Prime numbers have so many novel qualities, and are so enigmatic, that mathematicians have grown fetishistic about them. Twin primes are two apart. Cousin primes are four apart, sexy primes are six apart, and neighbor primes are adjacent at some greater remove.

From “Prime Curios!,” by Chris Caldwell and G. L. Honaker, Jr., I know that an absolute prime is prime regardless of how its digits are arranged: 199; 919; 991. A beastly prime has 666 in the center. The number 700666007 is a beastly palindromic prime, since it reads the same forward and backward. A circular prime is prime through all its cycles or formulations: 1193, 1931, 9311, 3119.

There are Cuban primes, Cullen primes, and curved-digit primes, which have only curved numerals—0, 6, 8, and 9. A prime from which you can remove numbers and still have a prime is a deletable prime, such as 1987.

An emirp is prime even when you reverse it: 389, 983. Gigantic primes have more than ten thousand digits, and holey primes have only digits with holes (0, 4, 6, 8, and 9).

There are Mersenne primes; minimal primes; naughty primes, which are made mostly from zeros (naughts); ordinary primes; Pierpont primes; plateau primes, which have the same interior numbers and smaller numbers on the ends, such as 1777771; snowball primes, which are prime even if you haven’t finished writing all the digits, like 73939133; Titanic primes; Wagstaff primes; Wall-Sun-Sun primes; Wolstenholme primes; Woodall primes; and Yarborough primes, which have neither a 0 nor a 1.

From -- Alec Wilkinson, "The Pursuit of Beauty: Yitang Zhang solves a pure-math mystery.