TITLE: Partitioning sets such that the sum of 2 elements is Prime
QUESTION [6 upvotes]: Given an $n >0$ is it possible to partition the set $\mathcal{P} = \{1,2, \cdots, 2n\}$ into $n$ pairs $(a_{i},b_{i})$ such that $a_{i} + b_{i}$ is a prime?

REPLY [13 votes]: Yes.  The proof is by strong induction.  The base case is obvious.  By Bertrand's postulate there exists a prime $p$ between $2n+1$ and $4n$, so pick the pairs $\{ p-2n, 2n \}, \{ p-2n+1, 2n-1 \}, ...$ and so forth.  Now it remains to pair up the numbers $\{ 1, 2, ... p-2n-1 \}$, which is possible by the inductive hypothesis.