Prime Tuples

Prime tuples are pairs, triples, quadruples ... of adjecent prime numbers. In a broader sense, prime pairs of the form (p,p+4) or (p,p+6) are also prime tuples. The program determines all in an interval [a, b].

Prime twins:

These are prime pairs of the form (p|p+2), i.e. with a difference of 2.

Prime cousins:

These are prime pairs of the form (p|p+4), i.e. with a difference of 4.

Sexy primes:

These are prime pairs of the form (p|p+6), i.e. with a difference of 6.

Prime triplets:

These are triplet primes of the form (p|p+2|p+6) or the form [p|p+4|p+6].
In the program, the two forms are distinguished by the type of parantheses and counted separately.
The triplet (3|5|7) is the only triplet of the form (p|p+2|p+4), since for p>3 one of the three numbers is always divisible by 3.

Prime twins between 1 and 200

(3|5) (5|7) (11|13) (17|19) (29|31) (41|43) (59|61)         
(71|73) (101|103) (107|109) (137|139) (149|151) 
(179|181) (191|193) (197|199) 

15 pairs of prime twins

Prime triplets between 1 and 100

(3|5|7) (5|7|11) [7|11|13] (11|13|17) [13|17|19]   
(17|19|23) [37|41|43] (41|43|47) [67|71|73] 

9 triplet prime triplets
4 of the form (p|p+2|p+6) and 4 of the form [p|p+4|p+6]

See also:

Prime factorization
Wikipedia: Prime number | Integer factorization