Suppose that $p_n$ is the $n$-th prime and $n \neq 1,2$.
To every $p_n$ we can associate $(n-2)$-tuple $(2p_n+p_{n-1},...,2p_n+p_2)$ and from some calculations that I have done it seems that it could be that at least one of the numbers from that $(n-2)$-tuple for every $p_n$ is a prime number.
Is that true?
If this conjecture were true then it would show that every even number of the form twice a prime is the difference of two primes.
I believe this is an open question for even numbers in general and so (although it looks highly likely) it is not likely to have an elementary solution.