# 22

This number is a composite. There is no prime in which the mean gap between the first n successive primes is exactly 22. [Dodson] It is unknown if there is a relatively prime pair of amicable numbers. If there is such a pair, their product must be at least 22 distinct primes. The smallest integer such that its sum of digits ± 1 and the product of its digits is prime, together with the fact that the sum of digits is the same as the product. [Russo] 22 is the smallest number which can be expressed as the sum of two primes in three ways. [Murthy] 1π(1) + 2π(2) + 3π(3) + 4π(4) + ... + 22π(22) is prime. [Poo Sung] The smallest multi-digit square-free number such that all the digits are primes. [Russo] The smallest multi-digit composite palindrome whose total number of prime factors (counting multiplicity) is square-free. [Russo] The longest known arithmetic sequence of primes is currently 22, starting with the prime 11410337850553 and continuing with common difference 4609098694200 (found by Pritchard, Moran, and Thyssen in 1993). 22 is the smallest multidigit composite palindrome such that the concatenation (211) as well as the sum (13) of its prime factors are prime. [De Geest] It is possible for a Queen to attack all 22 prime numbered squares on 9x9 Knight's Tour solution (Jacques Tramu, 2004). 22 is the smallest Hoax Number. The smallest dihedral semiprime. Note that it contains the smallest dihedral prime. [Capelle] The smallest sum of TWO primes with prime subscripts (PIPs) in TWO ways: 22 = 11 + 11 = 17 + 5. [Post] The number of 3-digit emirps with exactly 3 distinct digits. [Loungrides] There are 22 twin prime pairs between 22 cubed and the next consecutive cube. The smallest multidigit palindromic semiprime. [Silva] The divisors of the 22 can be added to 22 in various combinations to form a larger semiprime, e.g., 22+1+2=25=5*5, 22+11=33=3*11, 22+11+1=34=2*17, 22+11+1+2=35=5*7, 22+22+11=55=5*11 and 22+22+11+2+1=58=2*29. Is there another semiprime that can produce more than six ways? [Bergot] It remains unknown if there is a relatively prime pair of amicable numbers. If there is such a pair, their product must be divisible by at least 22 distinct primes. [Caldwell] Inserting a zero in the middle of the 22nd prime number produces the 22nd emirp. Note that the 22nd prime number is also an emirp. [Gaydos]

(There are 3 curios for this number that have not yet been approved by an editor.)