Bertrand's postulate proved by Chebyshev states that for any $x>1$ there is a prime $p$ in the interval $(x,2x)$. In 2012 I considered some refinements of this by imposing additional requirement involving the Legendre symbol. For example, I have the following conjecture.

Conjecture. (i) For any integer $n > 6$, there is a prime $p\in(n,2n)$ with $(\frac{−n} p) = −1$.

(ii) For any integer $n > 5$, there is a prime $p\in(n,2n)$ with $(\frac{2n}p)=1$.

I have verified this for $n$ up to $5\times10^8$, but I don't know how to prove the conjecture.

QUESTION: How to modify the known proofs of Bertrand's postulate to show the above conjecture?


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