# 12

This number is a composite. The sum of twin primes (except for the first pair) is divisible by 12. In the 19th century, the Russian mathematician Chebyshev proved that π(x) is greater than x/(12 log x). 12! + 34567 is prime. 12 is the smallest composite number whose sum of digits, product of digits and the sum of the sum of digits and product of digits are all prime. [Gevisier] The only number such that n ± 1 , n/2 ± 1 and n/3 ± 1 are all primes. [Murthy] The smallest multi-digit number exactly divisible by the square of a prime. [Russo] The concatenation of n!, (n-1)!, ..., 2!, 1! and 0! is prime for n=12, 6, 4, 3 and 2, but not for n=11, 10, 9, 8, 7, or 5 (nor for 13, 14, 15, ... up to at least 61). Note that 2, 3, 4, 6 and 12 are exactly the non-unit divisors of 12. [Hartley] 212 + 212 + 312 is prime. Note that 2*2*3 = 12. [Honaker] 12 is the largest known even number expressible as the sum of two primes in one way. [Firoozbakht] 12 = (1*2) * (prime(1)*prime(2)). Note that 12 is the only number less than 20000000 with this property. [Firoozbakht] π(12) = prime(1!) + prime(2!) [Firoozbakht] 12 is the smallest number n such that n and n! are product of distinct factorials of primes (12 = 2!3! and 12! = 2!3!11!). [Capelle] The smallest composite number ending in a prime digit. [Silva] The concatenation of the difference and the sum of number 12 and the 12th prime is prime (2549). Note that the concatenation of 12 and the 12th prime is an emirp. [Silva] π(12) = 11 + 22. [Kumar] 12 divides p^2-1 for all primes p>3. [Fellows] 12 is the number of prime digits appearing in the first 12 primes and 21 is the number of prime digits appearing in the first 21 primes. [Silva] The only multidigit number n up to 10000 such that 2^n+3 and 2^n-3, are primes. Another prime of this form, if it exists, will have more 3000 digits. [Bajpai] The smallest oblong number that is the sum of 2 successive primes. [Honaker] The largest known number n such that product of n and nth prime is a repdigit number(12*37=444). [Gupta] The 12th prime is the reversal of the 21th prime. [Hollesen] 12 is equal to 3*4 and at the same time equal to the 3rd prime plus the 4th prime. [Hollesen] The smallest n such that the concatenate exponents in the prime factorization of n = reversal(n). Can you find the next example? [Honaker]

(There are 6 curios for this number that have not yet been approved by an editor.)