# diamond geezer

### Thursday, March 08, 2018

Ten special types of prime number

Palindromic primes: Primes that remain the same when their digits are read backwards, e.g. 101, 383, 929, 14741, 71317, 143787341, 999998727899999 and 1234567894987654321.
Emirps: Primes that become a different prime when their digits are reversed, e.g. 13/31, 107/701, 149/941, 3257/7523, 943849/948349, 102435679/976534201 and 1301476963/3696741031.
Circular primes: Primes that remain prime when their digits are cycled, e.g. 113/131/311, 197/971/719, 199/991/919, 337/373/733 and 193939/939391/393919/939193/391939/919393.

Left-truncatable primes: Primes that remain prime when the leading digit is successively removed, e.g. 317/17/7, 853/53/3 and 62467/2467/467/67/7. The largest is 357686312646216567629137.
Right-truncatable primes: Primes that remain prime when the final digit is successively removed, e.g. 293/29/2, 599/59/5, 31193/3119/311/31/3. The largest is 73939133.
Two-sided primes: Primes that are both left-truncatable and right-truncatable. The only two-sided primes (over 100) are 313, 317, 373, 797, 3137, 3797 and 739397.

Repunit primes: Primes containing only the digit 1, e.g. 11, 1111111111111111111 and 11111111111111111111111. The next have 317 and 1031 digits.
Pandigital primes: Primes containing all 10 digits, e.g. 10123457689, 10123465789, and 10123465897. No pandigital primes are 10-digit numbers.

Beastly primes: Primes including the number of the Beast, e.g. 6661, 96661, 700666007 and 6660000000001.
Belphegor's prime: The palindromic beastly prime number 1000000000000066600000000000001, with 666 at its centre, surrounded on either side by thirteen zeroes.

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