If $ n = \left( \frac{p+q}{2 } \right) + p q :p,q \in\mathbb{P}-\left\{2\right\} $
Can we show there exists a prime number $\theta : \sqrt{2n} \leq \theta \leq n $ and $\theta$ is not a linear combination of $p$ and $q$ (restricted to positive coefficients)?
I know we can show there exists a prime number in the interval $[\sqrt{2n},n]$, if $\frac{\sqrt{n}}{\sqrt{2}} \geq2$ using Bertrand's postulate...