Sexy Primes

Since a Prime Number cannot be divisible by 2 or 3, it must be true that, for a Prime , . This motivates the definition of sexy primes as a pair of primes () such that (sexy'' since sex'' is the Latin word for six.''). The first few sexy prime pairs are (5, 11), (7, 13), (11, 17), (13, 19), (17, 23), (23, 29), (31, 37), (37, 43), (41, 47), (47, 53), ... (Sloane's A023201 and A046117).

Sexy constellations also exist. The first few sexy triplets (i.e., numbers such that each of is Prime but is not Prime) are (7, 13, 19), (17, 23, 29), (31, 37, 43), (47, 53, 59), ... (Sloane's A046118, A046119, and A046120). The first few sexy quadruplets are (11, 17, 23, 29), (41, 47, 53, 59), (61, 67, 73, 79), (251, 257, 263, 269), ... (Sloane's A046121, A046122, A046123, and A046124). Sexy quadruplets can only begin with a Prime ending in a 1.'' There is only a single sexy quintuplet, (5, 11, 17, 23, 29), since every fifth number of the form is divisible by 5, and therefore cannot be Prime.

Sloane, N. J. A. Sequences A023201, A046117, A046118, A046119, A046120, A046121, A046122, A046123, and A046124, in An On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html.
Trotter, T. Sexy Primes.'' http://www.geocities.com/CapeCanaveral/Launchpad/8202/sexyprim.html.