# How many prime numbers are there that can't be written as a sum of two composite numbers? [duplicate]

Obviously $$2,3,5,\ldots$$ but I'm not sure for what other numbers does it hold, or if there are infinitely many.

## marked as duplicate by lulu, Servaes, Song, J. M. is not a mathematician, ShaunMar 15 at 14:05

• Are you talking about only Natural numbers? Otherwise $2=12+(-10)$ or like that. – Love Invariants Mar 15 at 10:51
Any $$n\ge 13$$, prime or otherwise, is either $$8$$ or $$9$$ more than an even number $$\ge 4$$. Therefore, any such $$n$$ is a sum of two composite numbers. We can exhaustively check the only primes lacking such an expression are the primes from $$2$$ to $$11$$ inclusive, i.e. $$5$$ of them.