Question about $2p-1$ and $2p+1$, where $p$ is a prime. Let $x+1$ be any prime greater than $3$.  
By Bertrand's Postulate, there is at least one prime between $\frac{x}{2}$ and $x$.
Let $\{p_1,p_2,\dots, p_n\}$ be the primes between $\frac{x}{2}$ and $x$
In all cases that I've checked, there exists $p_i$ in this set where either $2p_i-1$ or $2p_i+1$ is a prime.
Is this always true?

Pretty sure it is open.  We know Sophie Germain is open, and this mentions a theorem of Chen regarding the $2 \cdot p - 1$ case which is strictly weaker (suggesting that infinite $2 \cdot p - 1$ primes is open).  Unless there is some way we can prove the disjunction without proving either case individually then this is also open.
