Sexy Primes

Sexy primes are pairs of primes of the form ( , ), so-named 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), ... (OEIS A023201 and A046117). As of November 2005, the largest known sexy prime pair starts with

(1)

where is a primorial. These primes have 10154 digits and were found by M. Fleuren, T. Alm, and J. K. Andersen (Andersen 2005).

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), ... (OEIS A046118, A046119, and A046120). As of October 2005, the largest known sexy triplet starts with

(2)

These primes have 5132 digit digits and were found by Davis (2005).

The first few sexy quadruplets are (11, 17, 23, 29), (41, 47, 53, 59), (61, 67, 73, 79), (251, 257, 263, 269), ... (OEIS A023271, A046122, A046123, and A046124). Sexy quadruplets can only begin with a prime ending in a "1." As of November 2005, the largest known sexy quadruplet starts with

(3)

These primes have 1002 digits and were found by J. K. Andersen (2005).

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.