Sexy Primes

Since a Prime Number cannot be divisible by 2 or 3, it must be true that, for a Prime $p$, $p\equiv 6\ \left({{\rm mod\ } {1,5}}\right)$. This motivates the definition of sexy primes as a pair of primes ($p,q$) such that $p-q=6$ (``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 $(p,p+6,p+12)$ is Prime but $p+18$ 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 $6n\pm 1$ is divisible by 5, and therefore cannot be Prime.

See also Prime Constellation, Prime Quadruplet, Twin Primes

References

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.