Sexy Primes

Sexy primes are primes that differ by 6, 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). In November 2005, the largest known sexy prime pair started with

(1)

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

A sexy prime pair found by Batalov (2024ab) on Nov. 23, 2024 consists of and , where

(2)

Both primes have 75274 decimal digits.

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 Oct. 2026, the largest known sexy triplet starts with

(3)

These primes have 16,330 decimal digits and were found by S. Batalov in June 2024 (Prime -Tuplets).

The first few sexy quadruplets are (5, 11, 17, 23), (11, 17, 23, 29), (41, 47, 53, 59), (61, 67, 73, 79), (251, 257, 263, 269), ... (OEIS A023271, A046122, A046123, and A046124). Except for (5, 11, 17, 23), sexy quadruplets can only begin with a prime ending in a "1." As of Oct. 2026, the largest known sexy quadruplet starts with

(4)

These primes have 5002 decimal digits and were found by S. Batalov in June 2024 (Prime -Tuplets).

There is only a single sexy quintuplet, (5, 11, 17, 23, 29). Among any five numbers , , ..., , one is divisible by 5. If all five are prime, that number must equal 5, which forces .