, 1487 ...

### The lesser of the cousin primes

The ID of the sequence in the Online Encyclopedia of Integer Sequences is A023200

The list begins: 3, 7, 13, 19, 37, 43, 67, 79, 97, 103, 109, 127, 163, 193, 223, 229, 277, 307, 313, 349, 379, 397, 439, 457, 463, 487, 499, 613, 643, 673, 739, 757, 769, 823, 853, 859, 877, 883, 907, 937, 967, 1009, 1087, 1093, 1213, 1279, 1297, 1303, 1423, 1429, 1447, 1483 ...

## Relation with other properties

### Related properties for pairs of primes

Property | Meaning | Comment |
---|---|---|

Twin primes | two primes that differ by | Both primes must be odd and they must be consecutive primes. |

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 |