A question in ring theory lead me to a system of diophantine inequalities. I have problems to solve it, but a positive answer of the following would bring me some effort:
Let $n \in \mathbb{N}$, $n \neq 9$ and $P = \max \{p \in \mathbb{P} \mid p \leq n\} $.
- Does there exist $r \in \{P+1,...,n\}$ such that there exists $q \in \mathbb{P}$ with $q | r$ and $q > n - P + 1$?
- Can we say something about the number of $r \in \{ P+1,...,n \}$ that satisfy this statement (in relation to the difference $n - P$)?
I have really no idea to approach this and I also cannot find a counterexample. I would be really glad if somebody could help me!
Thank you in advance!