# Cousin primes

## 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