6. Write all prime numbers that cannot be represented as the sum of two composite numbers.
Solution. The number 2 cannot be represented as the sum of two composite numbers (the number 1 is neither prime nor composite). From now on, we only consider odd prime numbers.

The difference between any prime number greater than 13 and the number 13 is an even number. Let this difference for a given prime number $p$ be $d$, i.e., $p-d=13$. The number 13 has the necessary representation as a sum, $13=9+4$. By adding the number 4 to the number $d$, we obtain the necessary representation of the prime number $p$ as the sum of 9 and the even number $4+d$.

By enumerating the possible sums, we find that none of the numbers $3,5,7,11$ can be represented as the sum of two composite numbers.

Answer. $2,3,5,7,11$.