Vipul: Created page with '==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 t…'

2010-05-03T19:57:04Z

Created page with '==Definition== Two odd primes that differ by <math>4</math> are called '''cousin primes'''. In other words, '''cousin primes''' are a pair of primes <math>(p,p+4)</math>. The t…'

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==Definition==

Two odd primes that differ by <math>4</math> are called '''cousin primes'''. In other words, '''cousin primes''' are a pair of primes <math>(p,p+4)</math>.

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

==Basic facts==

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

==Particular cases==

===The greater of the cousin primes===

{{oeis|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===

{{oeis|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===

{| class="wikitable" border="1"
! Property !! Meaning !! Comment
|-
| [[Twin primes]] || two primes that differ by <math>2</math> || Both primes must be odd and they must be consecutive primes.
|-
| [[Sexy primes]] || two primes that differ by <math>6</math> (with no prime in between) || Since this is a pair of successive primes, the prime gap is <math>6</math>.
|-
| [[Sophie Germain prime]] || a prime <math>p</math> such that <math>2p + 1</math> is also prime || the corresponding prime <math>2p + 1</math> is a safe prime
|-
| [[safe prime]] || a prime <math>p</math> such that <math>(p - 1)/2</math> is also prime || the corresponding prime <math>(p - 1)/2</math> is a Sophie Germain prime
|}

Cousin primes - Revision history

Vipul: Created page with '==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 t…'

Vipul: Created page with '==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 t…'