# 18

This number is a composite. It is possible for a Queen to attack all 18 prime-numbered squares on a closed Knight's tour. Finding a different solution without the aid of a computer is a religious experience! 18 is the smallest difference between an emirp and its reverse. [Poo Sung] 18 is the common difference in the arithmetic progression formed by the 5th, 10th, and 15th primes. [Rupinski] The smallest number C of the form 2a^2 such that C+1 and C-1 are both prime. [Hartley] 18 is the largest value of n less than a thousand such that if L(n) = length of n in base 10, then 2*n^n+1, 2*L(n^n)+1, and 2*L(L(n^n))+1 are all primes greater than 3 (as the expression 2*L(L(L(...(L(x))...)))+1 will converge at 3 for sufficient repetitions of L given any value of x). [Opao] 18 is the only two-digit number m , such that three numbers, m + prime(m), m^2 + prime(m^2) & m^3 + prime(m^3), are primes. [Firoozbakht] The sum of digits, digital product, and reversal of 18 are perfect powers of its prime divisors. [Silva] 18 equals the product of its prime divisors plus the product of their factorials. [Silva] There are a prime number (197699) of zeros in the set of all primes whose binary representation is no more than 18 bits, including leading zeros. [Post] The difference between any emirp pair is divisible by 18. [Green] 18 = π(81-18). The only number with this property. [Firoozbakht] The smallest Moran number, i.e., n such that n divided by the sum of digits of n is prime. There are only 18 primes that consist of distinct prime digits. Six of them yield three pairs of emirps. [Loungrides] The largest value of n less than ten thousand such that (3*2^n-1) and (2*2^n-1) are both prime. [Bajpai] 18 times the 18th prime has the sum of digits equal to 18. It is the largest such number with this property. [Bajpai] (18! plus the 18th prime) and (18! minus the 18th prime) are both prime. [Jha]

(There is one curio for this number that has not yet been approved by an editor.)