Adding prime numbers to get prime numbers. Do the following hold?

Question

Let $a,b,c,d$ be different primes.

Do the following hold?

$(2a+b)\in \Bbb P\vee(2a+c)\in\Bbb P$

$(2c+a)\in \Bbb P\vee(2c+d)\in\Bbb P$

$(2b+a)\in \Bbb P \vee (2b+d)\in \Bbb P$

$(2d+b)\in \Bbb P \vee (2d+c)\in \Bbb P.$

For example, $a=2,b=3,c=7,d=5$ satisfies the conditions.

I think it will eventually fail for some combination of primes.

Answer 1

To have a demotivational answer, let us consider the following six primes: $$ 211,\ 223,\ 229,\ 241,\ 271,\ 277\ . $$ Then a quick computer check shows that for any two $p,q$ values among the six above the number $2p+q$ is not a prime.

Answer 2

Here are all values of $(a, b, c)$ for which the first condition fails, such that $a<b<c<50$.

(2,5,11),(2,5,17),(2,5,23),(2,5,29),(2,5,31),(2,5,41),(2,5,47),(2,11,17),(2,11,23),(2,11,29),(2,11,31),(2,11,41),(2,11,47),(2,17,23),(2,17,29),(2,17,31),(2,17,41),(2,17,47),(2,23,29),(2,23,31),(2,23,41),(2,23,47),(2,29,31),(2,29,41),(2,29,47),(2,31,41),(2,31,47),(2,41,47),(3,19,29),(3,19,43),(3,29,43),(5,11,17),(5,11,23),(5,11,29),(5,11,41),(5,11,47),(5,17,23),(5,17,29),(5,17,41),(5,17,47),(5,23,29),(5,23,41),(5,23,47),(5,29,41),(5,29,47),(5,41,47),(7,11,13),(7,11,19),(7,11,31),(7,11,37),(7,11,41),(7,11,43),(7,13,19),(7,13,31),(7,13,37),(7,13,41),(7,13,43),(7,19,31),(7,19,37),(7,19,41),(7,19,43),(7,31,37),(7,31,41),(7,31,43),(7,37,41),(7,37,43),(7,41,43),(11,13,17),(11,13,23),(11,13,29),(11,13,41),(11,13,43),(11,13,47),(11,17,23),(11,17,29),(11,17,41),(11,17,43),(11,17,47),(11,23,29),(11,23,41),(11,23,43),(11,23,47),(11,29,41),(11,29,43),(11,29,47),(11,41,43),(11,41,47),(11,43,47),(13,19,23),(13,19,29),(13,19,31),(13,19,37),(13,19,43),(13,23,29),(13,23,31),(13,23,37),(13,23,43),(13,29,31),(13,29,37),(13,29,43),(13,31,37),(13,31,43),(13,37,43),(17,23,29),(17,23,31),(17,23,41),(17,23,43),(17,23,47),(17,29,31),(17,29,41),(17,29,43),(17,29,47),(17,31,41),(17,31,43),(17,31,47),(17,41,43),(17,41,47),(17,43,47),(19,31,37),(19,31,43),(19,31,47),(19,37,43),(19,37,47),(19,43,47),(23,29,31),(23,29,41),(23,29,47),(23,31,41),(23,31,47),(23,41,47),(29,37,41),(29,37,47),(29,41,47),(31,37,43),(37,41,43),(37,41,47),(37,43,47),(41,43,47)