# Primes $p\in(n,2n)$ with $(\frac{-n}p)=-1$

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?