The smallest composite number r such that n = 4r + 3 and m = 8r + 7 are primes. [Rivera]
Oliver Sacks' describes in his book 'The MAN Who MISTOOK HIS WIFE for a HAT,' a pair of autistic twin brothers who were swapping 20-figure primes.
The smallest number n such that neither 6n-1 nor 6n+1 are primes. [Necula]
20 ones plus 20 is prime. This is the largest such two-digit number. [Opao]
20^20 - 11...1 (20 ones) and 20 + 11...1 (20 ones) are primes. [Firoozbakht]
(20) = 2^3 + 0^3. It is the smallest number with this property. [Kumar]
The only number n < 1000, such that subtracting from it each of the first seven powers of 3, then all absolute differences that are obtained, 20-3^0, 20-3^1, …, 20-3^6 are primes, i.e., 19, 17, 11, 7, 61, 223, 709. [Loungrides]
The smallest number n whose sum of digits equals omega(n).
(There are 7 curios for this number that have not yet been approved by an editor.)