Cousin primes

Definition

Two odd primes that differ by $4$ are called cousin primes. In other words, cousin primes are a pair of primes $(p,p+4)$ .

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

Basic facts

For $p>3$ , if $(p,p+4)$ form a pair of cousin primes, then $p\equiv 1{\pmod {3}}$ (and hence $p\equiv 1{\pmod {6}}$ ) and $p+4\equiv 2{\pmod {3}}$ (and hence $p+4\equiv 5{\pmod {6}}$ ). In particular, $p+2$ is divisible by $3$ and cannot be prime. Hence, apart from the pair $(3,7)$ , every pair of cousin primes is a pair of consecutive primes.
For $p>3$ , it is not possible for both $(p,p+4)$ and $(p+4,p+8)$ to be pairs of cousin primes. Hence, the only prime that occurs in two pairs of cousin primes is the prime $7$ .

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 $2$	Both primes must be odd and they must be consecutive primes.
Sexy primes	two primes that differ by $6$ (with no prime in between)	Since this is a pair of successive primes, the prime gap is $6$ .
Sophie Germain prime	a prime $p$ such that $2p+1$ is also prime	the corresponding prime $2p+1$ is a safe prime
safe prime	a prime $p$ such that $(p-1)/2$ is also prime	the corresponding prime $(p-1)/2$ is a Sophie Germain prime